Effects of the Large-Scale Circulation on Temperature and Water Vapor Distributions in the Π Chamber

. Microphysical processes are important for the development of clouds and thus Earth’s climate. For example, turbulent ﬂuctuations in the water vapor concentration, r , and temperature, T , cause ﬂuctuations in the saturation ratio, S . Because S is the driving factor in the condensational growth of droplets, ﬂuctuations may broaden the cloud droplet size distribution due to individual droplets experiencing different growth rates. The small scale turbulent ﬂuctuations in the atmosphere that are relevant 5 to cloud droplets are difﬁcult to quantify through ﬁeld measurements. We investigate these processes in the laboratory, using Michigan Tech’s Π Chamber. The Π Chamber utilizes Rayleigh-Bénard convection (RBC) to create the turbulent conditions inherent in clouds. In RBC it is common for a large scale circulation (LSC) to form. As a consequence of the LSC, the temperature ﬁeld of the chamber is not spatially uniform. In this paper, we characterize the LSC in the Π chamber and show how it affects the shape of the distributions of r , T and S . The LSC was found to follow a single roll with an updraft and 10 downdraft along opposing walls of the chamber. Near the updraft (downdraft), the distributions of T and r were positively (negatively) skewed. S consistently had a negatively skewed distribution, with the downdraft being the most negative. show that the effects of the circulation are felt well into the bulk of the chamber, though, as expected, the amplitude of the circulation decreases towards the center. Our results also indicate that the circulation in the Π Chamber has a preferred orientation with pronounced azimuthal oscillations. We will now address the impact of the LSC on the temperature distributions in the bulk of the chamber, using the RTDs in the bulk ring.


Introduction
The effects that clouds have on Earth's climate system are quite sensitive to the details of processes that occur on scales much 15 smaller than the cloud as a whole. For example, two clouds with the same amount of liquid water can behave differently depending upon their droplet size distributions. If the liquid water content (LWC) is distributed over a large number of small droplets the cloud will be quite reflective and unlikely to precipitate. Conversely, a cloud with the the same amount of liquid water distributed over fewer droplets will be less reflective and more likely to precipitate (Twomey, 1977;Albrecht, 1989;Pincus and Baker, 1994). 20 The two principal processes that shape the cloud droplet size distribution are condensation/evaporation and collisioncoalescence. Condensation is driven by gradients in the saturation ratio, S, between the environment and the surface of the droplets and can result in a rapid increase in size for small droplets. However, because dr dt ∝ S r , where r is the radius of the The chamber operates under conditions of turbulent Rayleigh-Bénard convection (RBC), where the lower surface of the cell is set to a higher temperature than the upper surface. These conditions cause turbulent mixing due to the buoyancy difference between warm and cool air. As the air mixes it results in fluctuations in the temperature; the nature of these fluctuations in the scalar field have been studied intensely (see e.g. Ahlers et al., 2009;Chillà and Schumacher, 2012). As examples, fluctuations in temperature have been shown to depend on the geometry of the convection cell, the intensity of turbulence and the working 45 fluid.
The turbulent intensity and fluid properties are typically described using the Rayleigh number (Ra = αgH 3 ∆T κν ) and the Prandtl number (P r = ν κ ) respectively, where g is the acceleration due to gravity, ∆T is the temperature difference between the top and bottom plates separated by distance H, α is the coefficient of thermal expansion of the fluid, κ is its thermal diffusivity, and ν is its kinematic viscosity. For a gradient in both temperature and water vapor, the Rayleigh number becomes 50 (Niedermeier et al., 2018) where ≡ m d mv −1, m d is the molecular mass of dry air, m v is the molecular mass of water, and r is the vapor mixing ratio. Note that for the range of conditions explored in this paper, Ra is dominated by the first term in Eq. 1. Studies of the temperature profile on the vertical axis of cylindrical cells show a well mixed fluid with little gradient outside the boundary layer (Belmonte 55 and Libchaber, 1996;Sakievich et al., 2016;Xie et al., 2019). There are fewer studies of the off-center bulk temperature profiles (Liu and Ecke, 2011;He et al., 2019).
In turbulent Rayleigh-Bénard convection a structure forms in the fluid flow, referred to as the "mean wind of turbulence" or large-scale circulation (LSC). It is a mean, background flow within the overall turbulent motion in the chamber. For cells that have an aspect ratio (Γ ≡ D/H, where D is the cell diameter) near one or two, the LSC usually takes the form of a single 60 roll which spans the diameter of the chamber (Xia et al., 2008). This single roll has an updraft on one side of the cell that has a positive mean vertical velocity and a higher temperature than the center of the chamber. Along the opposite side of the cell the fluid typically has a negative vertical velocity and lower temperatures. A visualization of the circulation is shown on the left side of Fig 1. For cells with Γ 4, multiple convective rolls become the dominant circulation mode (Xia et al., 2008). We anticipate the circulation in the Π chamber will follow a single roll due to the chamber having Γ = 2. The updraft-downdraft associated with the large scale circulation typically adopts a specific orientation within the cell, but also has several oscillatory modes about that mean position. One of the primary oscillations is azimuthal, about a vertical axis that runs through the center of the cell (Brown and Ahlers, 2007b). Often the azimuthal oscillation at the top and bottom of the chamber are out of phase. The resulting oscillation is called the torsional mode (Funfschilling et al., 2008). In addition the LSC has been shown to oscillate side to side in what has been referred to as the sloshing mode (Brown and Ahlers, 2009). In cells 70 with very high symmetry, the LSC can spontaneously cease and reorient to a different angular location (Brown et al., 2005;Ahlers, 2006, 2007a;Xi et al., 2006). An asymmetry, such as tilting the cell, can fix the orientation of the LSC (Xi et al., 2009).
Our motivation for studies in the Π chamber is to understand cloud microphysical processes in the atmosphere; one emphasis is determining how fluctuations in the saturation ratio affect the drop size distribution and aerosol activation. For example, in 75 in S on droplet growth. We have recently refined the treatment of fluctuations in temperature and water vapor concentration using a one dimensional turbulence model (ODT) (Chandrakar et al., 2020a), which incorporates vertical variations. It should also be noted that a mean field approach captures many aspects of the microphysical processes in the chamber (Krueger, 2020).
While these models have provided valuable insights into processes in the chamber, the assumption of no spatial variability or of variability in only the vertical direction comes into question in the presence of an LSC in the chamber, where the mean temperature is horizontally nonuniform. It is necessary to measure the spatial and temporal variability in r, T and S in order to determine how closely the models of reduced dimensionality capture the true variability in the chamber.
In this paper we describe the basic characteristics of the flow in the chamber, including the large scale circulation because of the importance of these quantities on the distribution of temperature and water vapor, and thus on the saturation ratio. While 85 measurements of temperature in turbulent Rayleigh-Bénard convection are ubiquitous, as noted above, measurements of water vapor concentration are rare, and differ in some fundamental respects from measurements of temperature. We first describe how we compare measurements of water vapor and temperature through an exploration of how a path averaged measurement differs from an ideal point measurement. Next, we describe the behavior of the LSC in the chamber across several temperature differences. We then describe how the LSC changes the shape of the temperature distributions in the bulk of the chamber.

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Finally we present measurements of water vapor concentration, temperature and the saturation ratio, S, at different locations in the LSC of the Π-chamber for both dry (S < 1) and moist (S > 1) convection.

Methods
Our experiments were conducted in Michigan Tech's Π Chamber with the cylindrical insert in place; in those conditions, Γ = 2.
The cylindrical insert restricts the volume of the chamber to 3.14 m 3 . To induce convection, the top and bottom control surfaces 95 within the chamber are set such that T T op < T Bottom and T Sidewall = (T T op + T Bottom )/2. In the experiments reported here, data was recorded for temperature differences (∆T = T Bottom − T T op ) up to 16 K. These measurements are recorded at 1 Hz.
The chamber is described in greater detail in Chang et al. (2016).
We present measurements in two different conditions in the chamber; dry and moist convection. In our experiments the distinction between dry and moist convection is determined by the saturation ratio, S, defined as where r is the water vapor mixing ratio, r s (T ) is the saturated mixing ratio which is a function of the temperature T , e is the vapor pressure and e s is the saturation vapor pressure. In practice, the saturation values are calculated from an approximation of the Clausius-Clapeyron equation, using the measured value of T (Lamb and Verlinde, 2011). We define dry convection as a subsaturated condition (S < 1) in the chamber. In moist convection the chamber is supersaturated (S > 1). In moist conditions, 105 a cloud would form if aerosol particles were present, but for these experiments, we did not inject aerosols into the chamber which prevents the formation of cloud droplets.
A ring of eight RTDs, 1 cm away from the wall of the cylinder was used to determine the orientation and amplitude of the large scale circulation. The RTDs are on the horizontal midplane of the chamber and are evenly spaced such that the angular 110 distance between each one is π/4 radians. The setup for this experiment can be seen in Fig. 2. As noted above, the primary measurement in Rayleigh-Bénard convection has been temperature, which can be measured with a variety of sensors with the required accuracy and precision. Quantitative measurement of concentration is much less common. Measurements of water vapor concentration are limited by the dynamic range of the sensor and temporal resolution (e.g. capacitance hygrometers) or by the path over which the measurement is averaged (e.g. absorption hygrometers). We use 115 a LiCor LI-7500A infrared hygrometer at 1 Hz to measure water vapor. The path length, d, of the LiCor is 12.5 cm with a measurement volume of ≈ 12 cm 3 . To ensure that we measure T with the same temporal and spatial resolution as r (to get a reliable value of S) we use a high speed sonic temperature sensor (Applied Technologies, Inc.); it has a path length of ≈ 13 cm and was set to sample at 1 Hz. The sonic temperature sensor operates on the same physical principle as a sonic anemometer, using the Doppler shift in emitted and detected sound waves to derive the quantity of interest, using the known speed of sound.

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In this case, the temperature sensor is sensitive to the virtual temperature, T v (1 + 0.61r)T (Lamb and Verlinde, 2011), which can be converted to the actual temperature using measured water vapor concentrations from the LiCor. When the sonic temperature was near the center of the chamber, we calibrated the temperature derived from these two measurements against an RTD. Both water vapor and temperature sensors were collocated on a traverse system that allowed us to move the sensors along a line that bisects the chamber. The traverse system is located on the horizontal mid-plane (z = 0.5 m).

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In order to determine the behavior of the LSC we followed the method of Brown and Ahlers (2006) and Xi et al. (2009). We where T is the mean temperature in the ring, δ is the amplitude of the temperature variation among the eight RTDs and φ is the phase of the temperature variation. δ and φ are derived by fitting the measurements to Eqn 3. Notably φ represents the angular location of the updraft relative to the reference position. During dry convection we applied this procedure independently on two different sets of eight RTDs. The placement of the two sets of RTDs formed an outer ring (Wall) and inner ring (Bulk) which were located 1 cm and 30 cm from the sidewall of the chamber. In the moist convection experiments we only had the 135 wall ring in the chamber.
We can describe the characteristics of the temperature in the chamber using only RTDs. If this were our only objective we would only need to run the chamber in dry conditions. However, in studying cloud properties we also need to describe the distribution of water vapor and by extension the saturation ratio. Because our measurement of the water vapor mixing ratio is over a 12.5 cm path, we need to know how a volume/path averaged measurement will compare to an ideal (i.e. instantaneous, 140 point) measurement. Because we cannot perform such a measurement for the water vapor concentration, we used a large-eddy simulation (LES) to understand the effects of path averaging on water vapor concentration and temperature.

LES Results for Path Averaging
Our LES is the System for Atmospheric Modeling or SAM (Khairoutdinov and Randall, 2003), which has been adapted and  We use a single grid box as an 'ideal' measurement. We simulated the sensor's path length by averaging the temperature and water vapor of n adjacent points. We use the center bin as the reference and symmetrically average towards the ends of each line. The path length is then calculated by d = n * 3.125 cm along the x and y axes. On the diagonal the path length is calculated by d = n * 3.125 * √ 2 cm. The path length for a single point, denoted as d 0 , is the size of a grid box, 3.125 cm.
The result of path averaging is shown in Fig. 4, which shows a plot of the standard deviation of temperature for a path 155 length d, normalized by the standard deviation for d 0 . Not surprisingly, as the path length increases, the normalized standard deviation of the measurement decreases. Also note that the curves from the four different lines of numerical sensors collapse.
(The lines are denoted A, B, C, and D in Fig. 4 and Fig. 5). Note that although only results for T are shown in Fig. 4, the data for σ r (d)σ −1 r (d 0 ) and σ T (d)σ −1 T (d 0 ) are identical due to the the non-dimensional units, and the same advective equations and diffusivity for both scalars. The LES results indicate that, over the path length of the LiCor and sonic temperature sensors,  the standard deviations of T and r decrease by ≈ 8%. This result indicates that the measurements that we perform in the Π Chamber do not capture the true variability in the system, but capture over 90% of it.  The path averaged values for r(d) and T (d) were used with Eqn. 2 to calculate S(d). In Fig. 5 is shown plotted against d. Over the same path length as the LiCor and sonic temperature sensors, σ S decreases by ≈ 19%. The percent decrease in S over the path length of the LiCor is higher than 8% due to the combined averaging of r and T . 165 We have shown that a path averaged measurement will underestimate the turbulent fluctuations. This type of averaging likely acts as a low pass filter, with the high frequency fluctuations being artificially removed. In Fig. 6 the power spectra of T are shown for several different path lengths. As the path length is increased, the higher frequencies are proportionally removed at a faster rate, decreasing the slope of the spectra. Over the path length of the LiCor (≈ 12.5cm) the spectra are noticeably impacted by the path averaging, but represent the overall shape and magnitude of the spectra of temperature reported by the 170 single bin.

Determination of Basic Characteristics of the LSC
We first ascertain the characteristics of the circulation, using only measurements of temperature (i.e. in dry conditions). For these conditions, we do not need to place the traverse with LiCor hygrometer and sonic temperature sensor in the chamber, so we can use the second ring of RTDs in the bulk of of the chamber (30 cm away from the side walls). In previous studies 175 of Rayleigh-Bénard convection the first order moments of the circulation have been modeled as a single roll that spans the diameter of the cell using Eqn. 3. This roll takes the form of a warm updraft along one side of the chamber with the cooler downdraft located along the opposite side. Due to the positive correlation between the vertical velocity and temperature, either variable can be used to find where the mean updraft is located. The location of the updraft is then used to determine the orientation of the circulation. An example of the instantaneous temperature measured on the wall and bulk rings is shown in Fig. 7. In the figure, the temperature fluctuation, T = T − T is shown against Θ, where T is the mean temperature, averaged across all sensors in the ring and T is the temperature measured by a single sensor at time t. The solid line is the least squares fit to the temperature measurements using Eqn. 3. In both rings of RTDs, a sinusoid is a reasonable fit. The amplitude, δ, and the orientation, φ, were calculated from the fit; Fig. 8 shows the orientation of the circulation along the wall, φ W all (t), and in the bulk, φ Bulk (t), 185 as a time series. The difference between φ W all and φ Bulk is smaller than the uncertainty in the fit. Over the course of our measurements the mean orientation for both precesses by ≈ 0.3 radians. Both rings of RTDs show azimuthal oscillations of ≈ 0.6 radians. The time series also shows that φ W all and φ Bulk oscillate in phase.
In Fig. 9, the time series for δ W all and δ Bulk are shown. The LSC does not show any cessations, with δ never approaching zero. The amplitude along the wall is consistently higher than the amplitude in the bulk. δ for both rings fluctuates around 190 the mean by ≈ 0.1 C. The amplitude of the LSC being highest near the wall is consistent with the circulation predominately following the walls of the cell (Qiu and Tong, 2001). The strength of the LSC decreases towards the center.
As ∆T increases, the amplitude of the temperature on both rings increases, as shown in Fig. 10. For each ∆T , δ W all > δ Bulk . As ∆T increases, the standard deviation of δ (σ δ ) also increases, showing that the variability of the LSC depends on ∆T . Over the range of ∆T s we have investigated, the amplitude for both rings increases linearly. 195 Our data show that the effects of the circulation are felt well into the bulk of the chamber, though, as expected, the amplitude of the circulation decreases towards the center. Our results also indicate that the circulation in the Π Chamber has a preferred orientation with pronounced azimuthal oscillations. We will now address the impact of the LSC on the temperature distributions in the bulk of the chamber, using the RTDs in the bulk ring.    To minimize the effect of the chamber's temperature controls, which can fluctuate on the order of ten minutes, a high-pass 200 Fourier filter with a cutoff of around five minutes was applied to the individual RTDs in the inner ring. The cutoff at five minutes is roughly four times larger than the period of the large scale circulation (1/f 0 ). The angular deviation from the updraft was calculated from φ wall using 16 bins of size π/8 radians to minimize the effect of azimuthal oscillations of the circulation.
With this done, the normalized standard deviation (σ T /∆T ) is shown in Fig. 11. Near the updraft (| Θ − φ wall |= 0), σ T /∆T is lower than the rest of the chamber. The downdraft (| Θ − φ wall |= π) curiously has a normalized standard deviation that is 205 about twice the σ T /∆T in the updraft. In an ideal chamber σ T is likely the same for both the updraft and downdraft. The Π-chamber has several factors (for example viewing windows) that cause σ T /∆T to deviate away from the ideal case. Figure 11. σT normalized by ∆T as a function of the angular distance from the updraft. The temperature was filtered using a high pass Fourier filter with the cutoff at around 5 min.
In Fig. 12 the skewness of T is shown with the angular distance from the updraft of the LSC. The skewness is defined as For each ∆T the highest skewness is measured when the temperature is near the updraft (| Θ−φ wall |= 0) and lowest in the downdraft (| Θ−φ wall |= π). Perpendicular to the axis of the circulation the distributions become symmetric.

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∆T does not change the value of the skewness due to the normalization of the skewness. Multiple points in the same location are due to absolute value applied to Θ−φ W all . The same distance to the left and right of the circulation would then be expressed as two points at the same location away from the updraft.
The skewness is impacted greatly by rare events which likely contributes to the spread in values at each position from the updraft. In RBC rare events take the form of plumes that come from either the top or bottom boundaries. The positive skewness 215 in the updraft is a result of warm plumes from the bottom surface. Alternately, cold plumes are more likely to pass through the downdraft, causing a negative skewness. Ideally in positions perpendicular to the LSC we expect warm and cold plumes to pass a sensor at an equal rate resulting in zero skewness. In Fig. 12, the spread of values perpendicular to the LSC (| Θ−φ wall |= π) is likely due to the uncertainty in φ wall . Within the uncertainty, the off-axis region could be slightly closer to the updraft or downdraft. For example at Θ − φ wall = π warm plumes could be more frequent than cold plumes, despite being calculated at 220 a spot were they should have equal probability. Directly across from those sensors (Θ − φ wall = −π), the cold plumes may be more frequent than warm plumes. Both measurements would be represented at | Θ − φ wall |= π but the skewness would have the opposite sign.

Moist Convection Results
Having established the basic characteristics of the large scale circulation, using measurements of the temperature, we turn to 225 the scenario in which a difference in temperature and water vapor concentration between the top and bottom plates drives a convective flux of two scalars. The Rayleigh number is dominated by the temperature difference; the difference in water vapor concentration is small in comparison, which follows from Eqn. 1. Given that, the behavior of the temperature field in the bulk of the chamber will be comparable in moist and dry conditions. As noted in Sec 2, the inner ring of RTDs was removed to enable measurement of temperature and water vapor with the 230 sonic temperature sensor and LiCor respectively. These instruments were mounted such that they were probing roughly the same volume; additionally the sensors were mounted such that they could be moved across the chamber on a traverse. The time series of r, T and S are shown in Fig. 13 for ∆T = 12 K. The sensors were near the updraft and downdraft for 4 hrs each. The sensors were in the center for 8 hrs. The figure clearly shows that the variance of the scalars is a function of the position in the large scale circulation. Near the downdraft of the circulation, the variance is the highest. This is consistent with the standard 235 deviations near the downdraft shown in Fig. 11.
A Fourier analysis of r and T in the center of the chamber are shown on the right side of Fig. 13. The main peak is at a period of ≈ 72 seconds which is due to the large scale circulation frequency, f 0 . Harmonics of f 0 can be seen in both measurements.
For a more detailed analysis of Fourier spectra in the chamber, see Niedermeier et al. (2018).
As another perspective on these measurements, we show the probability distribution functions (PDFs) for T in Fig. 14. The   T and r reinforce the phenomenological picture that warm, humid plumes are more likely to be seen in the updraft region of the chamber. The opposite is true for the downdraft, where the presence of cold, low r plumes lead to both distributions being negatively skewed.    Because of the correlation between temperature and water vapor concentration in the chamber, a change in either r or T will not a priori lead to a change in S. A positive fluctuation in T could be associated with a positive fluctuation in r such that the ratio of r and r s do not change. (See Chandrakar et al. (2020a) for a more complete discussion of the correlation between r and T and the corresponding changes in S.) Our data show that the skewness of r and T are comparable in both sign and magnitude in the updraft region of the circulation. S however, is negatively skewed at each location in the chamber. The skewness of the 260 pdfs of the saturation ratio is -0.50, -0.64, and -0.18 in the updraft, middle and downdraft respectively.

Conclusions
The convection-cloud chamber at Michigan Tech, the Π Chamber, is a Rayleigh-Bénard convection cell, designed for studies of interactions between turbulence and cloud microphysics. Through measurements of the temperature in the chamber, we have shown that the large scale circulation is a single roll with a fixed overall orientation, but with pronounced oscillations about the 265 mean position, typical of the large scale circulation in Rayleigh-Bénard convection.
To determine the saturation ratio in the chamber, we measure water vapor concentration and temperature, simultaneously, to get the saturation ratio, S. Because point measurements of water vapor concentration are not currently possible, we have verified that our path averaged measurements capture an acceptable fraction of the true variance in the system, using a combination of Figure 16. The probability distribution of S − S middle near the updraft, downdraft and middle. For each region a high pass filter was applied to r and T with a cut off of ≈ 5 min. We have plotted the distributions S − S middle , not S to highlight the fact that the downdraft has a lower mean relative the middle and updraft. measurements and large eddy simulations. The LES shows that σ T and σ r decrease by ≈ 8% from their true values when the 270 measurement is averaged over approximately 12 cm, as ours are. The corresponding decrease in S is ≈ 19%. (Path averaging is more pronounced for σ S due to the combined averaging from r and T .) The LES shows that path averaged measurements do underestimate but still represent a sizable portion of the turbulent fluctuations of r, T , and S.
We show that water vapor concentration and temperature distributions in the updraft and downdraft are qualitatively similar.
In the updraft, both scalars have a higher mean than in the center and are positively skewed. Combining these measurements 275 into S shows turbulent fluctuations that are caused by fluctuations in r and T . S is consistently negatively skewed even in the updraft where both r and T are positively skewed. In the downdraft the distribution of S is more negatively skewed than the updraft.
As noted in the Introduction, one of our primary motivations to understand the spatial and temporal variability of the saturation ratio in the chamber is to then relate it cloud droplet growth. We have used zero and first order models of that variability in 280 our analysis of microphysics in the chamber to date (Chandrakar et al., , 2020a. The results presented here indicate that while these models capture the essential variability of T , r, and S in the chamber, spatial variations in the chamber may affect, for example, where droplets preferentially activate or evaporate. This is an ongoing topic for both experimental and modeling efforts. Future work will focus on how the fluctuations of S change upon the transition from moist to cloudy conditions. The in-cloud 285 saturation field is dependent on the initial S (the moist conditions that we have shown in this paper) and the influence of cloud droplets. The presence of droplets is expected to buffer the fluctuations in S, but the magnitude is currently unknown.