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  • View in gallery

    Power spectral densities of the saturation ratio S as a function of frequency f as measured for the ground-based observations and the two entire legs at 1130 and 1180 m. The black curves are averages over logarithmic equidistant bins; the blue curve provides a slope as predicted for classical inertial subrange scaling. The spectrum for the ground-based measurements (GND) has the original y scale; the other two spectra are vertically shifted by 103 and 106, respectively.

  • View in gallery

    Fluctuations of (top left) water vapor pressure and temperature and (bottom left) saturation ratio as observed from a 5-min-long record at ground level shortly before takeoff. Horizontal lines indicate the rms values. (right) The corresponding PDFs are displayed. A Gaussian fit and the standard deviation σ of each parameter are given.

  • View in gallery

    A photograph of the freshly evolving cumulus field taken from the helicopter (courtesy of B. Wehner).

  • View in gallery

    Vertical profiles of (a) potential temperature , (b) water vapor mixing ratio , and (c) aerosol particle number concentration . The red curve displays the ascent at the beginning of the flight, whereas the black curve displays the final descent. The two flight levels with cloud observations in z = 1130 and 1180 m are marked with blue horizontal lines. A red line at 750 m indicates the top of the mixed layer for the ascent.

  • View in gallery

    Time series of the cloud field observations measured at a height of 1130 m. The entire record covers a flight path of about 12 km. (a) The water vapor pressure e (green line), temperature T (red line), and saturation ratio (blue line). Also shown are the cloud parameters in terms of (b) droplet diameter d (blue points) and mean droplet diameter (orange circles) and (c) droplet number density (red circles).

  • View in gallery

    PDF of droplet diameter d. The PDF is calculated from 51 600 droplets during the two legs at 1130 and 1180 m.

  • View in gallery

    An enlarged portion of the record presented in Fig. 5 covering a flight path of 1 km. (a) Fluctuations of water vapor and temperature (left axis) and saturation ratio S (right axis). (b) The droplet diameter d, mean droplet diameter , and droplet number density ; and are mean values averaged over 1-s periods. (c) The vertical velocity w and the quasi-steady supersaturation . For small (e.g., close to zero), the values are out of scale.

  • View in gallery

    (left) Cloud data concatenated from quasi-homogeneous subrecords as observed during two flight legs at 1130 (blue curves) and 1180 m (orange curves). The ranges of each subrecord are marked on the lower x axis. Each subrecord was high-pass filtered by applying a third-order Savitzky–Golay filter with 201 filter elements (filter period of 2 s) before concatenation, resulting in fluctuations of the parameters. (right) The corresponding PDFs are displayed including a Gaussian fit and the standard deviation σ.

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Supersaturation Fluctuations during the Early Stage of Cumulus Formation

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  • 1 Leibniz Institute for Tropospheric Research, Leipzig, Germany
  • 2 Michigan Technological University, Houghton, Michigan
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Abstract

On time scales that are long compared to the phase relaxation time, a quasi-steady supersaturation sqs is expected to exist in clouds. On shorter time scales, however, turbulent fluctuations of temperature and water vapor concentration should generate fluctuations in supersaturation. The variability of temperature, water vapor, and supersaturation has been measured in situ with submeter resolution in warm, continental, shallow cumulus clouds. Several cumuli with horizontal extents of order 100 m were sampled during their first appearance and development to depths of ~100 m in a growing boundary layer. Fluctuations of the saturation ratio are observed to be approximately normally distributed with standard deviations on the order of 1%. This variability is almost one order of magnitude larger than sqs calculated using simultaneous measurements of the vertical velocity component and the droplet size distribution. It is argued that, depending on the ratio of the phase relaxation and the turbulent mixing time, substantial fluctuations in the supersaturation field can exist on small spatial scales, centered on sqs for the mean state. The observations also suggest that, on larger scales, fluctuations of the supersaturation field are damped by cloud droplet growth. Droplets with diameters of up to 20 μm were observed in the shallow cumulus clouds, whereas the adiabatic diameter was less than 10 μm. Such large droplets may be explained by a few droplets experiencing the highest observed supersaturations for a certain time. Consequences for aerosol activation and droplet size dispersion in a highly fluctuating supersaturation field are briefly discussed.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Holger Siebert, siebert@tropos.de

Abstract

On time scales that are long compared to the phase relaxation time, a quasi-steady supersaturation sqs is expected to exist in clouds. On shorter time scales, however, turbulent fluctuations of temperature and water vapor concentration should generate fluctuations in supersaturation. The variability of temperature, water vapor, and supersaturation has been measured in situ with submeter resolution in warm, continental, shallow cumulus clouds. Several cumuli with horizontal extents of order 100 m were sampled during their first appearance and development to depths of ~100 m in a growing boundary layer. Fluctuations of the saturation ratio are observed to be approximately normally distributed with standard deviations on the order of 1%. This variability is almost one order of magnitude larger than sqs calculated using simultaneous measurements of the vertical velocity component and the droplet size distribution. It is argued that, depending on the ratio of the phase relaxation and the turbulent mixing time, substantial fluctuations in the supersaturation field can exist on small spatial scales, centered on sqs for the mean state. The observations also suggest that, on larger scales, fluctuations of the supersaturation field are damped by cloud droplet growth. Droplets with diameters of up to 20 μm were observed in the shallow cumulus clouds, whereas the adiabatic diameter was less than 10 μm. Such large droplets may be explained by a few droplets experiencing the highest observed supersaturations for a certain time. Consequences for aerosol activation and droplet size dispersion in a highly fluctuating supersaturation field are briefly discussed.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Holger Siebert, siebert@tropos.de

1. Introduction

The difference between actual vapor pressure e and saturation vapor pressure (over a planar surface of water), usually expressed as a dimensionless supersaturation , can be considered one of the fundamental variables of cloud physics. It drives cloud droplet activation and growth but also responds to those processes through water mass conservation (Kostinski 2009; Rogers and Yau 1989). Yet s is almost never measured directly, in part because of the very small excess vapor pressures expected in clouds [of order 0.1%; e.g., Gerber (1991); Cooper (1989)]. Rather, supersaturation is more often estimated from microphysical properties and the vertical velocity for convective clouds (Politovich and Cooper 1988) or inferred from aerosol measurements (e.g., Ditas et al. 2012). The aerosol method is enabled by the sensitivity of cloud droplet activation to the mass and composition of cloud condensation nuclei (Gerber 1980). Both approaches, however, neglect the variability that one would expect must exist for a scalar quantity in a turbulent flow. The scalars contributing to the supersaturation field are water vapor pressure and temperature, the latter through the approximately exponential dependence of on temperature T. How large are supersaturation fluctuations, and to what extent do they depend on turbulent fluctuations of scalar fields e and T, fluctuations of microphysical variables like droplet number density n and mean diameter , or fluctuations of the vertical velocity component w? Motivated by these questions, we explore these fluctuations using high-resolution measurements made in small cumulus clouds just in their initial formation stage.

a. Quasi-steady supersaturation: Updraft and microphysics

The mean supersaturation in a cloud results from a quasi-steady balance between a source, typically taken as the result of an updraft within the cloud, and the sink resulting from droplet growth by vapor condensation (e.g., Cooper 1989; Rogers and Yau 1989; Korolev and Mazin 2003; Kostinski 2009). For sake of completeness, we give a brief summary here, and for convenience we adapt the notation of Cooper (1989). The supersaturation evolution equation is
e1
where w is the vertical velocity component, is the droplet number density, and is the mean droplet radius. The coefficients and are defined by Cooper (1989) and are sufficiently weak functions of p and T that we can treat them as constants for the sake of this study. Shortly after droplet activation, the droplet radius varies slowly, and it is possible to consider a quasi-steady balance with
e2
The quantity is often called the integral radius, and it is inversely proportional to the exponential relaxation time, called the phase relaxation time:
e3
For fluctuations in w, , or on time scales that are long compared to , the supersaturation is able to adjust to the quasi-steady value given by Eq. (2). On shorter time scales, the supersaturation is not able to achieve steady state and is therefore always in a transient response. Entrainment or other processes leading to reduction in and can lead to large and therefore efficient growth of the remaining droplets even if secondary activation is taken into account (Yang et al. 2016). At progressively smaller spatial and temporal scales, it is increasingly likely to find isolated regions with reduced droplet number density and therefore larger phase relaxation time and higher quasi-steady supersaturation. But as phase relaxation time becomes large, it eventually will exceed the eddy time scale for the given region, and therefore quasi-steady values will not be able to be reached. Basically, quasi-steady supersaturation is expected for the scales at which the phase relaxation time is of the same order or smaller than the eddy turnover time, usually on the order of centimeters to meters for typical cumulus conditions (Korolev and Mazin 2003; Lehmann et al. 2009).

b. Slow and fast contributions to supersaturation

In addition to the quasi-steady supersaturation resulting from dynamic equilibrium between updraft and droplet growth, supersaturation will also vary as a result of turbulent fluctuations in e and T caused by mixing or entrainment. Droplet activation and growth should tend to damp resulting supersaturation fluctuations on times scales that are long compared to , but much of the inertial subrange consists of eddies with shorter coherence times. Therefore, the supersaturation can be thought of as a superposition of slow, quasi-steady background and fast fluctuations due to turbulence. With this in mind, we rewrite the supersaturation evolution equation, Eq. (1) to include a rate of change due to turbulence. The fluctuation term can be considered to be of order [similar to the approach taken in a stochastic Lagrangian model (Pope 2000), although that formalism is not adopted here]:
e4
The physical interpretation becomes clearer when nondimensionalized with , yielding
e5
Then recognizing that the first term on the right-hand side is and that the last term can be defined as a Damköhler number , we have
e6

In the limit of , corresponding to slow turbulent fluctuations compared to the phase relaxation time, the steady supersaturation is . In the limit , corresponding to fast turbulent fluctuations, the steady supersaturation is , where it should be kept in mind that the term can be positive or negative. Part of the purpose of this work is to determine how the observed variability in supersaturation compares to that predicted from microphysical variability and whether any contribution is observed from additional, turbulent sources.

c. Supersaturation fluctuations due to turbulence

Even without direct knowledge of their origin (e.g., whether resulting from internal variability or from entrainment of environmental air), supersaturation is influenced by variations in both e and T. Here we review how those fluctuations contribute, laying the groundwork for identifying which are of most relevance in observed clouds. The approach is standard Reynolds decomposition, with the saturation ratio assumed to consist of steady and fluctuating components, . The saturation ratio is defined as , where is the saturation vapor pressure of water, obtained from the Clausius–Clapeyron equation. Following Kulmala et al. (1997), vapor pressure and temperature are decomposed into mean and fluctuating components, and , and second-order terms are retained and ensemble averaging is performed. Defining with R being the gas constant of water vapor, the squared mean of S can be written as follows:
e7
The overbrace in Eq. (7) is for convenience in the discussion later in this paper. The variance in saturation ratio is obtained using . The term is obtained using similar logic (i.e., retaining second-order terms and using ensemble averaging rules):
e8
It is worth noting that the variance of vapor pressure and temperature as well as the covariance of the two all contribute to the variance of saturation ratio. Again, for ease in the later discussion, these individual terms are identified with underbraces in Eq. (8).

We now proceed to the observations to identify the magnitude of as well as the expected variations in due to observed variability of w, , and . First, in section 2 we describe the instrumentation and sampling methods that allow for direct measurement of thermodynamic, turbulence, and microphysical properties with high resolution, and then in section 3 we present results from a specially chosen set of cloud measurements that allow for our picture of supersaturation variability to be evaluated. Finally, in section 4 we discuss and summarize the main results.

2. Experiment details

a. High-resolution measurements with ACTOS

Several flights in the cloudy and cloud-free boundary layer up to 2500 m were performed during the European Integrated Project on Aerosol, Cloud, Climate, and Air Quality Interactions (EUCAARI) in the Netherlands close to Utrecht in May 2008 (Kulmala et al. 2011). The flights were made with the helicopter-borne payload Airborne Cloud Turbulence Observation System (ACTOS; Siebert et al. 2006). ACTOS is a 200-kg autonomous measurement payload that hangs 150 m below the helicopter. The combination of the long rope with a true airspeed of 20 m s−1 ensures measurements are undisturbed from the helicopter downwash. The payload is equipped with several sensors for measuring standard meteorological, cloud, and aerosol properties. The combination of high temporal sensor resolution (at least 20 Hz) and the low true airspeed yields high spatial sensor resolution, making this setup suitable for observations of the finescale structure of small clouds. In the following, we provide a short introduction of the sensors that were used in this analysis.

The fast humidity measurements on ACTOS are mainly based on an open-path infrared absorption hygrometer (LI-7500 manufactured by LI-COR, Inc., Nebraska, United States), which is frequently compared to a small dewpoint mirror (TP3-S hygrometer manufactured by Meteolabor AG, Switzerland) to ensure long-term stability. This measurement approach is described in more detail in the following subsection. In-cloud temperature has been measured with the ultrafast thermometer (UFT; Haman et al. 1997). The sensing element—a small resistance wire with a diameter of 2.5 μm—is protected against impacting droplets with a shielding rod. In this analysis, absolute humidity a and temperature T both have a time resolution of 10 ms, although the LI-7500 has a low-pass filter at around 20 Hz.

Cloud droplets have been measured with a phase Doppler interferometer (PDI; Chuang et al. 2008), from which the droplet size distribution, mean droplet diameter , and droplet number density are derived. Because and require an adequate number of droplets to minimize counting uncertainties, these properties are calculated with a frequency of 1 Hz. Wind vector components are measured with an ultrasonic anemometer [Solent HS manufactured by Gill Lymington, United Kingdom; see Siebert and Muschinski (2001)] corrected for attitude and payload motion. A more detailed description of the ACTOS sensor equipment can be found in Siebert et al. (2003, 2006).

b. Fast humidity measurements in clouds with the LI-7500

The measurement principle of the LI-7500 is based on light absorption of infrared light with a wavelength of 2.59 μm for water vapor measurements and a reference line at 3.95 μm. The open-path absorption volume is 12.5 cm long, and the device was operated with a bandwidth of 20 Hz; in this study, only the analog outputs are available. That is, no housekeeping data have been stored. There are two different sources of performance degradation when operating the system in the presence of cloud droplets: (i) droplets on the optical windows might accumulate, resulting in a water film that finally can block the windows; and (ii) the droplets within the absorption volume can absorb and scatter the infrared light.

The first problem can be minimized by orienting the sensor head perpendicular to the mean-flow direction, as suggested in the instruments manual (LI-COR 2015). During EUCAARI, the LI-7500 was mounted horizontally and perpendicular to the mean ACTOS flight direction, which is typically limited to be within ±5° for the angles of attack and sideslip. That, together with the low observed liquid water contents and absence of precipitation, minimizes the risk of building a water film on the optical windows.

About the second problem, this is a known effect for large particles, such as raindrops or snowflakes. The instrument manual states (LI-COR 2015, p. 2-7), “The reason is that the droplets and flakes are moving, and if one is in the path for a sample measurement, but out of the path for a reference measurement (or vice versa), it will influence the reading.” During our measurements, no precipitation particles were present, but, because we wish to investigate small-amplitude variability, the possible influence of cloud droplets with size below d ≈ 20 μm might influence the signal.

We can make an estimate for fluctuations in the water vapor signal caused by the presence of water droplets by considering absorption by cloud droplets at the 2.59-μm signal and 3.95-μm reference wavelengths. Scattering of infrared radiation is also possible, but the droplets are sufficiently large that most light will be forward scattered into the detector; what light is lost because of scattering only decreases the signal-to-noise ratio and does not bias the differential measurement, so we treat only absorption losses here. The absorption coefficient for liquid water is of order 104 m−1 at both wavelengths (Hale and Querry 1973; Wieliczka et al. 1989), so a liquid water layer of order 100 μm thick would lead to significant reduction in the signal. If liquid water is present in both paths, it leads to a reduction in the signal-to-noise ratio, and the LI-COR instrument has an automated method for increasing the gain to account for this. In a turbulent cloud, especially where entrainment of external air is taking place, it is possible to have adjacent patches of clear and cloudy air even on the centimeter scale (Beals et al. 2015). As an extreme estimate, we consider one beam containing cloud with the peak liquid water content and the other beam being empty (the LI-COR beam diameter is 8 mm and beam path is 12.5 cm). A liquid water content of 0.1 g m−3 corresponds to a liquid-to-air volume fraction of 10−7, so the LI-COR pathlength corresponds to a liquid water layer depth of 0.01 μm, or 10−4 times the absorption penetration depth. Fluctuations in vapor pressure resulting from fluctuations in differential absorption can be estimated from a simplified version of the LI-COR calibration equation:
e9
where α is the absorptance and A = 5.42 mol m−3 and B = 4013.0 kPa mol m−3 are typical values for calibration constants [see section 4 in LI-COR (2015)]. Using the approximation , the variability in vapor pressure can be written as
e10
For T = 273 K and a corresponding typical value of absorptance for 100% relative humidity , we obtain δe ≈ 2 Pa for δα = 10−4. For the same temperature, es= 611 Pa and the supersaturation variability is therefore . Again, this should be considered an upper bound because it is based on the limiting scenario of cloud with peak liquid water content in one path and no cloud in the other path within the LI-COR instrument. We expect, therefore, that variability above the level is unlikely to be attributed to the presence of cloud droplets.

Finally, we note that spikes or enhanced variability due to different numbers of particles in the signal and reference paths would affect the power spectral density (here called “spectrum”) of the measured saturation ratio. For atmospheric conditions, one would expect that within a certain frequency range (classical inertial subrange scaling; e.g., Wyngaard 2010). Even a small number of distinct spikes would violate this scaling behavior, and a clean spectrum is a reliable support for spike-free data. In Fig. 1, we show the spectra as measured at ground level and for the two entire flight legs that included clouds (height of 1130 and 1180 m; see section 3 for more details). For a power spectral density, the integral over all frequencies results in the total variance of the signal. All spectra show inertial subrange scaling for frequencies up to about 10 Hz. For f > 10 Hz the spectra drop off because of a low-pass filter and for f < 1 Hz the upper spectrum tends to flatten because larger scales are not present close to ground (the LI-7500 was about 1 m above the ground during takeoff preparation). From the upper spectrum in Fig. 1, we estimate a spectral noise floor of S ≈ 10−7 Hz−1, which yields a standard deviation because of random noise of with the Nyquist frequency fNy = 50 Hz. In particular for higher frequencies (f > 1 Hz or so), the two spectra from the cloud data compare well with the spectrum observed under drop-free conditions. As suggested by the above estimates for differential absorption, fluctuations in number of cloud droplets between the signal and reference beams do not make a strong contribution to the power spectrum. Furthermore, we note that, even if on small scales (e.g., below 1 m) droplets influence the LI-7500 readings, the general data interpretation would not change because the variance is dominated by the larger scales because of the −5/3 slope of the spectrum. Finally, we will see in section 3 that a high local correlation of the temperature and moisture field is observed, even in the presence of cloud droplets; this further supports the conclusion that variability of the observed water vapor field is real and not caused by instrumental artifacts due to clouds. It should be kept in mind, however, that in cloud conditions where liquid water content is above 1 g m−3 the cloud droplet absorption contribution could become significant.

Fig. 1.
Fig. 1.

Power spectral densities of the saturation ratio S as a function of frequency f as measured for the ground-based observations and the two entire legs at 1130 and 1180 m. The black curves are averages over logarithmic equidistant bins; the blue curve provides a slope as predicted for classical inertial subrange scaling. The spectrum for the ground-based measurements (GND) has the original y scale; the other two spectra are vertically shifted by 103 and 106, respectively.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

3. Data analysis: Measurements in newly formed cumulus clouds

We present observations based on one flight (20 May 2008), which started under cloud-free conditions, but about 30 min after takeoff the onset of shallow cumulus convection was observed at a height of about 1100 m, allowing for several clouds to be sampled during the very early stages of their lives.

Before analyzing the cloud data, a 5-min-long record measured at ground level before takeoff is analyzed in order to evaluate the terms contributing to , as defined by Eq. (8). The results are shown in Fig. 2. The term labeled with in Eq. (7) was observed to be negligible and with and . The covariance is positive with a cross-correlation coefficient . The three terms in Eq. (8) are , , and , yielding a predicted . For the observed conditions, the fluctuations in the moisture field dominate over the fluctuations in the temperature field in contributing to the saturation ratio fluctuations. The cross-correlation and water vapor terms are of similar magnitudes and opposite signs, so a relatively small is in fact still significant. The predicted value of agrees well with the direct calculation of , and we are confident that Eq. (8) not only provides a reliable estimate of but also allows the relative contributions of , , and to be evaluated.

Fig. 2.
Fig. 2.

Fluctuations of (top left) water vapor pressure and temperature and (bottom left) saturation ratio as observed from a 5-min-long record at ground level shortly before takeoff. Horizontal lines indicate the rms values. (right) The corresponding PDFs are displayed. A Gaussian fit and the standard deviation σ of each parameter are given.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

The vertical extent of the observed clouds rarely exceeded 100 m, and the horizontal extent of coherent cloud structures was of the same size. Figure 3 gives an impression of the emerging cloud field and sparse coverage. The analyzed cloud data were sampled at two different heights: the first record was sampled at 1130 m, and the second one at 1180 m was observed about 2 min later. Figure 4 presents an overview of the stratification in terms of vertical profiles taken just after takeoff (ascent) and before landing (descent). During the ascent a well-mixed layer with height-constant values up to about 750 m (marked with a horizontal red line) is visible for all three variables: potential temperature , water vapor mixing ratio , and aerosol particle number concentration . Based on temperature (T = 12.2°C) and dewpoint (Td = 4°C) measured shortly before takeoff, the lifting condensation level is estimated to be , and, consistent with that number, no clouds were present during the ascent. About 20 min after takeoff, the first cumuli developed, which were sampled with ACTOS at a height of 1130 m. The age of these small cumuli is estimated to be about 5–10 min. The descent profile was taken 65 min after the ascent, and during this time it can be seen from and , and to some extent from , that the well-mixed layer has developed to approximately 1200 m. The strong fluctuations in and between 700 and 1200 m confirm that active mixing between the lower- and upper-level values is still taking place. This convective environment is the setting for the measurements made just during the birth stage of the small cumulus clouds.

Fig. 3.
Fig. 3.

A photograph of the freshly evolving cumulus field taken from the helicopter (courtesy of B. Wehner).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

Fig. 4.
Fig. 4.

Vertical profiles of (a) potential temperature , (b) water vapor mixing ratio , and (c) aerosol particle number concentration . The red curve displays the ascent at the beginning of the flight, whereas the black curve displays the final descent. The two flight levels with cloud observations in z = 1130 and 1180 m are marked with blue horizontal lines. A red line at 750 m indicates the top of the mixed layer for the ascent.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

Figure 5 shows an overview of the first cloud leg with a flight path of about 12 km. Figure 5a shows the water vapor pressure e, temperature T (both on the left scale), and calculated saturation ratio (right scale). Here, the empirical formula with T in degrees Celsius and in hectopascals was applied.The updrafts with can be clearly distinguished from the residual layer with S ≈ 0.5–0.6. The reason why S does not reach 1 (i.e., RH = 100%) for clouds is based on small absolute measurement errors for T and e. The longitudinal (along-flight direction) separation of 0.5 m between the two sensor locations and a constant time delay of 0.2 s for the LI-7500 analog output have been considered in the calculation of S. The lateral separation between the UFT and the LI-COR is about 10 cm, much smaller than the observed correlation length between T and a, which is a few tens of meters even in comparably uniform cloud regions. On the scale of updrafts, T and e are negatively correlated, with e being higher in updrafts but T decreasing inside the updrafts compared to the environment. Updraft regions with measured result in cumulus clouds with droplet diameters d typically ranging from 5 to 12 μm, with mean diameter (see Fig. 5b) and droplet number concentrations up to 1000 cm−3 (see Fig. 5c) but highly variable in time and space due to strong dilution. Phase relaxation times calculated from observed and in the least-diluted cloud regions are approximately 2 s. In later parts of this paper, therefore, we define the threshold to be the temporal boundary between slow and fast variability for turbulence and microphysics within these clouds. Following the discussion in section 1b, fluctuations in or slow compared to are expected to yield an approach to microphysical steady state and determine the quasi-steady supersaturation . Fluctuations faster than are expected to contribute to fluctuating supersaturation , as discussed in section 1c.

Fig. 5.
Fig. 5.

Time series of the cloud field observations measured at a height of 1130 m. The entire record covers a flight path of about 12 km. (a) The water vapor pressure e (green line), temperature T (red line), and saturation ratio (blue line). Also shown are the cloud parameters in terms of (b) droplet diameter d (blue points) and mean droplet diameter (orange circles) and (c) droplet number density (red circles).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

Figure 6 shows the droplet size distribution for the full flight path at 1130 m. The main mode at 7 μm is evident, but most surprising is the long, large-droplet tail, extending to diameters of 20–30 μm (we note that these large droplets have been cut off in Fig. 5b because they are so sparse). What is the origin of cloud droplets with d ≈ 25 μm in newly forming cumulus clouds only 100–200 m thick? We can be confident that these are not the result of measurement artifacts due to coincidence counts because the PDI method determines diameter not from the magnitude of scattered light intensity but from the phase difference between heterodyne Doppler bursts measured at three detectors [see chapter 5 in Wendisch and Brenguier (2013), as well as Chuang et al. (2008).]. The possible origin of these large droplets is further discussed in section 4.

Fig. 6.
Fig. 6.

PDF of droplet diameter d. The PDF is calculated from 51 600 droplets during the two legs at 1130 and 1180 m.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

In Fig. 7, an enlarged portion of the record in Fig. 5 is shown, corresponding to ≈1 km of flight path over a single updraft region with a diameter of 500 m and its immediate environment. In Fig. 7a the fluctuations and are shown together with . Again, droplets are observed only for (cf. Fig. 7b for the droplet data). One comparably homogenous cloud region over 90 m of flight path is labeled with a light gray box (on the right). This region is characterized by almost constant but a clear maximum of Nd ≈ 800 cm−3 located in the middle of the cloud region, decreasing to Nd ≈ 350 cm−3 at cloud edge. This behavior is typical of extremely inhomogeneous mixing (Lehmann et al. 2009). In Fig. 7c, the fluctuations of the vertical wind velocity in the range of ±1.5 m s−1 are shown. In the gray box indicating the cloud, is positive with values around zero at the cloud edge and a maximum of 1 m s−1 at the same location as .

Fig. 7.
Fig. 7.

An enlarged portion of the record presented in Fig. 5 covering a flight path of 1 km. (a) Fluctuations of water vapor and temperature (left axis) and saturation ratio S (right axis). (b) The droplet diameter d, mean droplet diameter , and droplet number density ; and are mean values averaged over 1-s periods. (c) The vertical velocity w and the quasi-steady supersaturation . For small (e.g., close to zero), the values are out of scale.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

The quasi-steady supersaturation with the two constants and is displayed in Fig. 7c with blue circles. The values are about a tenth of a percent, although for small the values can be extremely large and are therefore outside of the scale. These values are calculated at the 1-Hz resolution of the microphysical measurements in order to see the full range of variability. This is also close to the discussed earlier in this section, and we note that the smoothing resulting from a 2-s average does not significantly alter the resulting in the center of the cloud.

Next, the turbulent fluctuations that are fast compared to are characterized (i.e., those that do not contribute to quasi-steady properties). From the two flight legs at 1130 and 1180 m, several subrecords of 100–200-m length with statistically homogeneous conditions in S were selected and combined into two records (Fig. 8). Each subrecord was high-pass filtered applying a third-order Savitzky–Golay filter with 201 filter elements (filter period of 2 s, corresponding to ) before concatenation. The boundaries of the subrecords are displayed at the bottom of Fig. 8c. For (Fig. 8a), (Fig. 8b), and (Fig. 8c), the probability density functions (PDFs) are calculated and shown to the right of the time series, and the standard deviations σ for each parameter are given. All PDFs can be roughly approximated with a Gaussian fit, although some deviations are observed presumably because time series from cumulus clouds are never strictly stationary or homogeneous. For , we observe 0.012 for the lower leg and 0.005 for the upper leg, both greater in magnitude than the values in Fig. 7c. The reduced variability of in the upper leg is likely due to the closer location to the damping inversion. For the lower leg, the standard deviation due to the fluctuating character of T and e is about 10 times greater than the estimated values of , and more than 30% of the data (assuming a Gaussian PDF) show .

Fig. 8.
Fig. 8.

(left) Cloud data concatenated from quasi-homogeneous subrecords as observed during two flight legs at 1130 (blue curves) and 1180 m (orange curves). The ranges of each subrecord are marked on the lower x axis. Each subrecord was high-pass filtered by applying a third-order Savitzky–Golay filter with 201 filter elements (filter period of 2 s) before concatenation, resulting in fluctuations of the parameters. (right) The corresponding PDFs are displayed including a Gaussian fit and the standard deviation σ.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0115.1

For both heights, the two records in Fig. 8 show almost identical mean values from ° to , , and (see Table 1 for an overview.) The fluctuations of T and e are used to estimate again the three terms A, B, and C and the factor ζ in Eq. (8). The height in meters is used here as a subscript to identify the two records. The factor is about 7 times higher compared to , yielding a predicted for the observations in 1130 m for the higher level in 1180 m. That is, the fluctuations in the saturation field at 1130 m are still about 70% of the ground value, whereas 50 m higher in the cloud layer the values decreased to about 30% of the ground value. At both heights of the cloud observations the temperature and moisture field are positively correlated with for the lower leg and for the upper leg. Similar to the ground-based observations, the B term dominates in Eq. (8), whereas the A and C terms almost compensate each other for the two cloud legs. That is, the variability of the moisture field is the main contributor to the strong fluctuations of the saturation field.

Table 1.

Overview of all estimated parameters involved in Eq. (8) for the ground-based observations (GND) and the two cloud legs when only the statistically homogeneous parts are considered (cf. Fig. 8).

Table 1.

4. Discussion and summary

This paper opened by posing the following question: how large are supersaturation fluctuations, and to what extent do they depend on turbulent fluctuations of scalar fields e and T, fluctuations of microphysical variables like droplet number density and mean diameter , or fluctuations of the vertical velocity component w? The cloud birth case considered here has provided a unique opportunity to observe supersaturation fluctuations and droplet size distributions in shallow cumulus clouds during the initial, active growth stage, with low liquid water content and no precipitation. The fortuitous development of these clouds during a research flight with ACTOS allowed high-resolution measurements to be obtained. Furthermore, the low liquid water contents make possible the use of the LI-COR for evaluating supersaturation fluctuations with minimal noise from cloud droplets. The results show that the quasi-steady supersaturation in these small cumulus clouds (cf. Fig. 7) varies from less than 0.1% to approximately 0.4%, where is derived from fluctuations with time scales equal to and slower than 1 Hz. The turbulent fluctuations , derived from high-pass-filtered e and T data with a filter scale of , are found to vary from 0.7% to 1.3% (cf. Fig. 8). So indeed there is evidence that turbulence generates large fluctuations in supersaturation at all scales, both slow, quasi-steady variability, and more rapid variations on time scales shorter than the undiluted cloud phase relaxation time . This is consistent with the conceptual picture used to develop, for example, Eq. (6). Taken together, the full range of observed supersaturation variability is significantly larger than is often thought to exist in clouds: for example, typical calculations are based on mean values of s, and fluctuations are not considered. Further perspective can be gained by considering the fluctuations in supersaturation observed during the same zoomed section of the 1130-m flight leg shown in Fig. 7, but in a relatively stationary part of the time series just outside the cloud (shown by the left gray box). In the left gray box, where no droplets are present, , whereas at the same flight level just 10 s later, a value of is observed inside the cloud. Thus, we see that the cloud droplets do act as a buffer, reducing the magnitude of the supersaturation fluctuations through growth or evaporation, even if not fully to the level expected by the quasi-steady prediction.

The observation of relatively large supersaturation fluctuations has interesting implications for aerosol activation and droplet growth. We note, for example, that the observed magnitude of inside the shallow cumulus clouds is large in comparison to many of the physical and chemical effects given careful consideration in the theory of aerosol activation and CCN efficiency (McFiggans et al. 2006). This is not to imply any relative level of significance but simply that the measurements suggest that aerosol activation should be considered in the context of a supersaturation field that is highly variable in time and space. For example, the equilibrium assumption for aerosol particle activation is not always satisfied (Chuang et al. 1997), and possible biases would be expected to be more important at higher frequencies.

The equations for supersaturation evolution discussed in section 1b were expressed in terms of the dimensionless Damköhler number , so we return briefly to that context now. The turbulence correlation time is obtained from the autocorrelation function for the vertical velocity component and yields for the full flight leg at 1130 m (note that if this is calculated only for the “uniform” regions within a given cloud, this number is reduced by about a factor of 10, but we here consider the full field in order to capture the large-eddy structure). The phase relaxation time characterizing the least diluted cloud regions is , corresponding to . This suggests that these high-Nd portions of the cloud are in a “fast microphysics” regime and therefore would be expected to have damped supersaturation fluctuations because of the rapid buffering due to droplet growth and evaporation. The distribution of phase relaxation times, however, is extremely skewed in the positive tail: the 25th-percentile is representative of the thickest cloud regions, but the 50th and 75th percentiles are and . The mean value is strongly dominated by a small number of extremely large values that result from very small in highly diluted cloud regions. As pointed out in section 3, these young, actively growing clouds displayed strongly inhomogeneous mixing, with nearly constant and strongly fluctuating , and this in turn is directly responsible for the large, positive excursions in . The corresponding 25th, 50th, and 75th percentiles for Damköhler number are , , and (the average is because of the strong positive skewness). In nearly half of the cloud samples, therefore, the Damköhler number is of order unity or less. Therefore, in contrast to the cloud cores, the diluted cloud regions reside in the “slow microphysics” regime, in which supersaturation fluctuations caused by turbulent mixing are weakly buffered by droplet growth over the duration of their typical correlation time, and therefore droplets exposed to those fluctuations experience a wide variety of growth conditions [consistent with the findings of Chandrakar et al. (2016)].

The implications for droplet growth are especially compelling in light of the surprisingly broad cloud droplet size distribution observed in these shallow cumulus clouds (cf. Fig. 6). A wide range of research considering various mechanisms has considered the magnitudes of supersaturation expected for warm clouds (Politovich and Cooper 1988; Cooper 1989; Gerber 1991; Shaw 2000; Jeffery et al. 2007; Ditas et al. 2012; Pinsky et al. 2013; Korolev et al. 2013; Devenish et al. 2016; Chandrakar et al. 2016). To some extent, observations of broad droplet size distributions have been ascribed to instrument effects in the attempt to reconcile them with typically narrow modeled distributions (Devenish et al. 2012). However, as discussed already, the distribution width observed here cannot be explained in a straightforward way from instrumental effects. Can droplets as large as d = 20–25 μm be explained for shallow ~100-m-thick clouds? From the temperature profile in Fig. 4, we estimate T = 2°C at cloud base (zlcl=1037 m), which gives an adiabatic LWCa ≈ 0.16 g m−3 at a height 100 m above cloud base. Because , with being the density of water and Nd ≈ 1000 cm−3, we obtain a mean adiabatic droplet diameter of . This droplet diameter compares very well with the peak diameter of the droplet size distribution in Fig. 6. The challenge is therefore to interpret the large droplet tail.

Droplet growth by condensation is approximately described by (Rogers and Yau 1989): , with r being the droplet radius and and being thermodynamic terms describing the heat conduction and water vapor diffusion. For p = 900 hPa and T = 1°C, we find (Rogers and Yau 1989). If we now consider the supersaturation necessary for a droplet to grow to r = 10 μm at a height of about 100 m above cloud base and a mean updraft velocity of w = 1 m s−1 (t = 100 s), we find . This value is much larger than the calculated quasi-steady value but is, in fact, within the range of observed for the statistically homogeneous cloud records sampled at 1130 m.

These observations and findings leave us with many challenges for future work. It remains to be seen, for example, whether a fluctuating supersaturation field such as would be given by Eq. (6) can explain a size distribution tail such as that observed in Fig. 6. Theoretical work on condensation droplet growth in a turbulent environment has argued for both minor (Bartlett and Jonas 1972; Vaillancourt et al. 2002) and significant effects (Celani et al. 2005; Lanotte et al. 2009). The recent work of Sardina et al. (2015) and Chandrakar et al. (2016) seems mechanistically consistent with the current observations, however, and therefore provides some support for the Lagrangian stochastic approach. Certainly a better understanding of Lagrangian statistics within clouds would be helpful in this regard, because they determine the typical correlation time for fluctuations in scalar fields and therefore influence the extent to which size distributions can be broadened. Further measurements of supersaturation variability in clouds, under a wider variety of conditions, are also required in order to both motivate and constrain theory.

Acknowledgments

The measurement campaign was conducted within the European Integrated Project on Aerosol Cloud Climate and Air Quality Interactions (EUCAARI), coordinated by the University of Helsinki, Finland. Many thanks to our pilots Alwin Vollmer, Jan Nienhaus, and Karel Wiegman for the helicopter flights. We thank Dieter Schell and Christoph Klaus from the enviscope GmbH for their technical support during the campaign and KNMI for hosting the campaign. We thank Dennis Niedermeier for helpful discussions about the LI-COR. Participation by RS was supported by the U.S. National Science Foundation. We thank the three anonymous reviewers for their valuable suggestions which significantly improved the manuscript.

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