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Many types of radial basis functions (RBFs) are global in terms of having large magnitude across the entire domain. Yet, in contrast, e.g. with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coefficients. That is, changing a single data value mainly affects the coefficients of the RBFs which are centred in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well-conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coefficients for cardinal data, in both 1D and 2D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.

High altitude ER-2 aircraft observations in both hemispheres of the lower-stratospheric polar night jet stream reveal generalized scale invariance. The scaling exponent has systematic correlation with wind shear and temperature gradient showing across-jet persistence, along-jet anti-persistence, and an overall average value ≈5/9. The analysis provides a natural connection between generalized scale invariance and traditional large-scale dynamical meteorology. For one mission in the southern hemisphere composite variograms could be constructed for nitrous oxide and ozone, from which it was deduced that the former is a true passive scalar in the lower stratosphere, while sources and sinks are operative for the latter. Copyright © 2004 Royal Meteorological Society

The observed self-affine fractal behavior of ozone, wind, and temperature in the stratosphere is used to produce an indication of the rate of occurrence of chemistry relative to that of fluid mixing which, unlike the usual method, is independent of an assumed mechanism sequence of elementary chemical steps followed by application of the law of mass action to obtain differential equations describing the temporal and spatial evolution of the reacting gases. Rather, the occurrence of chemistry is deduced from the scaling of the observed variability; it is applied empirically to deduce the presence of chemical processes that are faster than fluid mechanical mixing above an altitude of 19 km during a balloon flight, and in the Antarctic spring, Arctic summer, and Arctic winter during high altitude aircraft flights. The method should apply to any volume of reacting molecules to which the equations of fluid motion apply and for which measurements can be made over a range of scales.

We apply a measure from fractal geometry to time series of airborne observations of ozone and meteorological quantities in the lower stratosphere. The Hurst exponent measure demonstrates that the lower stratospheric ozone, wind and temperature obey single exponent power law scaling to a high degree of fidelity. As a result, we suggest that ozone and horizontal windspeed behave as random, self-affine fractals, that vertical wind speed behaves antipersistently and that horizontal wind direction and temperature behave persistently. The persistent variables are candidates for multifractality.

Simultaneous in situ measurements of HCl and ClO have been made for the first time in the southern hemisphere, allowing a systematic study of the processes governing chlorine activation between 15 and 20 km in the 1994 Antarctic winter. Data for several other gases (O_3, NO, NO_y, OH, HO_2, N_(2)O, CH_4, CO, H_(2)O, CFCs), particulates, and meteorological parameters were collected from the ER-2 aircraft out of New Zealand as part of the 1994 Airborne Southern Hemisphere Ozone Experiment/Measurements of Atmospheric Effects of Stratospheric Aircraft (ASHOE/MAESA) campaign. Observations from the ER-2 in the fall (April–May), prior to polar night, show that chlorine activation begins with 60–75% of inorganic chlorine as HCl. By midwinter (July–August), near-total removal of HCl is observed. The wintertime loss of HCl in air recently exposed to extreme temperatures is found to be correlated with high levels of reactive chlorine (ClO and its dimer, Cl_(2)O_2) in the linear fashion expected from the stoichiometry of the heterogeneous reaction of hydrochloric acid with chlorine nitrate on polar stratospheric clouds (PSCs): HCl + ClONO_2 → Cl_2 + HNO_3. To constrain the role of different heterogeneous reactions and PSC types, we have used a photochemical trajectory model which includes heterogeneous sulfate and PSC chemistry. Model calculations of the evolution of reactive gases are compared with the in situ observations. In addition, simultaneous measurements of OH and HO_2 are used as a diagnostic for the occurrence of the heterogeneous reaction HOCl + HCl → Cl_2 + H_(2)O, which contributes to suppressed levels of HO_x inside the vortex. It is shown that the amount of chlorine activation is not strongly dependent on the composition of PSCs. However, HO_x levels exhibit different signatures depending on the type of heterogeneous surfaces that affected chlorine activation. Furthermore, this analysis implies that in the edge region of the Antarctic vortex, the observed near-total removal of HCl can result from latitudinal excursions of air parcels in and out of sunlight during the winter, which photochemically resupply HOCl and ClONO_2 as oxidation partners for HCl.

Fast response in situ measurements of a suite of chemical species made from the NASA ER2 high altitude aircraft, between 60°N and 70°S at potential temperatures up to 530 K from March to November 1994 at longitudes 115° W to 150°E, are considered for the view they offer of the Brewer-Dobson circulation in the lower stratosphere and upper troposphere. In the southern hemisphere, where most of the flights occurred, comparisons are made with measurements taken in August/September 1987 at longitudes 120° W to 60° W to examine temporal and longitudinal differences. Interpretations made suggest conceptual modifications to the simple construct of advection in a two dimensional long-term mean.

International audience; Observational data were taken in the 'vertical' structure at 2 Hz from research dropsondes for temperature, wind speed and relative humidity during the similar to 800 s it takes to reach the surface from the similar to 13 km altitude of the National Oceanic and Atmospheric Administration (NOAA) Gulfstream 4SP aircraft. The observations were made mainly through the depth of the troposphere above the eastern Pacific Ocean from 15 degrees N to 43 degrees N (dropsondes) and 60 degrees N (aircraft) in 2004. Grand averages of some key figures and of probability distribution functions (PDFs) were formed by compounding the data from the Winter Storms Projects 2004, 2005 and 2006, comprising 246, 324 and 315 (some dropped up to 60 degrees N) useable sondes, respectively. This sizeable data set was used to representatively characterize the statistical fluctuations in the 'vertical' structure from 13 km to the surface. The fluctuations are resolved at 5-10 m altitude, so covering up to 3 orders of magnitude of typical tropospheric weighting functions for passive remote sounders. Average 'vertical' statistical, multifractal, scaling exponents H, C(1) and alpha of temperature, wind speed and humidity fluctuations observed at high resolution were computed and are available as potential generators of representative, scale-invariant summaries of the vertical structure of the marine troposphere, for use in design and retrieval of remotely sounded observations.

HALOE observations of H2O and CH4 are compared with in situ techniques aboard the ER-2 aircraft during the northern winter of 1991/92, in particular for the dates 911208, 920108, 920217, 920222, and 920320 when the spatial coincidences are close, within +/- 1 deg latitude and +/- 12 deg longitude. The results reveal the limitations of comparing high resolution in situ aircraft data with a remote sounding limb scanner. Of the five comparison dates, three had HALOE/ER-2 coincidences which occurred near the edge of the Arctic vortex; the vertical variability in the HALOE results and the horizontal variability in the ER-2 observations on these days show that the vortex edge is not a region where exact agreement can be expected except by chance. On a 4th comparison date, there was substantial overlap away from the vortex edge, although for some species the aircraft data show considerable variability near the coincidence point. On the 5th comparison date, the ER-2 had no overlap in altitude with the lowest HALOE observations; however, a short linear interpolation over about 1 km altitude results in smooth composite profiles. Generally speaking the agreement between HALOE and the ER-2 at overlap altitudes is about 12 percent in the case of water vapor, which shows low horizontal and vertical variability. The agreement for methane, with limited data having altitude overlap, is better than 6 percent.

International audience; We would like to thank Esau (2009) for attempting to save the classical notion of stable layers; his argument is very close to one raised up by an anonymous reviewer of Lovejoy et al. (2008b). Since a similar argument is often invoked to justify atmospheric applications of linear gravity wave theories, it appears to be widespread in the community. We therefore hope this debate will clarify the issue.

International audience; We use 220 atmospheric profiles from state-of-the-art dropsondes to test the predictions of multiplicative cascade models of the atmosphere on the horizontal velocity, pressure, temperature, log potential temperature, log equivalent potential temperature, air density, humidity, and vertical sonde velocity. We found that the predictions were accurately verified (to within +/- 1 to +/- 2% over 10 m to 1 km for the statistical moments up to second order); the effective outer cascade scale L(eff) was in the range 1-30 km. In order to perform the analyses and to correctly interpret the results, we needed to overcome technical difficulties caused by the sonde's highly intermittent sampling. This intermittency is the result of both data outages and variable sonde fall speeds; we (surprisingly) found that the outages also had a cascade structure. The wide-range scaling of the sampling rate implies a variable sonde resolution, so that interpolation onto regular grids should generally be avoided (e.g., it would give rise to serious artifacts in estimating the corresponding spectra). In earlier studies, before the cascade nature of the outages was understood, interpolation was avoided by studying the fluctuations using all the pairs of measurement points; this was adequate for fluctuation scaling exponents in the range 0

[1] The notion of stable atmospheric layers is a classical idealization used for understanding atmospheric dynamics and thermodynamics. Using state of the art drop sonde data and using conditional, dynamical and convective stability criteria we show that apparently stable layers are typically composed of a hierarchy of unstable layers themselves with embedded stable sublayers, and unstable sub-sub layers etc. i.e. in a Russian Matryoshka doll-like fractal hierarchy. We therefore argue that the notion of stable atmospheric layers is untenable and must be replaced by modern scaling notions.

wave activity, high particulate surface areas (5–20 mm 2 cm �3 ), low ozone, and relatively high humidities (20–25 ppm of H2O, yet undersaturated with respect to ice), consistent with a heterogeneous mechanism on particles formed in recently lofted tropospheric air as it mixed into the lowermost stratosphere. These observations are similar to a previous one of chlorine activation on volcanic aerosol, suggesting a common heterogeneous chemical mechanism involving HCl, ClNO3, and HOCl that even in volcanically quiescent years can impact ozone photochemistry in regions of the lowermost stratosphere influenced by mixing from the tropopause region. Models not incorporating this chlorine activation process may be underestimating the impact on ozone in the midlatitude lowermost stratosphere by decomposition of very short lived halocarbon compounds, including substitutes for ozone-depleting compounds.

[1] The problem of turbulence is ubiquitous in the Earth sciences, astrophysics and elsewhere. Virtually the only theoretical paradigm that has been seriously considered is strongly isotropic in the sense that scaling exponents are the same in all directions so that any remaining anisotropy is “trivial.” Using 235 state-of-the-art drop sonde data sets of the horizontal wind at ≈5 m resolution in the vertical, we show that the atmosphere is apparently outside the scope of these isotropic frameworks. It suggests that anisotropy may frequently be strong requiring different scaling exponents in the horizontal and vertical directions.

[1] High-resolution (1 Hz at true airspeeds near 200 m s−1) observations of methane, ozone, water vapor, and temperature taken between the surface and 18 km from the WB57F aircraft near San Jose, Costa Rica (10°N, 84°W), are used to examine processes influencing the maintenance of these profiles. There was a clearly defined thermal tropopause on each of the eight profiles, accompanied by structure on methane-ozone scatterplots on the five profiles having methane observations. There was a well-defined decrease in methane mixing ratio between approximately 12 and 15 km in each of these five profiles, 2–5 km beneath the thermal tropopause, correlated with sharp changes in water vapor and equivalent potential temperature. The methane observations are interpreted as meaning that air is recirculated between the lower stratosphere and the upper tropical troposphere. At the point on each vertical profile where the water vapor had its minimum value, the air was never saturated or apparently supersaturated, although apparent supersaturation with respect to ice was observed in vertically extensive, near-adiabatic layers with tops some 200–300 m below the water vapor minimum on all profiles. One of the profiles also exhibited apparent supersaturation above its water vapor minimum, near 18 km. We examine the decrease in water vapor to minimal values as a four-stage process in which its mixing ratio was lowered from ∼10 to ∼3 ppmv, consider the role of solar and thermal evaporation of the smaller ice particles in the final stage of the dehydration process, and correlate features separating near-adiabatic layers above 150 hPa pressure altitude with local sea surface temperatures.

[1] Two long meridional flights of the WB57F high-altitude aircraft 1–2 km below the tropical tropopause are examined for ozone and total water variability in the context of a large suite of tracers having lifetimes ranging from 10−2 to 103 years, together with mass spectra of individual aerosol particles. It is concluded that local correlations are predominant; contributing processes to the maintenance of the composition are marine convection, continental convection, in situ chemical production, biomass burning, downward transport from the stratosphere and recirculation.

Exchange between the upper tropical troposphere and the lower stratosphere is considered by examining WB57F and ER-2 aircraft observations of water, ozone, wind, and temperature in the potential temperature range 360 < θ < 420 K. These processes are examined in part by using the technique of unified scale invariance on the airborne data, as has been done previously for the lower stratospheric polar vortex. Scale invariance is found, on scales from a few hundred meters to the maximum flown, 2700 km (25 great circle degrees). The results apply both to vertical exchange at the tropical tropopause and to isentropic exchange at the subtropical jet stream. All scales participate in the maintenance of the mean state, with substantial contributions from relatively infrequent but intense events in the long tails of the probability distributions. Past data are examined and found to fit this general framework. A unique mapping of tropical tropopause temperature to the total hydrogen content of the middleworld and overworld should not be expected; the head of the tape recorder is at 50-60 hPa rather than 90-100 hPa. The tropical tropopause is observed at potential temperatures θ T greater than the maximum moist static surface values θ W , such that θ T - θ W varies between 10 K in fall and up to 40 K in spring. The meridional gradient of θ T is directed from the subtropical jet stream to the inner tropics, with θ T declining by approximately 10 K from near 30°N to near 10°N in the vicinity of 95°W. The maintenance of these θ T values is discussed. Total water (measured as the sum of vapor and vaporized ice) and ozone, major absorbers of solar radiation and emitters/ absorbers of terrestrial infrared radiation, show scale invariance in the upper tropical troposphere. The implications of this result for the notion of a conservative cascade of energy via fluid dynamics from the largest to the smallest scales are discussed. The scaling exponents H z for total water and ozone in the upper tropical troposphere are not the value, 5/9, expected for a passive scalar, probably indicating the presence of sources and/or sinks operating faster than mixing.

[1] We consider the effects of power law scaling in the 1999–2000 Arctic lower stratospheric vortex from the point of view of the law of mass action and its application to the chemical kinetics of ozone loss embedded in a turbulent, macroscopic, fractal medium. The ER-2 observations of ClO obey power law scaling; the exponent varies with time in a manner shown to be consistent with the scaling of NOy and O3, via the influences of polar stratospheric clouds and actinic solar radiation. While the microscopic rate coefficient for ClO three-body recombination to the dimer applies as measured to three-dimensional volumes in which the sole transport mechanism is molecular diffusion, this cannot be true in the 2.56-dimensional space in which macroscopically fluctuating ClO reacts in the lower stratosphere. We show that the rate of loss of ozone via the ClO dimer mechanism is proportional to [ClO]2.20 in late January/early February and to [ClO]2.55 in March. Chemical ozone loss had already occurred by the date of the first flight, 20000120.

Airborne observations of ozone, temperature and the spectral actinic photon flux for ozone in the Arctic lower stratosphere April-September 1997 and January-March 2000 allow a connection to be made between the rate of production of translationally hot atoms and molecules via ozone photodissociation and the intermittency of temperature. Seen in the context of non-equilibrium statistical mechanics literature results from molecular dynamics simulations, the observed correlation between the molecular scale production of translationally hot atoms and molecules and the macroscopic fluid mechanical intermittency of temperature may imply a departure from Maxwell-Boltzmann distributions of molecular velocities, with consequences for chemistry, radiative line shapes and turbulence in the atmosphere, arising from overpopulated high velocity tails of the probability distribution functions (PDFs).

Measurements of HCl and CH4 taken by the aircraft laser infrared absorption spectrometer (ALIAS) on the ER-2 high-altitude research aircraft during the Southern Hemisphere winter of 1994 have been used to examine the abundance of HCl as a fraction of total inorganic chlorine. The fractional abundance of HCl shows a threshold behaviour as a function of temperature history; on a 10 day timescale, the abundance dropped sharply in those air parcels experiencing a temperature < 195 K, but little or no change was seen in parcels which stayed warmer than this temperature. The behaviour mirrors well the temperature behaviour calculated for the transformation of HCl into reactive forms (Cl2, HOCl) from laboratory studies of sulfate aerosols and polar stratospheric clouds. During the course of the winter, the fractional abundance of HCl outside the vortex decreased from its values in late May by about a third, while inside it dropped to near zero by early August. Some recovery was evident in October. Examples of the peel-off of low-HCl air equatorward of the wind maximum were evident in early June. Meteorological trajectories are used to show, in a case study of a flight in early August, that air parcels which experienced temperatures of < 195 K, and as a result had low fractional HCl abundances, did so largely poleward of the maximum in the polar night jet stream. Encountering temperatures of < 195 K during the previous 10 days was a necessary and sufficient condition for the transformation of HCl into reactive forms by heterogeneous reactions. The trajectories further showed that air arriving from sub-tropical latitudes had higher fractional HCl abundances than the air in the middle latitudes, and much higher fractions than the air at high latitudes. The resulting picture is one in which the fractional abundance of HCl in air at mid latitudes was the result of mixing of air from sub-tropical latitudes with air mainly from polward of the jet stream core which has experienced temperatures < 195 K. The sensitivity of the fractional abundance of HCl to the assumption that no HCl enters the stratosphere via the tropical tropopause is examined in the light of an observed profile near the equator with a volume fraction of 0.4 ppb HCl, zero ClO and tropospheric mixing ratios of CFCs at the tropical tropopause.

Vertical profiles of N2O and NO(y) taken by the ER-2 outside the vortex are used to construct average vertical profiles of F(NO(y)) = NO(y)/(A-N2O), where A is the tropospheric content of N2O three years prior to the measurements. The southern hemisphere had less nitrous oxide in the range 400 less than Theta less than 470 K, by up to 25% relative to the northern hemisphere. F(NO(y)) is the ratio of NOy produced to N2O lost in a stratospheric air mass since entry from the troposphere. The profiles of F(NO(y)) have the following characteristics: (1) Relative to 1991-1992, a year without denitrification inside or outside the vortex, the northern hemisphere in 1988-1989 showed denitrification outside the vortex ranging up to 25% and averaging 17% above Theta = 425 K. (2) Relative to the northern hemisphere in 1991-1992, the southern hemisphere in 1987 showed denitrification outside the vortex ranging up to 32% and averaging 20% above Theta = 400 K. (3) Below Theta = 400 K the southern hemisphere showed enhancements of F(NO(y)) relative to the northern hemisphere in 1991-1992 ranging up to 200% at Theta = 375 K, outside the vortex. Corresponding profiles of residual water, R(H2O) = H2O - 2(1.6 - CH4), are considered and shown to be consistent with those of F(NO(y)) in the sense that they show deficits outside the Antarctic vortex, which was both dehydrated and denitrified, but not outside the 1988-1989 Arctic vortex, which was denitrified but not dehydrated. R(H2O) is the water content of stratospheric air with the contribution from methane oxidation subtracted. Comparison of F(NO(y)) and R(H2O) below 400 K outside the Antarctic vortex leads to the suggetion that dehydration in the Antarctic vortex occurs by the sedimentation of ice crystals large enough to fall out of the stratosphere, whereas denitrification occurs mainly on mixed nitric acid-water crystals which evaporate below the base of the vortex at Theta = 400 K but above the tropopause.

The present study compares small-scale (less than 100 km) features in ER-2 measurements of ClO, O3, H2O, N2O, and NO(y) outside the lower stratospheric Arctic vortex of 1988-1989 with features on potential vorticity maps from ECMWF. The potential vorticity maps are obtained from T106 analyses and forecasts. Some of the plots were truncated to lower resolution (T63 or T42) which smooths out the finer-scale structure. Comparison of these lower resolution plots shows how much detail is lost by excessive smoothing. It is also evident that the forecast plots lose fine-scale structure due to dissipation in the model resulting mainly from horizontal diffusion. It is concluded that blobs of air on the maps at latitudes between the vortex edge and 25 deg N having potential vorticities characteristic of the vortex did indeed originate from the vortex, but that the real atmosphere is more sharply differentiated than the meteorological analyses, implying that the potential vorticity maps underestimate the amount of peeled-off material.

Due to both systematic and turbulent induced vertical fluctuations, the interpretation of atmospheric aircraft measurements requires a theory of turbulence. Until now virtually all the relevant theories have been isotropic or "quasi isotropic" in the sense that their exponents are the same in all directions. However almost all the available data on the vertical structure shows that it is scaling but with exponents different from the horizontal: the turbulence is scaling but anisotropic. In this paper, we show how such turbulence can lead to spurious breaks in the scaling and to the spurious appearance of the vertical scaling exponent at large horizontal lags. We demonstrate this using 16 legs of Gulfstream 4 aircraft near the top of the troposphere following isobars each between 500 and 3200 km in length. First we show that over wide ranges of scale, the horizontal spectra of the aircraft altitude are nearly k-5/3. In addition, we show that the altitude and pressure fluctuations along these fractal trajectories have a high degree of coherence with the measured wind (especially with its longitudinal component). There is also a strong phase relation between the altitude, pressure and wind fluctuations; for scales less than ≈40 km (on average) the wind fluctuations lead the pressure and altitude, whereas for larger scales, the pressure fluctuations leads the wind. At the same transition scale, there is a break in the wind spectrum which we argue is caused by the aircraft starting to accurately follow isobars at the larger scales. In comparison, the temperature and humidity have low coherencies and phases and there are no apparent scale breaks, reinforcing the hypothesis that it is the aircraft trajectory that is causally linked to the scale breaks in the wind measurements. Using spectra and structure functions for the wind, we then estimate their exponents (β, H) at small (5/3, 1/3) and large scales (2.4, 0.73). The latter being very close to those estimated by drop sondes (2.4, 0.75) in the vertical direction. In addition, for each leg we estimate the energy flux, the sphero-scale and the critical transition scale. The latter varies quite widely from scales of kilometers to greater than several hundred kilometers. The overall conclusion is that up to the critical scale, the aircraft follows a fractal trajectory which may increase the intermittency of the measurements, but doesn't strongly affect the scaling exponents whereas for scales larger than the critical scale, the aircraft follows isobars whose exponents are different from those along isoheights (and equal to the vertical exponent perpendicular to the isoheights). We bolster this interpretation by considering the absolute slopes (|Δz/Δx|) of the aircraft as a function of lag Δx and of scale invariant lag Δx/Δz1/Hz. We then revisit four earlier aircraft campaigns including GASP and MOZAIC showing that they all have nearly identical transitions and can thus be easily explained by the proposed combination of altitude/wind in an anisotropic but scaling turbulence. Finally, we argue that this reinterpretation in terms of wide range anisotropic scaling is compatible with atmospheric phenomenology including convection.