The absolute dispersion σi2 describes the spread of a scalar relative to a fixed origin along the coordinate axis i. Mathematically, σi2 is the variance of the 1-D mean concentration distribution along the considered axis. demonstrated that the absolute dispersion is directly linked to the Lagrangian autocorrelation function of the motion of one particle. According to Taylor's theory and assuming homogeneity and an exponential autocorrelation function see e.g. the evolution of the absolute dispersion with time t in the vertical coordinate z is modelled as σz2(t)=2⋅w′2‾⋅TL2tTL-1-exp⁡-tTL, with the vertical velocity w(t) and the vertical Lagrangian timescale TL. Assuming homogeneity, the variance of the vertical velocity σw2=w′2‾ can be obtained from the velocity monitored by a sonic anemometer placed at the source location. Given the very short range of our current measurements this is an acceptable approximation. The Lagrangian timescale TL cannot be measured directly by a fixed point measurement, instead the Eulerian timescale TE can be obtained from such measurements. assumed that the Lagrangian and Eulerian time scales have a fixed ratio β=TL/TE. The proportionality constant β can be found using the relationship proposed by , βi≈0.7, where i=σwu‾ is the turbulence intensity in the along wind direction with mean velocity u‾.

The absolute dispersion of an ensemble of puffs (or clusters of particles) can be assumed to be partitioned between two statistically independent components: the meandering of the puffs as a whole with respect to the source location, and the spread of the puffs around their centre of mass, called relative dispersion. This is sketched in Fig. . In mathematical terms, the variance of the mean concentration distribution σi2 is decomposed as a sum of the variance of the centre of mass distribution σm,i2 and the variance of the concentration of the puff relative to its centre of mass σr,i2, σi2(t)=σm,i2(t)+σr,i2(t). Experimentally, the variances are obtained by averaging over multiple realisations of single puffs.

A cluster of particles released at the same time from a finite source will follow slightly different paths and form a distribution around its centre of mass. The relative dispersion is therefore influenced by the source size r0, i.e. the initial separation of the particles. For an initial particle separation (puff size) in the inertial subrange of turbulence, i.e. larger than the Kolmogorov length scale and smaller than the length scale of (local) energy containing eddies, the particle separation will be first influenced by the source size and then become independent of the initial separation e.g.Eqs. A1–A6. Based on inertial range scaling arguments e.g. , the characteristic timescale of the source is given by ts=(r02/ϵ)1/3, where ϵ is the mean dissipation of turbulent kinetic energy. The following Eqs. (4)–(6) are valid for puff sizes in the inertial subrange of turbulence, which was observed in our experiment (see Appendix  for details).

showed that for t≪ts the spread of a puff, or cluster of particles, is dominated by the initial velocity differences between the particles (“ballistic regime”) 〈r2〉=113Ckϵ2/3r02/3⋅t2+r02 for t≪ts, where Ck is the Kolmogorov's constant for the longitudinal structure function in the inertial subrange. Here, r is the 3-D separation between two particles of the cluster and 〈r2〉 is the ensemble mean square separation between all particles of the cluster. In homogeneous isotropic turbulence, 〈r2〉 is related to the 1-D relative dispersion as σr,i2=〈r2〉/6 see, e.g. . Equation () reduced to the vertical component reads then σr,z2(t)=σr,z02+6-2/3⋅113Ckϵ2/3σr,z02/3t2, with the 1-D initial vertical separation σr,z02=r02/6.

For larger times t≫ts, the rate of change of particle separation becomes independent of the initial separation, and the spread of the puff is proportional to the Richardson–Obukhov constant Cr according to the Richardson–Obukhov scaling e.g.. 〈r2〉=Cr⋅ϵ⋅t3+r02 for t≫ts The value of the Richardson–Obukhov constant is uncertain, as it is difficult to estimate from experiments and numerical simulations see for a detailed discussion. However, Cr and the directly related relative dispersion are important for models as the relative dispersion defines the effective rate of mixing of a puff and therefore the decay rate of concentration fluctuations e.g..

Videos of column-integrated concentrations (CIC) of an instantaneous release of a passive tracer can be used to measure different aspects of turbulent dispersion, especially when simultaneous images from different directions are available. The CIC images contain direct information about the puffs' position and spread projected to the image plane (see Fig.  for a sketch and Fig.  for an example image). The image plane is spanned by two discrete coordinate axes i=[1,..,Ni] and j=[1,..,Nj], describing the image columns and rows. We define a rectangular extension of the projected puff, the so-called region of interest (ROI), to distinguish different puffs that may be present in the image, and to reduce the impact of noise. Then, the total signal Stot of the puff (or, in statistical terms, the zeroth moment of the column-integrated concentration PDF) is given by Stot=∑i,j in ROIS(i,j), where S(i,j) is the CIC at pixel (i,j). The centre of mass (CM) of the puff in the image plane (first moment of the column-integrated concentration PDF) is given by icmjcm,=1Stot∑i,j in ROIS(i,j)ij. The spread of mass around its centre as given by the variance (2nd moment of the column-integrated concentration PDF) is described by the weighted covariance matrix C. The diagonal elements of C are the spreads of the SO2 puff in the image plane along the image columns and rows, respectively. Accordingly, the horizontal spread along pixel columns is given by C1,1=1Stot∑i,j in ROI(i-icm)2⋅S(i,j)=1Stot∑i,j in ROIi2⋅S(i,j)-icm2, The spread along pixel rows C2,2 is calculated equivalently.

However, retrieving quantitative dispersion parameters such as the total mass from the camera images requires that the pixel dimensions in the virtual object plane, containing the puff, are known. A pixel is, strictly speaking, a solid angle defined by the focal length of the camera lens f. Thus, for knowing the apparent width of the pixel at the position of the puff, the distance d of the SO2 puff to the camera needs to be known. Then, the apparent width of a pixel sp(d) is given by sp(d)=si⋅d-ff, where si is the physical width of the pixel on the CCD sensor. The height of the pixel in the virtual object plane is calculated analogously and it is equal in case of a sensor with square pixels. In the following, square pixel are assumed for simplicity.

When the puff's 3-D extension is small in comparison to the distance from the puff's CM to the camera, differences in distance over the puff's extension can be neglected and a constant scaling can be assumed for the whole ROI. Scaling the CIC images with the pixels' apparent area sp2, relates the image to a global reference system. It follows for the total mass M of a puff M=sp2(d)⋅Stot, and the horizontal puff spread in square metres C1,1[m2]=sp2(d)⋅C1,1[pixel2].

The spread describes the mass distribution relative to the centre of mass and projected to the image plane. It is hence connected to the relative dispersion. Depending on the relative orientation of the mean wind direction and the camera's optical axis, it can equal the vertical, along- or across-wind direction in some cases. In other cases, assumptions of the plume shape have to be made (e.g. Gaussian plume) or a 3-D reconstruction of the distribution is necessary. When detecting the puff's CM with more than one camera, the CM position in a global coordinate system can be reconstructed. For analysing the statistical nature of the turbulent dispersion, an ensemble of puff releases is required. Then, the meandering is calculated from the variance of the 3-D CM positions and the relative dispersion is connected to the measured puff spread.

The SO2 camera method is based on the principle of absorption spectroscopy of backscattered sunlight. Gaseous SO2 molecules exhibit a distinct, wavelength-dependent absorption cross section in the ultraviolet σ(λ), where λ is the wavelength. The relationship between the light intensity before and after passing through a SO2 cloud – I0(λ) and I(λ) – is described by the Beer–Lambert law I(λ,L)=I0(λ)⋅exp⁡-∫0Lσ(λ)⋅c(l)dl,=I0(λ)⋅exp⁡-σ(λ)⋅S. where c(l) is the SO2 concentration at position l along the light path l∈[0,L] through the SO2 cloud and S=∫0Lc(l)dl is the SO2 slant column density (SCD) along this light path. Generally, radiative transfer effects (e.g. multiple-scattering inside the SO2 cloud and light dilution ) have to be taken into account when translating the slant column density to the column-integrated concentration. However for this study, the effects are negligibly small due to the absence of aerosol and the small extension and short distance of the SO2 puffs to the cameras. Therefore, the slant column densities correspond nearly exactly to the column-integrated concentrations and are used as such throughout the publication.

The SO2 cameras record intensity images of the SO2 cloud I(λ). Images of the clear sky intensity I0(λ) can be measured in the same direction when the SO2 cloud is not present (i.e. before or after a release experiment). The SO2 slant column density S is proportional to the optical density τ(λ), which is retrieved from the two images by τ(λ)=-ln⁡I(λ)I0(λ)=σ(λ)⋅S. Using a narrow bandpass filter in the ultraviolet (typically 310 nm), a narrow spectral band of strong SO2 absorption is selected. While high-precision laboratory measurements of the SO2 absorption cross section σ(λ) are available e.g., calibration from τ(λ) to S is nevertheless necessary due to uncertainties of the exact filter function. The measured optical density images τ are approximated to SO2 SCDs by linear regression using absolute measurements of the SCDs S=aτ+b,=-aln⁡I(λ)I0(λ)+b, where a and b are calibration constants. Such measurements are available from images of gas cells containing a known amount of SO2 and/or from spectra of a built-in spectrometer . Making use of the differential optical absorption spectroscopy (DOAS) technique, a time series of precise point measurements of the SO2 SCD corresponding to a small pixel area within the camera images can be retrieved and correlated to the image time series.

Moving meteorological clouds behind the SO2 cloud can change the illumination of backscattered sunlight between the two images I0(λ) and I(λ). This leads to artefacts in the retrieved SCD images which can be of the same magnitude as the SO2 signal. While SO2 camera measurements under cloudy conditions should therefore be avoided if possible, we could obtain only such measurements due to the weather conditions during the experiments.

In this publication, the background images I0 were taken from the same direction between two puff releases and the images were calibrated using the built-in spectrometer. Note that, contrary to typical applications e.g., measurements at only one wavelength (λ=310 nm) can be used for the analysis due to the absence of broadband absorption from additional aerosol in the SO2 cloud. More details on the retrieval steps used in this publication can be found in the Appendix .

The position and spread of individual SO2 puffs are tracked from the release point automatically. For that, rectangular ROIs containing the full puff need to be detected. Such a detection can be difficult for several reasons. (1) The images partly contain up to two puffs and artefacts from clouds, which can imitate SO2 absorption. (2) Small fractions of the puffs can separate completely from the puffs. (3) The images are noisy, making correct identification of pixels with low SO2 values at the edges of the puffs difficult. In consequence, the ROI has to be large enough to contain the full puff but small enough to exclude additional puffs and clouds.

To overcome these challenges, we choose an approach combining iterative tracking from the release point and applying signal thresholds to two noise-reduced versions of the original image. In this way, the ROI could be detected robustly and the total signal, CM and spread of the puff could be retrieved from the original image. Details of the detection algorithm can be found in Appendix . Further, this approach allows the tracking of several puffs in the same image frame as long as they are separable. Single clouds can be ignored if they are not at the same position as the puffs and even the position of a puff in front of an overcast sky can be constrained spatially, even if not fully detected.

The previously retrieved CMs projected to the image planes of the cameras can be used to retrieve the 3-D trajectories of the CM in the global coordinate system. These allow for calculating the distances between a puff and the individual cameras at any given time. Subsequently, the scaling factor (Eq. ) for the other moments of the PDF (Eqs.  and ) can be determined.

The individual images of the six cameras are recorded at irregular time intervals due to differences in exposure and read-out times. Combining the irregular image times, the derived 3-D trajectories in the global coordinate system were retrieved on an arbitrary-chosen discrete, regular time grid. Here, 250 ms was chosen so that at least one image of every camera lies within each interval. The time series of the CM image coordinates of the six cameras are synchronised and interpolated to this common time grid.

For every time step and each camera, the line-of-sight line from the position of the camera through the detected CM in the image plane at (icm,jcm) is determined by calculating the azimuth and elevation angle. The azimuth angle of the CM is the sum of the camera's azimuth angle α of the optical axis and the relative azimuth angle of the CM to the optical axis αcm=α+arctan⁡(icm-ic)⋅sif, where icm is the pixel column of the CM, ic=Ni2 is the pixel column containing the optical axis (approximated by the central pixel), si is the physical pixel width on the CCD sensor and f is the focal length. The elevation angle is calculated analogously based on the camera's elevation angle and the pixel row jcm of the CM.

At every time step, the position of the CM in the global reference system is then calculated based on the line-of-sight lines of all available cameras using a least squares optimisation: the CM is the point in the global reference system which minimises the square distance to all lines. The CM can be calculated for every time step for which data from at least two cameras were available. However, in this analysis data from at least three cameras were used in order to reduce discontinuities caused by uncertainties in the cameras' position and pose. The reconstructed 3-D trajectories can then be used to determine the distances between the cameras and the puffs at any given, individual image time.

The total SO2 mass of the puffs is conserved, since loss mechanisms (e.g. dry deposition and oxidation of SO2) can be neglected on such short time scales as the ones observed. A change of the measured absolute mass and differences between the signals of different cameras are indications of measurement biases and limitations. These include besides others the cameras' detection limits, incomplete detection of the puff by the derived ROI, additional signals (both negative and positive) from cloud artefacts, uncertainties in the trajectory retrieval and thus the scaling parameter, and radiative transfer effects.

The upper panel in Fig.  shows the total mass of the puffs as observed by four of the six cameras. Cameras 5 and 6 were excluded due to overly pronounced additional signals from the cloudy sky. The background images including the cloud cover for each camera were optimised for the time of the second displayed puff (indicated by shaded area). For this puff, the retrieved total SO2 masses from the four cameras show good agreement: the mass first increases to circa 1.2 g SO2 while the puff is released and then stays constant for all cameras until the puff is no longer tracked. For the other puffs – and thus increasing time difference to the background images – the relative differences between the cameras increases (up to 50 %).

The total mass is strongly affected by clouds, which add both negative and positive signals to the total mass. For camera 3, single clouds are visible along the full pathway of the puff, resulting in a generally overestimated signal. For cameras 2 and 4, single clouds appear only from the middle of the image. Thus in this case an underestimation of the total mass starts only a few seconds after the release. Cameras 5 and 6 (not shown), however, fail to reproduce the released total mass even for the second puff. Camera 1 observes the puffs free of additional signal from clouds and hence catches the correct mass. However, due to it's frontal alignment to the puff's propagation direction, subsequent puffs might overlap. This was the case for the three puffs between 10:30:20 and 10:30:50 UTC. For these puffs no separate mass or spread information can be extracted.

As the mass cannot be retrieved accurately for all data points, it can be assumed that the puff spread would be affected in a similar way by the additional signal due to clouds or overlapping puffs. Therefore, such data points should be discarded from the analysis of the turbulent dispersion. Only measurements for which the total mass lies within a physically reasonable range (here, 1.0 to 1.3 gs-1) are included for further discussion.

Figure  shows the puff spread (Eqs.  and ) in the image plane for four cameras. It is pointed out that these puff spreads are projected to the camera's object plane at the position of the puff. Hence only the puff spread perpendicular to the camera's optical axis is measured (see Fig. ).

In the horizontal, the cameras' relative orientation lead to different projections and thus not directly comparable puff spreads. Camera 1 views the puffs almost frontal and thus the retrieved puff spreads are across-wind in first approximation. Cameras 2 and 3 view the puffs nearly perpendicular to their propagation direction, hence they measure approximately the along-wind spread and their results agree reasonably well. The limited comparability of the cameras and the short data set of only six puffs does not allow for a further analysis in terms of horizontal dispersion.

The elevation angles of the cameras are comparably small (2.3–3.9∘). The vertical projection to the image plane is negligibly small for these elevation angles (cos(3.9∘)=0.998). Hence the measured vertical puff spreads correspond to the real vertical spread of the puff and thus are comparable between the cameras. The measured values of the four cameras agree with each other. In the following discussion only the vertical puff spread is considered for simplicity.

The vertical meandering σm,z2 was calculated as the variance of the ensemble average of the CM trajectories. The shortest trajectory of the six puffs extended over 8 s after release. The ensemble average was calculated for every time step up to this time in order to give a constant weight to all detected trajectories (i.e. at every point in time, the same number of trajectories is averaged). Figure  shows the meandering for the six puffs and, additionally, it shows the meandering when additional puff trajectories from the full duration of the experiment are included.

This enabled an assessment of the uncertainty of the meandering estimate. The number of included trajectories was varied by simultaneously reducing the minimum trajectory length and increasing the time interval. Including a different number of puffs can lead to both a higher and lower σm,z2. The meandering is generally larger when more trajectories are included, particularly in the first few seconds all values lie above the meandering for the six puffs only. The meandering calculated from the full time period at medium trajectory lengths (7 s) was up to two times higher. The increase might originate from atmospheric variability or from the poor statistics. Additionally, a decreasing trend with increasing minimum trajectory length can be observed. This might be explained by the experimental setup. Some trajectories could get discarded during the data processing due to, e.g. clouds in the background or the puff moving out of the field of view. This leads to an effective data reduction to only certain directions and therefore an underestimation of the vertical meandering. In conclusion, the meandering shows a high dependence on the included trajectories, which can be only resolved if a higher number of puffs is available.

The relative dispersion is the spread of the SO2 distribution around its centre of mass. It can therefore be estimated for each individual puff. The spread of the six puffs, averaged over cameras 1–4, and their ensemble average are plotted on a double logarithmic scale in Fig. . The observed relative dispersion does not show a clear transition from the t2 to the t3 regime. In facts, the slope suggests that only the initial t2 regime is observed. That means that the largest observed puff length scales are still affected by the initial separation and, consequently, the puff dispersion according to the Richardson–Obukhov scaling (t3 regime) could not be observed in this experiment. A wider field of view of the cameras would result in longer observation times, which enable an estimate of the Richardson–Obukhov constant by fitting Eq. () to the extended data using the measured value for the energy dissipation.

The well-defined t2 expansion regime (the linear part with a slope of 2 in Fig. ) allows for estimating the effective vertical source size. Following Eq. (), the resulting vertical source size is fitted to σr,z0=8.3 cm and compares to the radius of the release outlet (6.25 cm). The increased number can be explained by the jet created at the source by the blower. Assuming an isotropic source, the source timescale was estimated to ts=(6⋅σr,z02/ϵ)1/3=2.6 s from the vertical source size and energy dissipation rate ϵ. The resulting time lies in the middle of the observed time period, making it possible to theoretically observe the onset of the transition to the inertial subrange.

The absolute dispersion describes the spreading of particles relative to a fixed origin. It is calculated as the sum of meandering and relative dispersion (Eq. ). In Fig.  the 1-D absolute dispersion in the vertical dimension is displayed. The figure contains two parameterizations (w′2‾, TL) of Taylor's theorem (Eq. ). In both cases, w′2‾ is taken from the sonic anemometer data close to the source. The estimate of the Lagrangian timescale differs: TL is either modelled from the measured Eulerian timescale TE,z=3.07 s from the same anemometer data using the empirical constant β=6.87 (Eq. ) or fitted to the absolute dispersion retrieved from the image data. The modelled Lagrangian time is TL,zmodel=21.1 s and the fitted one is TL,zfit=5.9 s. The fitted Lagrangian timescale relates to the measured Eulerian timescale with βfit=TL,zTE,z=1.9 and lies within the previously reported range of 1 to 10 see, e.g..

Here we report the absolute dispersion during the first 8 s after the release. Hence, all measurements were recorded at times below both the modelled and the measured Lagrangian timescales. For times much smaller than the true Lagrangian timescale, the absolute dispersion can be approximated by a quadratic relation, σz2≈σw2‾t2, independent of the Lagrangian timescale, hence making an estimation of the latter nearly impossible for short observation times. Therefore, even if a retrieval of the Lagrangian timescale from the current image data is possible, it is not reliable since the puff observation time does not exceed the Lagrangian timescale.

Further, the absolute dispersion was observed close to the source when it is dominated by the meandering (σm2≈10σr2). The absolute dispersion has therefore an uncertainty similar to that of the meandering (see Sect.  above).