Using stellar scintillation for studies of turbulence in the Earth’s atmosphere

1. Introduction

Ages ago observers noticed twinkling of stars in the night sky. Twinkling stars have been a source of inspiration for many artists and astronomers. While artists have created beautiful paintings and poetry, astronomers have been inspired to understand the causes of the twinkling and how to eliminate it. In science, the technical term ‘scintillation’ is used for the twinkling of stars.

It took a long time to understand the true reason for twinkling of stars. First theories by Aristotle [1] and Ptolemy attributed twinkling of stars to a weakness of human vision. They assumed that seeing is related to some human flux going from the eye to the target. The farther the target, the stronger the scintillation, and, as a consequence, planets are twinkling less than stars. Leonardo da Vinci also believed that the reason for stellar scintillation belongs to the human eye and is an optical illusion [2]. In the late sixteenth century, Tycho Brahe related twinkling of stars to the movement of the stars in the celestial sphere, while Johannes Kepler believed that stellar scintillation is caused by actual changes in the brightness and colour of the stars [3]. It was Isaac Newton who first found, in the early eighteenth century, the true cause of stellar scintillation. He realized that atmospheric turbulence is responsible for the twinkling of the stars [4]. In the nineteenth century, scintillation studies entered a new phase and obtained scientific interpretation thanks to the works and observations by Arago [5], Donati [6], Dufour [7], Montigny [8–10], Wolf [11] and Respighi [12].

It is well known nowadays that scintillation of stars is the result of interaction of stellar light and air density (and, consequently, refractive index) irregularities. In a simplified qualitative explanation, scintillation can be considered as random focusing and defocusing of stellar light. This phenomenon is analogous to the light pattern at the bottom of a sun-lit swimming-pool. The theory of stellar scintillation and its relationship with atmospheric turbulence has been a subject of intensive research.

This review is dedicated to using stellar scintillation for studies of atmospheric turbulence. By ‘turbulence’, we mean any kind of fluctuations that are responsible for the observed scintillations. These include mostly usual turbulence (isotropic, strongly nonlinear) and gravity waves (GWs; strongly anisotropic and weakly nonlinear); other kinds of fluctuations can also be present. We restrict the scope of this review to consider the scintillation of stars only (while there are also scintillations of planets, radio sources and even interstellar scintillation of radio waves) with application to the Earth's atmosphere only (although the technique can also be applied to the atmosphere's of other planets, either from an orbit around the planet, or from the Earth). We present an overview of ground-based, air-borne and satellite stellar scintillation measurements in the Earth's atmosphere, discuss the approaches to data analyses aimed at quantifying the parameters of atmospheric turbulence, and give an overview of the main geophysical results.

2. Atmospheric sources of scintillation

Scintillation is the result of interaction of a light wave with atmospheric air density irregularities. The atmosphere is a complicated fluid: it is stably stratified but nearly everywhere (and always) at the border of instability. It contains a broad range of scales and a broad range of physical phenomena, most of which are nonlinear. It is anisotropic, inhomogeneous and non-stationary. Small-scale air density irregularities, which cause the observed scintillations, are produced by turbulence, internal GWs and instabilities of different kinds.

In the stochastic description of turbulence, the atmosphere is described as a base state with superimposed fluctuations. The fluctuations are assumed to possess the nature of a stochastic field. The base state of the atmosphere is close to some steady state (this is common for ‘natural’ phenomena). The fluctuations mainly result from instabilities in this steady state or instabilities in the ‘perturbations’ of this steady state (such as GWs). Furthermore, the long-term action of the perturbations can affect the base state.

Atmospheric air density fluctuations define a random field of relative fluctuations of refractivity ν=(N−〈N〉)/〈N〉 (here N=n−1 is refractivity, n is the refractive index, and angular brackets hereafter denote the statistical mean). For ‘optical’ wavelengths, 0.2<λ<20 μm, the mean refractivity is defined by

2.1

In general, the field of refractivity fluctuations is described by random functions of three spatial coordinates (or the spatial vector r=(x,y,z), z being the vertical coordinate) and time. However, for application to scintillations, the turbulence can be considered as ‘frozen’, as it is observed by a sensor (Taylor's frozen-field approximation). This allows consideration of the random field depending on the spatial component only. The random refractivity field is assumed to be locally homogeneous, which is the spatial equivalent of a random process with stationary increments. Locally homogeneous fields are characterized usually by the structure function:

2.2

2.3

If the field of refractivity fluctuations is assumed to be locally isotropic [15,16], Φ(κ) in this model can be written as

2.4

2.5

The model (2.4) assumes that all directions in the turbulent atmosphere are treated equally. This assumption is justified if the influence of the buoyancy forces on the formation of refractivity irregularities can be neglected. Under the conditions of stable temperature stratification with a low gradient of wind speed, for example, such an assumption cannot be valid. Local vertical becomes a distinguished direction. The simplest way to adjust the spectrum Φ_n(κ) to the presence of a distinguished direction is to introduce the notion of the anisotropy coefficient η [16]. As a result, the three-dimensional spectrum and the structure function take the following forms:

2.6

2.7

2.8

A model for the spatial three-dimensional spectrum of temperature irregularities generated by internal GWs, which originate and propagate in a stably stratified atmosphere, was suggested in [24]. This is also a model with constant anisotropy, which gives the same power law for the corresponding one-dimensional vertical and horizontal wavenumber spectra.

However, systematic measurements [25–27] have shown that the horizontal spectra are close to a dependence over a large interval of wavenumbers. To account for different slopes of one-dimensional vertical and horizontal spectra, a model with variable anisotropy was proposed in [28]. This model develops the ideas discussed in [29,30]. The ratio of the horizontal and vertical scales—that is, the anisotropy coefficient η—characterizes the shape of the irregularities. The anisotropy coefficient is larger than one, η≥1, and increasing with the vertical scale of irregularities. In [28], it is suggested that η depends only on the vertical scale of irregularities: η=η(κ_z/κ_w), where 2π/κ_w is some characteristic scale of transition towards isotropy. The authors have proposed a model of the three-dimensional spectrum of temperature irregularities [28]. The one-dimensional vertical spectrum corresponding to this model behaves as , while, if one assumes η∼(κ_w/κ_z)², the horizontal spectrum has the slope −5/3 over a wide range of wavenumbers, which agrees with observations. Although more accurate, the models with variable anisotropy have not yet been used for interpretations of scintillation measurements (see also the discussion in §5e).

Although there exist models of the three-dimensional spectrum of atmospheric irregularities that have been successfully used for interpretation of scintillation measurements, the small-scale structure of atmospheric irregularities is very complex and still incompletely described. Measurements of the atmospheric microstructure reveal a fairly large number of very strong (positive) temperature gradients (approx. 30–100 K km⁻¹) within very thin layers (approx. 3–20 m). Such data are obtained with fast-response temperature sensors on balloons, radar echoes and scintillations of binary stars [31–34]. The ‘sheets’ are observed at various altitudes, most of them being in the lower stratosphere. In situ measurements with sensors some distance apart show that the ‘sheets’ are neither flat nor horizontal. Mixing within these thin layers has been detected [33,35]. Other relatively recent analyses apply multifractal models to describe the atmospheric structure [36,37]. For a review of these analyses and the resulting space–time cascade model, which is based on strongly anisotropic and intermittent generalizations of the classical turbulence laws, see [38]. Spatial inhomogeneity and intermittency (the atmosphere is not everywhere and not always turbulent; when turbulent, the parameters change from region to region) introduce further complication into the description of atmospheric turbulence.

The theory of turbulence in stably stratified media is still under development and is the subject of active discussions. Intensive work is being done towards the development of new turbulence closure theory [39–41] and the quasinormal scale elimination model [42–44].

3. Theory of scintillation

The general theory of scintillation for ground-based and spacecraft observations was developed by Tatarskii [15] and Ishimaru [16]. Stars can be considered as point sources of light, and wavefronts can be assumed to be a plane at the entrance of the atmosphere. Atmospheric irregularities have a random structure, producing random effects on stellar light that passes through the atmosphere. These require a statistical description, by either autocorrelation functions, spatial wavenumber spectra or structure functions. These descriptions are integral transforms of one another, thus being equivalent in some sense. Theoretical approaches relate the characteristics of a model of the turbulent medium and the observed scintillations. They have been developed for both wave-optical treatment and geometrical-optics approximation, using the power spectrum of refractive-index fluctuations or corresponding structure functions. A number of monographs exist on wave propagation in random media: classical ones by Tatarskii [15,45], as well as [16,46–48]. A general theory of scintillation observed from space is developed in [16,49–52]. The phenomena associated with star scintillation as observed from the ground was reviewed by Dravins et al. in three papers [53–55].

As was mentioned above, the frozen-field hypothesis is used for describing scintillation phenomena. Within the Rytov smooth perturbation approximation [47], there is a fairly simple connection between the (three-dimensional) spectrum of refractive index fluctuations and the two-dimensional scintillation spectrum in the observation plane. So far, these relations have been used to retrieve the parameters of the spatial spectrum of refractive index fluctuations from satellite and in situ scintillation measurements [19,20,21,56–58]. The Rytov method is valid when the monochromatic scintillation variance is small, (weak fluctuations). For more general conditions (strong scintillations), the moment equations are so difficult that they are solved only for some special cases and approximations. A variety of techniques have been proposed such as the parabolic equation method [16,59], extended Huygens–Fresnel principle [16,60], multiple phase screen [61], the path integral [62,63] and numerical techniques [64,65]. Among these, the parabolic equation method with the Markov approximation [59,66] occupies an important place.

The theory of strong scintillations applied to the stellar occultation measurements has been further developed by Gurvich et al. [67]. That paper considers the problem of remote sensing of stratospheric irregularities under strong scintillation conditions. The authors have analysed calculated scintillation spectra and have concluded that the parameters of the three-dimensional spectra of stratospheric irregularities can be retrieved under the conditions of relatively strong scintillation (characterized by scintillation variance computed with weak fluctuation assumption below 1.5–1.6).

Since the scintillation theory is based on the theory of wave propagation in random media, several studies have been dedicated to checking its validity. In particular, the probability density function of scintillations has been examined. Weak perturbation theory predicts starlight to be lognormally distributed, as the refractive-index fluctuations, which can be considered as a large number of independent events, modulate the intensity in a multiplicative manner. For small and moderate scintillation amplitudes, the experimental distributions follow the lognormal law, but for greater ones, departures occur, as expected [21,53,68].

The scintillation method, as a tool for studying atmospheric turbulence, has specific features and limitations. First, scintillation measurements are not direct: in order to get information about small-scale processes, a functional relationship between the equivalent parameters of atmospheric irregularities and parameters of observed scintillation or solution of the inverse problem is needed. Conceptually, the problem is simpler in two dimensions, but observations are mostly one-dimensional. The amount of all currently available scintillation measurements is not sufficient for reconstructing the structure of air density via solving the tomography problem (as described, for example, in [69]). Instead, spectral properties of atmospheric irregularities are assumed, so that a relationship between the structure of air density irregularities and the parameters of the observed scintillations can be found. Second, depending on the measurements, scintillations are insensitive to some parameters of the atmospheric spectral model. This is discussed in more detail in sections dedicated to particular types of measurements. Third, the parameters of the spectral model (, for instance) estimated from scintillation measurements represent ‘effective’ parameters averaged along a ray with the weight function determining the effective region of interaction between the light wave and the turbulent atmosphere. Fourth, the smallest scale observable from scintillation is the diffractive Fresnel scale, , where L is the distance from the observer to the point representing the interaction region of the light wave and the turbulent atmosphere.

For recording scintillations, important are star magnitude and the size of the telescope (in order to get sufficient signal-to-noise ratio), the sampling rate of the detector (in order to get sufficient resolution) and the chromatic width of the optical filter (chromatic refraction in the atmosphere leads to changes in the bandwidth scintillation spectrum, particularly, to attenuation of variance).

The specifics of data, processing and retrieved parameters depend on the geometry of line of sight. In the following, we review the data processing and the results from ground-based, satellite and air-borne measurements of stellar scintillation.

4. Ground-based measurements of stellar scintillation

(a) Introductory notes

Quantitatively, scintillation observed on the ground can be explained as follows. Let us imagine one optical turbulent layer moving at altitude h above the ground, as sketched in figure 1. After passing through the turbulent layer, the wavefront, which was initially a plane, becomes distorted. Light rays propagate perpendicularly to the wavefront towards the ground, converging in some places and diverging in others, giving rise to bright and dark speckles. It is important to recall that scintillation is related to light intensity fluctuations and not to phase or complex amplitude. In figure 1, if one puts one's eye, or a photoelectric receiver, just after the turbulent layer, one will see the star wandering, if the eye pupil is less than the spatial coherence of the wavefront. The receiver will not yet notice any intensity variation, because light rays do not have enough propagation space after crossing the turbulent layer. At ground level, after propagation over a distance h, light beams converge and diverge to produce speckle patterns, known as ‘shadow band patterns’. This is well described in [45,70] in the framework of Fresnel diffraction.

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For ground observations, where the line of sight is nearly vertical, the strongly anisotropic fluctuations (generated mostly by GWs) can be completely ignored. The reason for the elimination of the anisotropic component is that scintillation results from refractivity gradients perpendicular to the line of sight. Because of the rather large horizontal scales of the GW component (many kilometres) and of its very large anisotropy, the corresponding gradients can be ignored completely; thus turbulence can be regarded as isotropic (or nearly so). Furthermore, because of the strong anisotropy, this approximation is valid up to large angles from the vertical (as large as approx. 75^°), thus covering all the astronomical applications. In analyses of ground-based stellar scintillation measurements, the Kolmogorov spectrum is assumed.

(b) Multi-dimensional scintillation analysis

Since the 1960s, most scintillation studies have been performed on one-dimensional temporal signals I(t), which reflect all the perturbations from the whole atmosphere. Even if one makes the stationary assumption along with the frozen turbulence hypothesis, there is no way to retrieve detailed vertical profiles of turbulence from the temporal autocorrelation of the scintillation observed at the ground. This is because of the fact that the atmosphere is never moving as a whole with a constant velocity. Hence, the ground intensity is the superposition of many scintillation patterns from turbulent layers moving at various velocities in different directions. A few attempts have been made to obtain spatial power spectra using different spatial filtering in the pupil plane [71–73]. Measurements by Protheroe [72] agreed well with a layer propagating at an altitude of approximately 10 km. Owing to the above-mentioned superposition of many layers, neither the fine structure of the atmosphere nor any information about the ground layer is attainable.

The idea of using a ‘multi-dimensional analysis’ of the scintillation in order to obtain information about the layered structure of the atmosphere has been proposed in [31,74,75]. In this approach, the scintillation intensity I is considered as a multi-dimensional function of time, wavelength λ, position of the observer r(x,y) and angular position of the star (α,β):

4.1

By finding the cross-correlation function of scintillation with respect to several of these variables, it is possible to retrieve the turbulence parameters corresponding to different layers with different and different velocities v. For example, the spatio-wavelength analysis allows optical turbulence profiles to be retrieved (§4b(i)). The spatio-angular technique, first developed by Rocca et al. [74] and then by Vernin & Roddier [75], has opened the era of the Scidar (scintillation and ranging, by analogy with radar) technique, which has become a standard tool for remote sensing of the Earth's atmosphere. This method is discussed in §4b(ii). The ‘generalized Scidar’ technique (§4b(iii)) allows the optical turbulence also at the ground level to be detected. The spatio-temporal technique constitutes the basis for data analyses using the so-called single star Scidar (SSS); this is discussed more in §4b(iv).

(i) Spatio-wavelength analysis: the differential refraction

The spatio-wavelength analysis can be applied for bright stars close to the horizon, as was demonstrated by Caccia et al. [76]. When stars are approaching the horizon, scintillation becomes chromatic. This phenomenon is related to the refractive index dependence on wavelength, thus resulting in different bending. A cut of the cross-correlation of the scintillation pattern at λ₁ and λ₂, C_I(0,y,λ₁−λ₂)=〈I(0,y′,λ₁)I(0,y′+y,λ₂)〉, assuming that the y-axis is perpendicular to the horizon, allows us to retrieve vertical profiles of optical turbulence. Since this experiment works only with very bright stars (Sirius) close to the horizon, the method was abandoned.

(ii) Spatio-angular analysis: the Scidar

The Scidar principle (figure 2a) was first proposed by Rocca et al. [77]. Two stars separated by an angle θ cast onto the ground two identical speckle patterns (or scintillation), but displaced from one another by a distance d=θh, h being the altitude of the layer. The distance d is measured by computing the autocorrelation function of the scintillation patterns, which peaks at d. The intensity of the peak is related to the refractive index structure characteristic . Each turbulent layer produces a bump in the measured autocorrelation function (since the width of the autocorrelation function is proportional to the Fresnel scale, the higher the layer, the wider the bump).

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For a continuous distribution of turbulent layers, the relationship between the measured autocorrelation function of the scintillation produced by a double star C_**(r) and the distribution is

4.2

where C_*(r,h) is the autocorrelation function of the scintillation produced by a single star and a layer at altitude h, and a and b are related to the magnitude difference Δm between each component of the double star:

4.3

Equation (4.2) is an integral relationship between C_**(r) and with the kernel K=aC_*(r,h)+b[C_*(r−θh,h)+C_*(r+θh,h)]. Avila et al. [78] first mentioned the use of the maximum entropy method to invert this relationship and retrieve the  profile.

(iii) The generalized Scidar

The Scidar experiment was tested in many observatories and validated with comprehensive comparisons with other techniques such as a Differential Image Motion Monitor (DIMM) and instrumented balloons measuring the optical turbulence [79]. In its classical version, the Scidar is still insensitive to low-layer turbulence, which can largely affect the astronomical seeing (or image degradation) within a 50–80% proportion.

Fuchs et al. [80] first proposed analysing the scintillation in a virtual plane, below the pupil plane by a distance h_gs, as sketched in figure 2a. This plane disposition allows a layer at ground altitude to have enough Fresnel propagation to give scintillation, because h_gs≠0. Using a laboratory demonstrator, these authors have shown that the variance of the scintillation was positive when a phase screen (turbulent layer) was close to the pupil plane, goes to zero at a distance h=h_gs (gs for generalized Scidar), and then increases again at larger distances. Equation (4.2) is changed to

4.4

where H=h−h_gs and h_gs is the virtual altitude in the generalized Scidar mode. This altitude, which increases the Fresnel propagation distance, needs to be larger than zero but sufficiently low in order to avoid a saturation effect of the scintillation variance.

The generalized Scidar became a standard instrument for remote sensing of the atmosphere for astronomical purposes as well as for atmospheric studies. There still remains a strong restriction on the size of the entrance pupil of the telescope, D_telescope, which needs to be larger than θh_max, h_max being the maximum altitude that one wants to sample. Generally, for available bright double stars with a small difference in magnitude Δm, one needs a 1.5 m telescope, which limits the use of the Scidar to existing observatories already equipped with large telescopes.

(iv) Spatio-temporal analysis: the single star Scidar

The spatio-temporal analysis, i.e. the analyses of the auto- and cross-correlation of time evolution of scintillation,

4.5

has found its application in the so-called SSS technique [74,81,82,83,84]. Figure 2b shows the SSS principle for a single turbulent layer. In the real atmosphere, many layers are moving at different velocities v_i leading to many cross-correlation peaks situated at d_i=v_iτ. One might think that a simple cross-correlation function Γ_*(x,y,τ) should be sufficient to retrieve and v(h) related to each turbulent layer: the position of the bump giving the velocity, its amplitude giving and the size of the bump giving the altitude of the layer through its proportionality to the Fresnel scale. However, Caccia & Vernin [84] have demonstrated that the latter assumption is not true, because the dispersion of the wind velocity σ_v enlarges the size of the cross-correlation peak. This can be described as a convolution of the natural correlation by a Gaussian function whose width is proportional to the time lag τ. Since the decorrelation due to velocity dispersion does not have the same effect on the enlargement of the autocorrelation as that due to altitude , the inversion is possible only if the cross-correlation is measured at various lags. The inversion is an ill-posed problem, which can be solved, for example, with the simulated annealing technique [85].

The SSS concept was validated in several places before being sent to Dome C on the high Antarctic plateau, where routine observations have been performed during 2006 and 2007 polar nights. The instrument is made of a portable 40 cm telescope, an optical bench with a stop field and a 10 mm collimating lens and a fast readout charge-coupled device (CCD) camera. Under clear sky conditions, vertical profiles of , velocity components v_x(h) and v_y(h), and the wind velocity dispersion σ_v(h) are obtained every 11 s, during hours of continuous observations, tracking numbers of bright stars. SSS is a reliable, transportable instrument based on sound theory, carefully tested, allowing calibrated and wind velocity profiles from the ground up to 25 km.

5. Satellite stellar scintillation measurements

(a) Introductory notes

Stellar scintillation from space through the Earth's atmosphere is observed in limb-viewing geometry (figure 3a). These observations have special characteristics: first, an observer is located far beyond the atmosphere, and second, the influence of the denser atmospheric layers below the perigee point of the line of sight is eliminated naturally. This distance leads to a substantial enhancement of scintillation during the free-space propagation. As a result, the observed fluctuations of the light flux have relatively large amplitudes even in the case of very weak perturbations of the air density (an example of scintillation records is shown in figure 3b). The second feature is also favourable for studying air density variations at high altitudes that are not exploitable from the Earth's ground because of the masking effect of the dense and turbulent boundary layer.

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Measurements conducted from space through the Earth's atmosphere bear a certain similarity to ground-based astronomical observations of scintillation of the stars during their occultation by planets [90–93]. However, the large distance between the observer on the Earth and the planet results in a limited spatial resolution, which is of the order of the corresponding Fresnel scale. Even for the planets nearest to the Earth, the spatial resolution is no better than a few hundreds of metres. In observing stars from a low orbiting satellite through the Earth's atmosphere, the Fresnel scale is tens of centimetres.

(c) Approaches to data analyses

The method that allows information to be obtained about the parameters of the three-dimensional spectrum of air density irregularities using one-dimensional scintillation spectra was first proposed by Gurvich & Kan [56] and then developed further by Gurvich & Chunchuzov [19] and Sofieva et al. [20]. In this approach, the spectrum of the air density irregularities is parametrized and the parameters of the spectral model are retrieved via fitting experimental scintillation spectra. Large satellite velocity ensures the validity of the frozen-field approximation. The spectrum Φ_ν of refractivity fluctuations is represented as a sum of two statistically independent components, i.e. anisotropic, Φ_W, and isotropic, Φ_K:

5.1

For parametrization of the anisotropic component Φ_W, which corresponds to anisotropic irregularities generated by a random ensemble of internal GWs, the spectrum (2.7) has been used in [19,20]. In [56], the analogue of (2.7) but without an outer scale was used. For parametrization of Φ_K, which corresponds to isotropic irregularities generated by turbulence appearing as a result of GW breaking and/or different instabilities of atmospheric motions, the Oboukhov–Corrsin model of locally isotropic turbulence [111] has been used:

5.2

where is the structure characteristic of isotropic fluctuations (related with the structure characteristic of temperature fluctuations as and κ_m is the inner scale of isotropic irregularities. The structure characteristic is proportional to ε_ν/ε^1/3, where ε_ν is the rate of dissipation of refractivity fluctuations and ε is the kinetic energy dissipation rate.

Theoretical relations that allow the calculation of a one-dimensional spectrum of relative fluctuations of intensity at the observation plane from the given spectrum of air density irregularities (the forward model) have been developed in [52,56] in the phase screen approximation under the weak scintillation assumption. In the framework of the phase screen approximation, the effect of the extended atmosphere on the light wave passing through it is replaced by that of a virtual thin phase screen. It is assumed that this screen produces the same phase modulation on the propagated waves as the extended atmosphere [16].

The spectral model of refractivity irregularities has altogether six parameters: for the anisotropic component, they are the structure characteristic C_W, the anisotropy coefficient η, the inner scale l_W and the outer scale L₀; for the isotropic component, they are the structure characteristic and the Kolmogorov scale l_K. However, not all of them can be retrieved from scintillation spectra. Numerical simulations [52] have shown that the anisotropy coefficient cannot be estimated from scintillation spectra, except for very oblique occultations. The Kolmogorov scale can be retrieved from scintillation measurements only with sufficient (greater than 10 kHz) sampling frequency of the photometer, and only if it is larger than the Fresnel scale. This was possible with the MIR photometer having the maximal sampling frequency of 16 kHz (the vertical resolution is a few decimetres), but impossible with the GOMOS sampling frequency of 1 kHz photometer (the vertical resolution is about 1 m). Thus, it was possible to retrieve five parameters of refractivity spectra from MIR scintillations, and four parameters from the GOMOS measurements. Typical values for the non-retrieved parameters, i.e. the anisotropy coefficient η=30 and the mean value l_K=0.25 m for the Kolmogorov scale [57], have been used in retrievals [20]. The detailed description of the algorithm for reconstruction of GW and turbulence spectra parameters is presented in [20]. The inversion procedure is based on a nonlinear fitting of the experimental scintillation spectra by the modelled ones. An example of the experimental scintillation spectrum and the retrieval results is shown in figure 3c. The developed algorithm is rather fast, and thus it is suitable for processing of vast amounts of satellite data.

The turbulent structure characteristic (or C²_T) can be reconstructed from scintillation data without solving the inverse problem described above, as the scintillation variance corresponding to the isotropic component σ²_I,iso is proportional to . The value of σ²_I,iso can be estimated using the simplified spectral analysis, as described in [58,105]. This technique uses the fact that scintillation spectra at scales from l_W to the scale corresponding to the Nyquist frequency are defined mainly by the turbulent component and they are nearly flat.

When chromatic scintillation measurements are available, additional information can be obtained using the cross-spectra and coherence spectra. Details of these analyses and their application to GOMOS data are discussed in [108,109].

(d) Geophysical results

The scintillation measurements with the EFO-2 photometer on board the MIR station have provided estimates of GW and turbulence spectra parameters for altitudes of 25–70 km [57] for the first time, as these altitudes are very difficult to explore with other methods having comparable vertical resolution. Typical values of the parameters and their altitude dependence have been obtained. It was found that the Kolmogorov scale l_K increases monotonically with height from a few decimetres at altitudes of 27–30 km to 1–1.5 m at 63–67 km. The inner scale of the anisotropic component l_W (proportional to the buoyancy scale) increases more rapidly from 1–2 m at altitudes of 27–30 km to 10–30 m at 45–53 km. Between the scales l_K and l_W, there is a scale range whose width increases with height. The retrieved estimates of the structure characteristic of the anisotropic component C_W for altitudes of 27–53 km are found to be in good agreement with predictions based on the assumption that the regime of saturated internal GWs exists within this altitude range [22,112]. The estimates of the turbulent structure characteristic C²_T increase almost exponentially with height from (2–10)×10⁻⁵ K² m^−2/3 at 27–30 km to (0.5–2)×10⁻² K² m^−2/3 at 65–67 km. For altitudes of 60–67 km, these estimates are in agreement with in situ measurements [113].

The retrieved Kolmogorov scales allow for the estimation of the rate of kinetic energy dissipation ε to heat. For the heights 60–67 km, the obtained values ε=(1–5)×10⁻³ m² s⁻³ agree with the results of in situ measurements [113]. As altitude decreases, the dissipation rate decreases monotonically to (0.5–10)×10⁻⁷ m² s⁻³ at 27–30 km. The simultaneous measurements of ε and C²_T have allowed for the estimation of the dissipation rate of air density irregularities through molecular diffusion. The calculated height dependence of the Bolgiano–Oboukhov scale [111] has shown that the assumption (used in the model) of local isotropy of small-scale irregularities is certainly justified above 35 km.

The MIR scintillation data have also allowed for the retrievals of the outer scale of the anisotropic component [19] and for the detection of quasi-periodic disturbances in atmospheric air density and temperature [114].

Observations of stellar scintillations by GOMOS have confirmed the estimates of GW and turbulence spectra parameters obtained from MIR. Furthermore, the global coverage of GOMOS measurements has allowed for the study of spatio-temporal distributions of these parameters, including the polar regions. Already, first analyses have shown surprising results. Strong increases in scintillation variance and C²_T at high latitudes in winter were observed [58,105]. It was shown that the maximum of these enhancements can be associated with the polar night jet. The simplified spectral analysis has shown the transition of scintillation variance towards small scales with altitude, which is probably related with turbulence appearing as a result of wave breaking. The breaking of GWs in the polar night jet seems to start in the upper stratosphere. This previously predicted feature [115,116] was confirmed by the GOMOS observations for the first time.

In [58], the first estimation of global distribution of C²_T at altitudes of 30–50 km for the four seasons of year 2003 is presented. As a general rule, the values of zonal mean C²_T increase with altitude. The largest C²_T values are achieved in polar regions in winter, where they may reach 0.006 K² m^−2/3 in the Southern Hemisphere, i.e. values that are comparable with those observed in the turbulent boundary layer (fig. 2 in [58]). Weaker turbulence is observed at low latitudes. Minimal values of C²_T are less than 10⁻⁴ K² m^−2/3 there and it has a pronounced zonal structure. The average turbulence intensity at altitudes of approximately 42 km follows the sub-solar point, with enhancements located mainly over continents (figure 4, note that uncertainty of the estimate is much smaller than their variability). Turbulence enhancements in tropical regions are not related with orography. Despite a good correlation with typical regions of deep convection, the overall distribution of C²_T displays a more complicated structure.

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GW spectra parameters at altitudes of 30–50 km observed in 2003 are discussed in detail in [106], where their global distributions are presented. The use of outer scale estimations has enabled the estimation of an important GW parameter, the potential energy E_P per unit mass [106]. Enhancements of E_P in equatorial regions are clearly observed, analogous to those found in global analysis of radio-occultation data. They are caused by large values of GW outer scale at these locations. Enhanced E_P is observed also in the Southern Hemisphere winter at high latitudes, but there is no symmetric E_P enhancement in the Northern Hemisphere winter at high latitudes. Sofieva et al. [106] suggested that it is probably caused by the fact that occultations at Northern Hemisphere winter high latitudes were carried out in January and in December 2003, when sudden stratospheric warmings (SSW) lasted a significant part of the observational period. It coincides with previous lidar observations [117], which have shown reduced GW activity during SSW.

6. Air-borne measurements of stellar scintillation

The scintillation of stars observed through the Earth's atmosphere was also reported from balloon-borne investigation of the nocturnal vertical distribution of stratospheric minor constituents [118]. Compared with satellite observations of stellar scintillation, the geometry (and the physics) is very similar, but the relative velocity of the line of sight with respect to the atmosphere is of the order of 10 m s⁻¹ or less while it exceeds 1 km s⁻¹ for the satellite observations. Consequently, the small-scale fluctuations are much easier to observe and characterize from a balloon-borne instrument. The altitude range where scintillation can be analysed is practically contiguous—or slightly overlapping—thus allowing the observation of the whole stratosphere. While the analysis of satellite measurements is currently limited to altitudes above 30 km by the strong scintillation, the balloon-borne observations are limited upwards by their maximal float altitude, around 30–40 km. In the case of satellite measurements, the distance between the observing instrument and the dominant refractivity fluctuations is large (approx. 3200 km for GOMOS) and does not change significantly with the altitude of the ray perigee. For balloon-borne observations, this distance increases from zero up to 400–500 km when the tangent point decreases from the float altitude down to the troposphere.

During the preparation of the gomos instrument, a balloon-borne experiment especially dedicated to the multi-spectral recording of scintillation, amon-ra, was performed in February 1999 and analysed [119]. The results of this balloon experiment allowed for the validation of the hypothesis and parameters underlying the scintillation correction used during the gomos data processing. The fact that the characteristic scintillation frequencies recorded from a balloon-borne gondola are much smaller than from a satellite allowed the scintillation to be recorded simultaneously with 10 Hz sampling frequency and within 14 spectral channels covering the full visible range (see figure 5, adapted from [119]). The observed intensity peaks—light focusing—produced by small-scale refractivity fluctuations are observed at all wavelengths, but with slightly different arrival times because of the refractive dispersion. Since the refraction angle is nearly proportional to the refractivity, the shape of these peaks reproduces the wavelength dependence of standard air refractivity [120].

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The experiment amon-ra was flown again in March 2003 during an Envisat (and thus gomos) validation campaign [21]. The duration of continuous recording was extended from 60 to 600 s, thus allowing easier and deeper statistical and spectral analysis of the scintillation time series. In order to retrieve the spectral parameters of the small-scale atmospheric dynamics, an analytical formulation of the two-dimensional scintillation spectra was developed. This spectral model is fully compatible with the model used for the analysis of the gomos scintillation (see §6) except that the layered structure of the atmosphere is explicitly taken into account, with values of the local parameters variable with the altitude. This analytic model can also handle the case where the light source is at a finite distance (including within the atmosphere) from the observing instrument. This analytic model needs to be numerically integrated in order to be compared with experimental data.

The statistical distribution of recorded intensities follows closely the lognormal model when the scintillation is weak, as expected. Progressive divergence appears when the scintillation becomes moderate, while strong scintillation seems to follow a decreasing exponential law. The intensity distribution is independent of wavelength for the weak scintillation regime. The time spectra of the recorded scintillation are in agreement with the analytic model, and the position of the cut-off allows one to estimate the characteristic inner scale of the anisotropic fluctuations 10–20 m, in general agreement with values deduced (at higher altitudes) from satellite measurements. Finally, the normalized intensity variance closely follows the values predicted by the model, except for the strong scintillation case, where the model is known to be inaccurate and where the chromatic refraction smoothing should be taken into account.

The description and analysis of the scintillation recorded from a stratospheric balloon is qualitatively comparable to the case of the satellite measurements. However, some parameters describing the geometry or characteristic velocities of these measurements are significantly different, thus allowing exploration of different ranges for altitudes or model parameters. As usual, when comparing local measurements with satellite ones, the global coverage and long-period monitoring capabilities of satellite observations are unique characteristics that also apply to scintillation measurements. One of the potential applications of such experiments might be using them for validation purposes.

7. Summary and discussion

The scintillations observed through the Earth's atmosphere result from interaction of light waves and the turbulent atmosphere, thus carrying important information about small-scale processes in the atmosphere. The scintillation can be observed in different viewing geometries, when the line of sight is nearly vertical (ground-based astronomical observations) or nearly horizontal (occultation geometry). Since the scintillation measurements belong to remote sensing measurements, inverse problems are required in order to get information about atmospheric turbulence. The data analyses rely on using spectral models of air density (hence, refractivity) fluctuation, the theory of wave propagation in random media and the inverse problem in order to retrieve some parameters of the spectral model.

There are several limitations and specific features inherent in the scintillation method. For interpretation of scintillations, a spectral model of air density irregularities should be assumed. This model need not necessarily be a comprehensive model of atmospheric dynamics. For good agreement with observed scintillations, it is sufficient to use a model with important (and meaningful) parameters. This model can even be partly or completely empirical. The analyses of the scintillation data provide the values (and variability) of the parameters in different atmospheric conditions. These values may be used in order to evaluate models of atmospheric dynamics and validate different hypotheses.

An important limitation of scintillations as an investigation tool for atmospheric dynamics is that they are insensitive to some parameters of the spectral model (e.g. ground-based scintillation measurements are insensitive to anisotropic irregularities; in the case of satellite stellar occultation measurements, the change of anisotropy is not important for scintillation if the anisotropy is large).

When comparing the parameters retrieved from scintillations with other kinds of measurements, it is important to take into account that the scintillations provide ‘effective’ parameters, which are averaged along a ray, with the weight function determining the effective region of interaction between the light wave and the turbulent atmosphere.

Stellar scintillation measurements in occultation geometry are characterized by a good spatial resolution, which is limited by the Fresnel scale or by the scale corresponding to the sampling rate of the detector. This also imposes a limitation: information at scales smaller than the Fresnel scale becomes unavailable even with a sufficient sampling rate of the detector.

Nowadays, scintillations have provided useful information about turbulence in the atmosphere from the ground to the mesosphere. The Scidar technique has become a standard experiment for remote sensing of the Earth's atmosphere. Satellite observations of stellar scintillation provide information about the structure of air density irregularities in the altitude range of 25–50 km, where other measurements at such small scales are very scarce. The important strength of scintillations in occultation geometry is that the wavenumber range of the transition of the GW spectrum to turbulence is covered. This opens up the possibility for monitoring GW breaking in the stratosphere. The scintillation is affected by small vertical scale GW (with vertical scales from a few metres to a few kilometres), which are of high importance for parametrization of GW effects in global circulation and climate models. Perhaps the most valuable application of the turbulence parameters retrieved from scintillation measurements is using them to optimize the turbulence schemes in atmospheric models. This becomes increasingly important due to their tendency to higher altitudes and smaller scales.