Limitations of Bin and Bulk Microphysics in Reproducing the Observed Spatial Structure of Light Precipitation

Mikael K. Witte aNaval Postgraduate School, Monterey, California
bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
cJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Hugh Morrison dNational Center for Atmospheric Research, Boulder, Colorado

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Anthony B. Davis bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Joao Teixeira bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California
cJoint Institute for Regional Earth System Science and Engineering, University of California, Los Angeles, Los Angeles, California

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Abstract

Coarse-gridded atmospheric models often account for subgrid-scale variability by specifying probability distribution functions (PDFs) of process rate inputs such as cloud and rainwater mixing ratios (qc and qr, respectively). PDF parameters can be obtained from numerous sources: in situ observations, ground- or space-based remote sensing, or fine-scale modeling such as large-eddy simulation (LES). LES is appealing to constrain PDFs because it generates large sample sizes, can simulate a variety of cloud regimes/case studies, and is not subject to the ambiguities of observations. However, despite the appeal of using model output for parameterization development, it has not been demonstrated that LES satisfactorily reproduces the observed spatial structure of microphysical fields. In this study, the structure of observed and modeled microphysical fields are compared by applying bifractal analysis, an approach that quantifies variability across spatial scales, to simulations of a drizzling stratocumulus field that span a range of domain sizes, drop concentrations (a proxy for mesoscale organization), and microphysics schemes (bulk and bin). Simulated qc closely matches observed estimates of bifractal parameters that measure smoothness and intermittency. There are major discrepancies between observed and simulated qr properties, though, with bulk simulated qr consistently displaying the bifractal properties of observed clouds (smooth, minimally intermittent) rather than rain while bin simulations produce qr that is appropriately intermittent but too smooth. These results suggest fundamental limitations of bulk and bin schemes to realistically represent higher-order statistics of the observed rain structure.

© 2022 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: Mikael Witte, mikael.witte@nps.edu

Abstract

Coarse-gridded atmospheric models often account for subgrid-scale variability by specifying probability distribution functions (PDFs) of process rate inputs such as cloud and rainwater mixing ratios (qc and qr, respectively). PDF parameters can be obtained from numerous sources: in situ observations, ground- or space-based remote sensing, or fine-scale modeling such as large-eddy simulation (LES). LES is appealing to constrain PDFs because it generates large sample sizes, can simulate a variety of cloud regimes/case studies, and is not subject to the ambiguities of observations. However, despite the appeal of using model output for parameterization development, it has not been demonstrated that LES satisfactorily reproduces the observed spatial structure of microphysical fields. In this study, the structure of observed and modeled microphysical fields are compared by applying bifractal analysis, an approach that quantifies variability across spatial scales, to simulations of a drizzling stratocumulus field that span a range of domain sizes, drop concentrations (a proxy for mesoscale organization), and microphysics schemes (bulk and bin). Simulated qc closely matches observed estimates of bifractal parameters that measure smoothness and intermittency. There are major discrepancies between observed and simulated qr properties, though, with bulk simulated qr consistently displaying the bifractal properties of observed clouds (smooth, minimally intermittent) rather than rain while bin simulations produce qr that is appropriately intermittent but too smooth. These results suggest fundamental limitations of bulk and bin schemes to realistically represent higher-order statistics of the observed rain structure.

© 2022 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: Mikael Witte, mikael.witte@nps.edu

1. Introduction

Large-scale models of the atmosphere [including general circulation models (GCMs)] typically operate at spatial scales of order 10–100 km, much larger than some of the physical processes they must represent. This means that smaller-scale processes must be parameterized as a function of grid-resolved variables while also accounting for subgrid variability. In addition to the scale discrepancy, many (if not most) parameterized processes are nonlinear which, according to Jensen’s inequality, implies that the grid mean of a generic nonlinear process rate f(x)¯ is not equal to the process rate obtained using a grid-resolved input variable, f(x)¯f(x¯). If f(x) is convex f(x)¯f(x¯), and if f(x) is concave f(x)¯f(x¯). Our study is specifically concerned with warm cloud microphysical processes (i.e., autoconversion and accretion) for which the governing equations in standard bulk microphysics schemes (e.g., Khairoutdinov and Kogan 2000, hereafter KK) are convex, thus implying that process rates are systematically underestimated if only grid-mean input variables are used.

Since the late 1970s, statistical approaches (e.g., Manton and Cotton 1977; Mellor 1977; Sommeria and Deardorff 1977) have been used to find the saturated portion of a grid box and predict cloud macrophysical properties (i.e., cloud fraction and mean condensate amount) using the joint probability distribution function (PDF) of relevant thermodynamic fields (e.g., moist conserved temperature and total water mixing ratio). A hallmark of these schemes was the use of simple PDFs for analytical tractability, but in recent years schemes have evolved to reflect observational constraints (Larson et al. 2002; Tompkins 2002; Molod 2012). They have also grown in complexity to unify the representation of small-scale turbulence, subgrid-scale convective motions and cloud macrophysics (e.g., Golaz et al. 2002; Siebesma et al. 2007; Park 2014). Yet it was not until the early 2000s that comparable approaches were developed to account for subgrid variability in GCM microphysics parameterizations, based on the insight that a more physical approach can be taken to compensate for underprediction of cloud process rates than the use of ad hoc tuning parameters to bring some model evaluation metric (e.g., top-of-atmosphere radiative balance) in line with long-term observations (Rotstayn 2000; Pincus and Klein 2000). Many microphysics schemes now use analytical PDFs of cloud and rain properties to derive what is termed an “enhancement factor” that modulates process rates obtained using grid-mean inputs (e.g., Morrison and Gettelman 2008; Bennartz et al. 2011; Weber and Quaas 2011; Larson and Griffin 2013; Xie and Zhang 2015).

Beyond the selection of an appropriate distribution shape, distribution parameters relevant for microphysics (e.g., of saturation excess, cloud drop number concentration Nc, cloud water mixing ratio qc, or rainwater mixing ratio qr) have been shown to vary with spatial scale and numerous observational studies have been undertaken to quantify the scaling behavior of, for example, cloud and rain liquid water mixing ratios (among others, Lebsock et al. 2013; Boutle et al. 2014; Xie and Zhang 2015; Kahn et al. 2017; Wu et al. 2018; Witte et al. 2019b, hereafter W19b; Zhang et al. 2019). These observational studies have used a wide range of available measurements (including aircraft in situ, ground-based remote sensing, and spaceborne platforms), yet each platform has weaknesses that limit the relevance of their diagnosis of scaling behavior. Aircraft in situ probes have limited sample volume and little to no context regarding the instantaneous vertical structure of horizontal variability (although some studies have attempted to work within these limitations; i.e., Zhang et al. 2021). Aircraft in situ measurements and ground-based remote sensing rely heavily on Taylor’s frozen turbulence hypothesis to convert from time series to transects in space. Remote sensing retrievals (ground based or spaceborne) of Nc, qc, and qr rely on a series of assumptions that are not universally valid, nor are all relevant retrieved quantities independent of one another. Space-based remote sensing retrievals may also have unresolved subpixel variability due to coarse resolution (of order hundreds of meters) and passive sensors must assume cloud vertical structure. Finally, all of these platforms are effectively one-dimensional in the horizontal, with the exception of passive satellite sensors (e.g., from geostationary or low-Earth-orbiting imagers) that do not sample in the vertical. A notable exception is ground-based scanning radar, which can retrieve cloud or rain properties in an effectively two-dimensional plane.

Given the limitations of observational diagnosis of assumed distribution parameters, it is appealing to use numerical models to quantify the statistical inputs required to compute enhancement factors. Fine-scale model output from, e.g., large-eddy simulation (LES), is desirable such that the convective motions driving cloud formation and organization are adequately resolved. Cloud structure is well resolved in three dimensions; thus, explicit knowledge of gridscale microphysical variability is available. For example, Larson and Griffin (2013) use LES of an idealized stratocumulus case to quantify variances and covariances for an “upscaled” bulk microphysics scheme. Yet these variability diagnostics are themselves dependent on parameterized microphysics and subgrid-scale turbulence/mixing. In particular, microphysics schemes are uncertain because of limitations in process understanding and numerical representation (Xue et al. 2017; Morrison et al. 2018, 2020). Furthermore, clouds and precipitation can be considered as stochastic phenomena that are poorly described by Gaussian statistics (Davis et al. 1999). Thus, one might question the suitability of simple, lower-order statistical measures (e.g., variance) to describe cloud and rain variability.

While stochastic parameterizations are becoming increasingly common in coarse-gridded models for weather and climate prediction (Berner et al. 2017; Palmer 2019), this is not generally true of bulk and bin microphysical parameterizations (with a few exceptions; e.g., Kuell and Bott 2014; Qiao et al. 2018; Stanford et al. 2019)—a particularly ironic point given the dependence of warm rain formation on the so-called stochastic collection equation (SCE; Pruppacher and Klett 2010). It is worth noting that the Lagrangian particle-based approach to microphysics does, in fact, stochastically solve the collection equation (for a detailed discussion, see Grabowski et al. 2019). For example, Lagrangian methods can account for “lucky drops” that arise due to statistical fluctuations and covariances in the drop size distribution (DSD) that cannot easily be represented by bin microphysics.

The concerns raised above lead to the central question addressed by the present study: How do model uncertainties limit the applicability of model-derived microphysical fields for informing GCM parameterizations across scales? We pose two hypotheses related to this question:

  1. To reproduce observed cloud/rain variability and covariability, a model must be able to produce mesoscale organization. The simulated domain must therefore be large enough (and the horizontal grid spacing fine enough) for such organization to develop. In stratiform precipitation, mesoscale cellular patterns also likely play an important role in determining precipitation structure.

  2. More degrees of freedom in microphysical parameterization formulation (i.e., bin versus bulk) will produce a better match with observed variability.

To evaluate the first hypothesis, we perform a series of LES runs of a well-known case of precipitating nocturnal stratocumulus with varying horizontal domain size and cloud drop number concentration (as a proxy for mesoscale organization) using a bulk microphysics scheme. Varying Nc exerts a strong influence on cellular organization because, all else being equal, lower Nc leads to stronger precipitation, for which enhanced evaporation below cloud base leads to the formation of cold pools. Convergence of cold pool boundaries is in turn associated with a dynamical transition from a closed-cell cloud field dominated by broad, weak updrafts and narrow, strong downdrafts to an open-cell cloud field with narrow, strong updrafts and broad, weak downdrafts (Feingold et al. 2015). The same case is simulated using a drop size-resolving bin microphysics scheme to address hypothesis 2. A method for quantifying variability across scales, bifractal analysis, is then applied to one-dimensional horizontal transects of simulation output and the results are compared with observed bifractal parameters from aircraft in situ measurements and ground-based remote sensing of marine boundary layer clouds. This study represents a first attempt at evaluating the scaling behavior of modeled microphysical fields to inform parameterizations of precipitation processes.

The remainder of the paper is organized as follows. Summaries are given of the bifractal analysis framework in section 2 and results from its application to observations in section 3. The LES model, experimental setup, and analysis approach are described in section 4. Simulation results are described in section 5, followed by discussion of the model–observation comparison in section 6. Concluding remarks are given in section 7.

2. Analysis methodology

Fractals are self-similar sets or functions in space and/or time that exhibit the same geometry regardless of scale. Monofractals are characterized by a single fractal dimension or scaling exponent that quantifies the complexity of the function, while multifractals are characterized by an infinite family of scaling exponents. Fractal and multifractal analyses are frameworks to evaluate two-point variability (e.g., structure) across scales. A general review of the history and mathematics of multifractals is beyond the scope of this paper; it is sufficient to begin with the application of the method to in situ observations of clouds, beginning with the work of Davis et al. (1994, 1996, 1999) and Marshak et al. (1997). More specifically, see Davis et al. (1994) for a graphical introduction to the structure function-based analysis used here, and for an overview of previous applications of the method to observations of stratocumulus clouds and drizzle, see Davis et al. (1999) and Wood (2005), respectively.

Most geophysical systems exhibit variability across a broad range of scales, do not vary smoothly and are statistically nonstationary—atmospheric turbulence being a canonical example. On the other hand, geophysical systems are often scale invariant over some subrange(s), e.g., the inertial subrange of turbulence. Practically speaking, scale invariance means that the power spectral density of a signal as a function of frequency, E(f), follows a power law over some range, i.e., E(f) ∝ fβ, where β is known as the spectral slope. As discussed in W19b, one attractive quality of scale-invariant fields is that they can be used to characterize variance analytically at a given frequency ϕ since var(ϕ)=ϕE(f)df.

Spectral analysis provides a valuable diagnostic but from the perspective of understanding physical processes, it has limited explanatory power. Davis et al. (1996) examined what meaningful information can be gained from traditional spectral analysis and found that power spectra of aircraft-observed liquid water content (LWC; simply qc divided by air density) were highly ambiguous and yielded little insight into the underlying processes responsible for producing observed LWC distributions. Through the lens of spectral analysis, different processes (stochastic or otherwise) can arrive at similar power spectra; thus, a means of disambiguating the contribution of various candidate processes is required. Davis et al. (1996) suggest that spectral analysis is instead optimally employed as a preliminary step to identify suitable regimes that exhibit scale invariance, to which multifractal analysis can then be applied to characterize scale-invariant processes more precisely. This philosophy is adopted in the present study.

A popular kind of multifractal analysis relies on structure functions, which describe variability in a field as a function of spatial location and separation. While structure functions can be evaluated over any number of dimensions, for comparison with published analyses of aircraft observations we limit the analysis here to one (horizontal) dimension. We then define the qth-order structure function as in Ma et al. (2017):
Sq(r)=|f(x+r)f(x)|q,0rL,0xLr,
where f(x) is a generic variable (in this study, cloud or rainwater mixing ratio) along dimension x, r is spatial separation distance, angle brackets 〈⋅〉 denote an ensemble mean, and L is the maximum dimension of the system. We define L and ensembles (here this term denotes a set of realizations of observed or simulated horizontal transects through cloud) differently for aircraft data and simulation output. For aircraft data, L is the length of a level flight segment—either an entire leg or some subdivision thereof (subdivision increases the number of ensemble members)—and the ensemble is a suite of discontinuous flight legs that may span several weeks or months. This compositing approach is taken to maximize ensemble size and minimize statistical noise, not because it represents an objectively “correct” method. Effectively, it is taken to circumvent the inherent limitations of aircraft sampling. In contrast, L is the horizontal domain size for model output and ensembles are composited over all horizontal transects in the x and y directions through each level of the model domain, sampled at some interval throughout a simulation. We note that horizontal transects from the model are correlated to varying degrees, but this is not a deficiency; as with the aircraft data, this approach is intended to maximize sampling of the domain. Examination of LES-derived structure functions defined individually over the x and y horizontal dimensions showed small differences for the simulations analyzed in this study, although we speculate that our horizontal aggregation procedure may be less defensible in the presence of significant shear.
For scale-invariant fields satisfying 1 < β < 3 (signifying a nonstationary field with stationary increments), the qth-order structure function Sq(r) is also expected to exhibit scale invariance with respect to r within the scaling regime identified from spectral analysis:
Sq(r)rζ(q),
where ζ(q) is a family of scaling exponents. When ζ(q)/q is nonconstant the field is multifractal, while a monofractal field exhibits constant ζ(q)/q. The exponent H1 = ζ(1) of the first-order (q = 1) structure function is known as the Hölder or Hurst exponent and characterizes the smoothness and nonstationarity of a signal. The exponent H1 takes values [0, 1] with a value of 0 corresponding to stationary noise (e.g., random processes with β < 1 such as white or colored noise) and an extreme value of 1 corresponding to an (almost) everywhere differentiable function (Davis et al. 1999). Stochastic cascade models can span the H1 dimension with a random additive process η(t), e.g., for an input signal x(t) and η(t), the noised output signal x(t)˜=x(t)+η(t).
The degree of multifractality (or, put another way, the deviation from monofractal scaling) can be understood to correspond to intermittency, which is essentially a measure of the tendency of extreme values or “spikes” to dominate a signal. Intermittency can be quantified using a number of methods. Davis et al. (1994) and Marshak et al. (1997) use a singularity analysis approach that involves normalizing and coarse-graining structure functions, while Pierrehumbert (1996) empirically fits a hyperbolic branch to ζ(q):
ζ(q)=q1/a+q/ζ,
where a ∈ (0, ∞] measures the average level of fluctuations and the asymptotic value ζ = ζ(∞) ∈ (0, ∞]. Pierrehumbert’s approach is much less expensive to compute than singularity analysis while retaining high accuracy; thus, we use this approach in our study. For a monofractal field, ζ = ∞ and the field is not intermittent, while for a multifractal field ζ < ∞ and the field is intermittent. Finally, we convert from parameters (a, ζ) to (H1, C1) as in Davis et al. (1999) and Ma et al. (2017):
H1=ζ1+ζ/a,and
C1=ζ(1+ζ/a)2,
where H1 quantifies smoothness as discussed above and C1 ∈ [0, 1] quantifies intermittency. Here, C1 = 0 corresponds to no intermittency (i.e., ζ = ∞) and C1 = 1 corresponds to a singular function, e.g., Dirac-delta or Heaviside functions. For the limiting case of a Dirac delta at (H1, C1) = (0, 1) in the bifractal plane, max(x)/x¯. Complementary to H1, stochastic models can span the range of C1 by applying multiplicative cascades from a random noise process ξ(t) to a generic input signal y(t), e.g., y(t)˜=ξ(t)y(t).

3. Findings from observational analyses

Nominally assuming H1 and C1 to be orthogonal, we can plot (H1, C1) pairs in what is termed the “mean multifractal” or “bifractal” plane [introduced in Davis et al. (1994) and Marshak et al. (1997), respectively]. An example of the bifractal plane is shown in Fig. 1, with well-known functions populating the vertices of the plane and passive scalars in well-developed turbulence spanning a range of intermittency values (0.05 < C1 < 0.15) with H1 = 1/3. Cloud and rainwater properties from observations are also plotted in Fig. 1. Most of the observational data points are derived from aircraft in situ measurements with a few additional points from shipborne remote sensors (EPIC 2001; Bretherton et al. 2004). FIRE’87, Physics of Stratocumulus Top (POST), and VAMOS Ocean–Cloud–Atmosphere–Land Study (VOCALS) were flown in subtropical stratocumulus (Sc) over the Pacific Ocean; ASTEX in subtropical Sc and shallow cumulus; and Southern Ocean Cloud Experiment (SOCEX) in Southern Ocean boundary layer clouds. Results from these campaigns were originally given in other publications: ASTEX in Davis et al. (1994; cloud) and Wood (2005; rain), FIRE’87 in Davis et al. (1996) and Marshak et al. (1997), SOCEX in Davis et al. (1999), EPIC 2001 in Wood (2005), and POST in Ma et al. (2017). In situ qc measurements were taken at high frequency (f > 100 Hz) with a Gerber Particulate Volume Monitor (PVM)-100 probe (Gerber et al. 1994) for all campaigns except FIRE’87, which used a King hotwire probe (King et al. 1981) at 20-Hz sampling frequency. The ASTEX rain points from Wood (2005) used 1-Hz optical array probe measurements. Note that different moments of the observed DSD with respect to diameter are presented: all in situ cloud water data points are from liquid water content (third moment) measurements, while two of the in situ rainwater points are derived from sedimentation rate (between fourth and fifth moment depending on drop size) observations, and radar reflectivity corresponds to the sixth moment.

Fig. 1.
Fig. 1.

Observed values of (H1, C1) from aircraft in situ and ground-based remote sensing measurements. MWR/LWP stands for microwave radiometer/liquid water path, q is condensate mixing ratio (subscript c denotes cloud, r rain), R is rain/sedimentation rate, and Z is radar reflectivity factor. Vertices are labeled with well-known functions corresponding to extrema in the bifractal plane. Each data point represents an aggregate over an entire field campaign of observations. Gray ellipses underlying data points denote the observed “cloud” and “rain” populations referenced in the text and subsequent figures. References are given in the text.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Novel cloud and rainwater values from the VOCALS field campaign are also shown in Fig. 1 using the campaign-average high-rate cloud probe data presented in W19b. Cloud and rainwater were partitioned using a threshold drop diameter of 50 μm, the maximum droplet size measured by the PVM-100 probe according to the manufacturer; a similar threshold is applied to bin model output. While 50 μm is an admittedly arbitrary value—for example, the SB scheme used in the University of California, Los Angeles, large-eddy simulation (UCLA-LES) uses a threshold diameter of 80 μm—it is not possible to test the sensitivity of the observational results to threshold size because the cloud water probe only measures bulk liquid water content (i.e., it does not resolve drop size). The scaling regime analysis closely follows W19b with one important difference: the slope of the compensated power spectral density [i.e., the spectral slope of E(f)fβ] is removed from the cost function used to diagnose scale breaks, which effectively places greater weight on spanning the largest possible scale range. While this minimizes the impact of the change in β at l ≈ 150 m found in the observed cloud water spectra by W19b, the upper scale breaks diagnosed without the compensated slope criterion match better with previous results and those from the simulated qc spectra (not shown).

There are several important features to note in Fig. 1. First, cloud and rain data cluster in two distinct populations (see annotation of Fig. 1) and we will use this separation to evaluate simulated cloud and precipitation structure. Cloud water is smooth with H1 > 0.2 and C1 < 0.15. A few campaigns map very closely to passive scalars in turbulence (e.g., SOCEX) while more recent results from POST (Ma et al. 2017) and VOCALS are significantly rougher. Advances in aircraft data acquisition systems in the 20+ years since the ASTEX, FIRE, and SOCEX observations were taken give greater confidence in the more recently acquired data. It is unclear why the cloud remote sensing point (liquid water path observed by microwave radiometer during the 2001 EPIC campaign) is smoother than passive scalars in turbulence, although an obvious hypothesis is that vertical integration results in smoothing.

Rain water in the bifractal plane is both rougher and more intermittent than any of the cloud water points in Fig. 1. One interpretation of the decrease in rainwater H1 is that it is due to the sample volume limitations of in situ drizzle probes, but the consistency of H1 between the in situ and radar data suggests robust measurement statistics. Of greater interest regarding rainwater is the increasing trend in C1 from qr (third moment of DSD) to sedimentation rate (fourth–fifth moment) to radar reflectivity (sixth moment). Higher moments of the DSD are increasingly sensitive to the largest, rarest drops found on the long right tail of the DSD. This relationship between moment order and sensitivity to increasingly rare drops provides a simple, intuitive explanation for the increase in C1 and will be further examined in future work.

One of the primary limitations of in situ airborne measurements is the trade-off between extensive horizontal and vertical sampling. Ma et al. (2017) attempted to overcome this trade-off by using 1-kHz measurements over relatively short horizontal extents (7.2 km for level legs in cloud, ≈112 m for sawtooth legs across cloud top) to evaluate the dependence of scaling properties on altitude. Level legs were simply classified “cloud base” or “cloud top” since no simultaneous remote sensing was available to constrain instantaneous vertical location in cloud more quantitatively. They found that H1 and C1 both increased with height, although this trend only emerged clearly at the highest level of aggregation (i.e., full field campaign average). To our knowledge, no other study has examined the vertical structure of cloud or rain bifractal parameters and only a few have analyzed simulated horizontal cloud geometry through the fractal lens (e.g., Siebesma and Jonker 2000; Luo and Liu 2007).

4. Model configuration

Idealized simulations of the nocturnal drizzling stratocumulus case observed during research flight 2 of the Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS-II RF02) (Ackerman et al. 2009) are performed using the UCLA-LES model (Stevens et al. 2005). Initial profiles of θl, qt, and horizontal winds are shown in Fig. 2. The model configuration is similar to that used in Morrison et al. (2018) and Witte et al. (2019a) in terms of dynamics and subgrid-scale diffusion. Note that subgrid-scale diffusion is not applied to prognostic microphysics variables. We use the simple longwave radiation parameterization suggested by Stevens et al. (2005). The UCLA-LES results use the bulk microphysics scheme of Seifert and Beheng (2001, 2006, hereafter referred to as the SB scheme) or the Tel Aviv University (TAU) bin scheme (Tzivion et al. 1987, 1989) as implemented in Witte et al. (2019a). For the TAU scheme, we use the standard mass-doubling size grid without coupling of turbulence and collision–coalescence rates. Note that the constant cloud/aerosol concentrations used in the simulations are given in mixing ratio units (per unit mass) instead of the more typical units of concentration (per unit volume) because the microphysics parameterizations use mixing ratios for internal computations. These quantities are roughly equivalent since air density ρa ∼ 1 kg m−3 throughout the boundary layer. Further details on the microphysics schemes are given in the next paragraph. We follow the nudging strategy of Zhou et al. (2018) and nudge potential temperature θ and water vapor mixing ratio qυ to their initial profiles to maintain a constant inversion height (zi = 795 m) and quasi-steady precipitating state. All simulations were performed using the Cheyenne system at the National Center for Atmospheric Research (Computational and Information Systems Laboratory 2019). Finally, to demonstrate the robustness of the analysis we also examine model output first presented in Feingold et al. (2015), who simulated the DYCOMS-II RF02 case using a different model [System for Atmospheric Modeling (SAM); Khairoutdinov and Randall 2003] with coarser horizontal grid spacing (dx = 200 m), different numerics, and the KK bulk scheme with time-varying Nc.

Fig. 2.
Fig. 2.

Initial conditions for UCLA-LES simulations. (left) Liquid water potential temperature θl, (center) total water mixing ratio qt, and (right) horizontal wind components u (solid line) and υ (dashed line).

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

The SB bulk scheme uses constant Nc and is configured to prognose one moment of the cloud DSD: the mass mixing ratio qc, and two moments of the rain DSD: mass mixing ratio qr and number mixing ratio Nr. It is an option to use constant Na and prognose Nc but this was not employed for this study. The KK bulk scheme used in the SAM simulation is configured similarly (prognostic variables: qc, Nr, qr). The main difference between the SB and KK schemes is in the formulation of process rates (i.e., for autoconversion and accretion). The TAU bin scheme holds Na constant and prognoses two moments of the cloud and rain DSDs (Nc, qc, Nr, and qr). While the bulk schemes assume DSD shapes for both the cloud and rain drop size regimes, the bin scheme explicitly calculates collision–coalescence rates from the full bin-resolved DSD. Thus, the important differences between TAU and the bulk schemes are that 1) drops are activated from a constant background aerosol field and 2) process rates are explicitly calculated over the full DSD.

Simulations span a range of domain sizes and constant aerosol/cloud drop number concentrations (depending on which microphysical scheme is employed) as shown in the simulation list in Table 1. A constant horizontal grid spacing dx = 50 m is used for the entire set of simulations, such that each successive doubling of domain size corresponds to a quadrupling of the total number of grid points. A variable vertical grid with nz = 193 layers is employed similar to Ackerman et al. (2009) with dz = 5 m grid spacing near the surface and in the cloud layer, and a stretched grid elsewhere. The model top is at 1500 m. For this study, a deeper layer of constant dz (from 400 to 1000 m) is used than in Ackerman et al. (2009). Simulations with SB microphysics are run for 18 h for each domain size and value of Nc to test the hypotheses posed in the introduction. Due to the substantially greater computational expense of the bin scheme, the simulation duration is reduced to 12 h and only the 28.8 km (nx = 576) domain is run for each value of Na.

Table 1.

List of simulations performed. Here, Nc is cloud drop number mixing ratio, Na is aerosol number mixing ratio, nx is number of horizontal grid points, and L is horizontal domain size. All simulations except SAM_nx200_KK were run with UCLA-LES. The asterisk in the Nc entry for the SAM simulation denotes variable cloud drop number mixing ratio.

Table 1.

Procedure for analyzing simulation output

It is appropriate to use different procedures to analyze observations and simulation output because of the vastly greater data coverage in the model. We use half-hourly three-dimensional output starting at t = 2 h and analyze cloud and rainwater mixing ratio (qc and qr, respectively) on a level-by-level basis. We calculate one-dimensional structure functions for cloud and rainwater at each level where the cloud and rain fractions (defined as the fraction of grid points per horizontal plane with cloud/rain mixing ratio greater than 0.01 g kg−1) exceed threshold values of fc > 0.2 and fr > 0.015, respectively. Results are insensitive to small perturbations to the selection of threshold condensate fractions. To augment data aggregation, structure functions are then averaged over all transects in both the east–west and north–south directions; no major differences are found by analyzing the horizontal components independently, although uncertainty in diagnosed (H1, C1) is increased.

To determine the scaling regime to which the family of scaling parameters ζ(q) is fit, a uniform lower bound of spatial separation rmin = 6dx = 300 m is specified. The upper bound rmax is determined by optimizing a similar cost function as that applied to the observational fits, but with equal weight given to the coefficient of determination R2, the log–log slope of the compensated structure function S1(r)rζ(1) and the root-mean-square error. An example of the mean structure functions from the bulk N10_nx1152 simulation normalized at the height of maximum qc is shown in Fig. 3 with rmin and rmax given for reference. For both cloud and rainwater, rmax ranges from 3 to 4 km for bulk simulations with nx > 144 with no obvious dependence on Nc. The bin simulations generally exhibit larger rmax, spanning 3–8 km for both condensate species. As this study is primarily concerned with the scale-invariant regime, the reader is referred to Zhou et al. (2018) for a more detailed discussion of the characteristic scales of organization using simulations of the same case study.

Fig. 3.
Fig. 3.

Domain-mean structure functions of integer order q = [1, 10] from the bulk N10_nx1152 simulation at the height of maximum mean qc (z = 732.5 m), t = 14 h runtime. Curve shading represents order q, with progression from dark to light indicating increasing q.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

The linear fits of logSq(r) versus logr between rmin and rmax are thus obtained for a large number of values of q, yielding for each case a scatterplot of ζ(q) values versus q. In turn, these data are used in a nonlinear fit to the model for ζ(q) in (3). Finally, (H1, C1) are computed from the fit parameters (a, ζ) using (4)–(5).

5. Results

a. General results from the bulk microphysics simulations

Time series of domain-mean relevant cloud and precipitation quantities for the bulk simulations are given in Fig. 4. In general, output is clustered by drop concentration with minor variations across different domain sizes and agrees well with the results of Ackerman et al. (2009). Domain-maximum cloud water mixing ratio qc, max(qc), displays the greatest dependence on domain size, with larger domains exhibiting higher max(qc). This is consistent with the results of Kazil et al. (2017), who found that the long tail of the liquid water path distribution widens with increasing domain size. Another notable feature is the approximately one to two orders of magnitude decrease in mean precipitation rate from low to high cloud drop number concentration. Using the KK bulk scheme, Zhou et al. (2018) found a logarithmically decreasing steady-state surface precipitation rate with increasing Nc, spanning from roughly 0.1 mm day−1 for Nc = 65 mg−1 to 1 mm day−1 for Nc = 10 mg−1. The SB bulk scheme exhibits a similar relationship between Nc and rain rate but spans a wider range of rain rate for about the same Nc, with mean rain rates from 0.003 mm day−1 for Nc = 55 mg−1 to 1 mm day−1 for Nc = 10 mg−1 from UCLA-LES. This illustrates the increased sensitivity to cloud drop number concentration of the SB scheme compared to KK and is consistent with the findings of Savic-Jovcic and Stevens (2008). Overall, the time series view of UCLA-LES output demonstrates the near-steady state of cloud and precipitation properties reached in all simulations by 12–14 h.

Fig. 4.
Fig. 4.

Time series of UCLA-LES output with bulk microphysics: surface precipitation rate, liquid water path (LWP), domain-maximum liquid water mixing ratio, cloud cover, and inversion height, shown from top to bottom. Shaded regions underlying the surface precipitation and LWP panels denote the observed (dark gray) and simulated (light gray) ranges from Ackerman et al. (2009). The inversion height is defined as the highest altitude for which total water mixing ratio qt > 8 g kg−1.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Profiles of cloud and rain properties from the bulk simulations are shown in Fig. 5. Note that bin microphysics simulations are also included in this figure, but will not be discussed in detail until section 5d. The nonprecipitating cases (defined here as quasi-steady-state surface precipitation rate less than 0.1 mm day−1; N35 and N55 bulk simulations) produce comparable profiles of liquid water mixing ratio qc and cloud fraction, and very little rain. Despite a steady-state surface precipitation rate R ∼ 1 mm day−1, the Na75_nx576 bin simulation stands out as the only “borderline” precipitating case in terms of rain fraction. The lowest concentration simulations (N10 and Na35) have lower peak qc and cloud fraction (maximum qc about 50% lower, cloud fraction about 0.2 lower), leading to lower mean LWP. Interestingly, the precipitating (Na35 and Na55) bin simulations exhibit a bimodal distribution of qr with one peak in cloud and another of comparable magnitude just below z ∼ 350 m. The lower peak is caused by drizzle falling through open cellular convection, which forms as a result of moistening and cooling due to evaporation of precipitation during the first 4 h of the heavily drizzling bin simulations. Bulk simulations have a single peak in qr near 700 m and a monotonic decrease toward the surface as well as a higher cloud base (400–500 m for bulk, 250–350 m for bin).

Fig. 5.
Fig. 5.

Mean profiles of (top) cloud and (bottom) rain properties from UCLA-LES simulations with bulk and bin microphysics. (left) Mean condensate mixing ratio, (center) condensate area fraction (defined as q > 0.05 g kg−1), and (right) frequency of area fraction exceeding an arbitrary threshold to be included in the bifractal analysis.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

b. Bifractal characteristics of cloud and rainwater from the bulk simulations

Next, we analyze the bifractal characteristics of qc and qr from the bulk simulations. Figure 6 and Table 2 show time-mean multifractal characteristics from bulk simulations with the SB scheme and one additional simulation using SAM with time-varying Nc and nx = 200 (Feingold et al. 2015). Note that the cloud water point for the N35_nx1152 simulation is not easily visible underneath other markers at (H1, C1) = (0.27, 0.05). The majority of points in Fig. 6 fall in the ellipse encompassing observed cloud water (H1, C1), with the smallest domain (nx144 and nx200) simulations appearing both smoother and more intermittent than their large domain counterparts. These deviant values for simulations with nx < 288 are attributed to reduced accuracy in the structure function fitting algorithm in domains with a small number of grid points. For the largest domain size, the spread in values as a function of concentration is quite small with mean cloud (H1, C1) = (0.29, 0.06) over all three nx1152 simulations. This suggests that simulated bulk cloud water effectively behaves like a passive scalar in turbulence. Note that the precipitating N10_nx1152 simulation is slightly but robustly more intermittent than the effectively nonprecipitating N35_nx1152 and N55_nx1152 runs. We interpret this increased intermittency as a consequence of precipitation more efficiently converting cloud to rainwater. Finally, it is worth noting that the SAM results (green stars) are comparable to those from UCLA-LES, although there is a slightly different relationship between qc and qr.

Fig. 6.
Fig. 6.

Domain-mean, time-mean (t > 2 h) bifractal characteristics from simulations with bulk microphysics. Filled symbols denote cloud water, open symbols rainwater. Error bars are the temporal standard deviation of vertical-mean (H1, C1). Note that simulation SAM_nx200_KK uses both a different model (SAM) and different microphysics scheme (KK) than the UCLA-LES bulk configuration. The N35 and N55 bulk runs did not produce sufficient precipitation to analyze qr.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Table 2.

Domain-mean, time-mean multifractal properties of qc and qr from LES.

Table 2.

For all cases with significant precipitation and in contrast to the observations, bulk rain is remarkably “cloud-like” in the bifractal plane. In particular, rain from UCLA-LES bulk simulations is smoother and less intermittent than cloud water. The SAM simulation has the smallest separation between cloud and rain of all simulations analyzed, with nearly identical intermittency. These are surprising results given that rain is significantly rougher and more intermittent than cloud water in the observations. It is unlikely that this “cloud-like” rain is a numerical artifact given the convergence of N10 simulations with nx > 144. It is possible that the full domain average view masks vertically varying bifractal properties; for example, perhaps “in-cloud” qr resembles cloud, but subcloud qr better matches observed behavior. To address this question we turn next to vertical profiles of cloud and rain bifractal properties.

c. Vertical profiles of cloud and rainwater bifractal properties from the bulk simulations

Time-mean profiles of H1 and C1 for qc are shown in Fig. 7 and for qr in Fig. 8. The vertical extent of the profiles in Fig. 7 can be interpreted as a rough estimate of cloud depth in normalized altitude. Beginning with qc in Fig. 7, we first note the increased smoothness and greater intermittency for nx144 simulations relative to the other runs throughout cloud depth while the larger domain simulations tend to cluster by precipitation state (i.e., profiles from the precipitating N10 simulations are similar and simulations from the nonprecipitating N35 and N55 simulations are similar). Despite the greater magnitude of H1 and C1, the nx144 simulations display a similar trend with respect to height as the larger domain simulations. Profiles of H1 and C1 for the higher concentration simulations (N35 and N55) exhibit a minimum near z/zi = 0.9 and increase moving downward. The H1 then decreases below z/zi = 0.65 while C1 monotonically increases. This trend is opposite that of Ma et al. (2017), who found lower (H1, C1) at cloud base than top, although the increasing trend in both H1 and C1 for z/zi > 0.9 [an approximation to the sublayers analyzed by Ma et al. (2017), corresponding to about 8 vertical levels in our model] is consistent with their results.

Fig. 7.
Fig. 7.

Profiles of (left) H1 and (right) C1 for bulk cloud water qc as a function of inversion height-normalized altitude z/zi.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Fig. 8.
Fig. 8.

Profiles of (left) H1 and (right) C1 for bulk rainwater qr as a function of inversion height-normalized altitude z/zi. The N35 and N55 bulk runs did not produce sufficient precipitation to analyze qr.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

The qc(H1, C1) profiles for the N10 bulk simulations are markedly different, with H1 nearly constant from 0.7 < z/zi < 0.9 and a generally decreasing trend in C1 from cloud top to cloud base. Both C1 and H1 are greater for the precipitating cases for z/zi = 0.7, consistent with the idea that precipitation influences both the horizontal and vertical distribution of qc. Interestingly, all simulations with nx > 144 have similar bifractal characteristics for qc near cloud base. We interpret this result as evidence of similar cloud-base dynamics/turbulence for all configurations, which we speculate could be unique to stratocumulus in which there is a balanced mixture of updrafts and downdrafts versus cumuliform clouds for which cloud base is more specifically associated with updrafts. At cloud top, on the other hand, precipitation formation and associated feedbacks with entrainment cause greater contrasts across different concentrations. The profiles from SAM (gray dotted curve) are qualitatively different than any of the UCLA-LES simulations. This is a consequence of temporally averaging over a simulation with time-varying domain-mean Nc, such that it is difficult to interpret the time-mean profiles unambiguously.

Turning next to the profiles of (H1, C1) for qr in Fig. 8, note that only the N10 and SAM simulations are plotted. The N35 and N55 simulations with SB microphysics did not exceed the threshold criterion for analysis (rain fraction > 0.015) at any level or time. In contrast, KK produces appreciable precipitation at both of these concentrations (as demonstrated by, e.g., Feingold et al. 2015; Zhou et al. 2018). The profile shapes are relatively uncomplicated compared to those for cloud water. The H1 increases from a minimum value ≈ 0.2 near cloud top and saturates at H1 ≈ 0.35 for z/zi < 0.7. The C1 oscillates about a roughly constant value throughout the profile. As with qc, the nx144 simulation has greater H1 and C1 compared to the other runs and the SAM output is qualitatively somewhat different, most notably with an increase of qr H1 near the surface. The spike to large C1 at z/zi = 0.3 is likely a numerical artifact from uncertainty in the structure function fitting procedure. Overall, there is relatively little vertical variability for qr(H1, C1) as compared to the mean profiles of qr and rain fraction (Fig. 5), and there is clustering of the larger domain results near the theoretical prediction for passive tracers in turbulence, (H1, C1) ≈ (0.33, 0.05).

d. General bin microphysics simulation results and comparison to bulk counterparts

Simulation output with bin microphysics is not expected to match perfectly with that from bulk simulations for several reasons. Bulk cloud and rainwater typically assume an analytic functional form for the drop size distributions (gamma for cloud, exponential for rain; Seifert and Beheng 2001). Because these distributions depend on only a few parameters (e.g., 2 or 3), the degrees of freedom to describe precipitation formation processes (i.e., autoconversion and accretion) are much more limited than for bin microphysics. Thus, there is variability in process rates in the bin simulations even for the same qc and Nc. Additionally, constant cloud drop number concentration is imposed for the bulk simulations, whereas for the bin simulations cloud droplet concentration evolves based on the source from activation of aerosols and sink from collision–coalescence and evaporation. This further limits the degrees of freedom in the bulk simulations. These details lead to significant differences in the evolution of micro and macrophysical properties in the bin runs compared to bulk runs for the same domain size (e.g., profiles of qc, qr and cloud/rain area fraction and time series of surface precipitation rate and LWP; see Fig. 5 for profiles and Fig. 9 for time series).

Fig. 9.
Fig. 9.

Time series of UCLA-LES output with bin and bulk microphysics for domains nx ≥ 576. Panels follow Fig. 4. Shaded regions underlying the surface precipitation and LWP panels denotes the observed (dark gray) and simulated (light gray) ranges from Ackerman et al. (2009).

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Interestingly, Fig. 10 shows that both bulk and bin simulations display similar nonmonotonicity in the LWP–rainwater path (RWP) relationship with increasing drop/aerosol concentration. However, commensurate with increased LWP (Fig. 9), bin runs exhibit much greater precipitation rates than the SB bulk runs for a given drop number concentration (Fig. 10). In addition to greater rain rates, there are also differences in cloud morphology visible in LWP snapshots (Fig. 11) from the bulk N10_nx576 and bin Na55_nx576 simulations at 6-h run time when both simulations have domain-mean RWP ≈ 10 g m−2. Based on all of these considerations, it is expected that bin and bulk qc and qr will exhibit different multifractal characteristics.

Fig. 10.
Fig. 10.

Comparison of (a) time-mean, domain-mean cloud liquid water path and rainwater path and (b) aerosol and cloud number mixing ratio from UCLA-LES bulk (red markers) and bin (black markers) simulations with nx = 576. Time averaging was performed over hours 3–12.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Fig. 11.
Fig. 11.

Snapshots of cloud LWP at 6 h for the (a) bulk N10_nx576 and (b) bin Na55_nx576 simulations. Domain-mean rainwater path is approximately 10 g m−2 for both simulations at this time.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

e. Bifractal characteristics of bin cloud and rainwater

Figure 12 shows time-mean, domain-mean multifractal properties of bin qc and qr with (H1, C1) from the bulk nx576 and nx1152 simulations for reference. First, bin cloud C1 is quite variable and demonstrates an inverse dependence on Na while H1 falls in a narrow range (0.34 < H1 < 0.41). Notably, only the Na75_nx576 qc point falls within the range of observed (H1, C1) values. This is, perhaps not coincidentally, the aerosol number mixing ratio used by Ackerman et al. (2009) for the “large” aerosol mode (mode radius r = 0.06 μm) to arrive at domain-mean Nc ≈ 55 mg−1. This rather large sensitivity of cloud C1 to Na suggests that the modeling framework used to simplify the microphysical representation (i.e., constant aerosol, no aerosol processing) and maintain steady state (by nudging T and qυ) may lead to different multifractal behavior than for a transient state, particularly since multifractal properties are thought to depend on mesoscale cloud organization.

Fig. 12.
Fig. 12.

Mean bifractal characteristics from simulations with bin and bulk microphysics. As in Fig. 6, filled symbols denote cloud water and open symbols rainwater. Error bars are the temporal standard deviation of vertical-mean (H1, C1).

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

The statistics of bin qr, on the other hand, demonstrate improvement over the bulk simulations with significantly higher C1—in fact, somewhat higher than observed. While bin qc exhibits an inverse relationship between Na and C1, bin qr shows an inverse relationship between Na and H1. Given the lack of precipitation in the Na35 and Na55 bulk cases, it is not possible to infer whether a similar relationship exists for bulk output. Nevertheless, rain H1 from both bin and bulk simulations is too smooth compared to observations, and for both configurations bears more of a resemblance to cloud H1. This raises an interesting question: If bin and bulk schemes both produce smoother rain H1 than observed, is it possible that the observations are “wrong”? If the only observations available were from aircraft, a plausible argument could be made that sample volume limitations prevent accurate diagnosis of rain (H1, C1). However, the consistency of airborne observations with radar, which has a comparable sample volume to an LES grid cell and is sensitive to the largest/rarest drizzle drops, suggests that the observed rain (H1, C1) is robust.

f. Vertical profiles of bin cloud and rainwater bifractal properties

Figures 13 and 14 present profiles of bin cloud and rain (H1, C1), similar to Figs. 7 and 8. The bulk nx576 simulations are included for reference. As shown in Fig. 12, bin qc is both smoother and more intermittent on average than the bulk runs. Note also that bin runs have “cloud” water extending nearly to the surface; this is a consequence of using a simple size threshold to separate “rain” and “cloud” categories. There are some parallels that can be drawn between the profiles of bin and bulk cloud (H1, C1): the lowest drop/aerosol concentration simulations for each scheme exhibit the highest (H1, C1) values, and the shape of the profiles for the lowest concentrations are qualitatively different than for the higher number concentration, more lightly drizzling configurations. For the Na35_nx576 case, H1 monotonically decreases below z/zi = 0.85 and there is a marked peak in C1 at z/zi = 0.7 that is totally absent from all other cases (both bulk and bin). This might reflect an overly vigorous rain formation process pushing the Na35_nx576 simulation into an unrealistic region of bifractal space.

Fig. 13.
Fig. 13.

Profiles of (left) H1 and (right) C1 for bulk and bin cloud water as a function of inversion height-normalized altitude z/zi.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Fig. 14.
Fig. 14.

Profiles of (left) H1 and (right) C1 for bulk and bin rainwater as a function of inversion height-normalized altitude z/zi.

Citation: Journal of the Atmospheric Sciences 79, 1; 10.1175/JAS-D-21-0134.1

Bin rain produces much noisier profiles of (H1, C1) than the bulk N10_nx576 simulation (Fig. 14), but the vertical variation of H1 matches reasonably well with bulk output in cloud. Descending from cloud top, bin rain H1 increases through cloud before shifting to lower values around z/zi = 0.5 and is relatively constant below. The profiles of bin rain C1 are considerably more variable than the bulk results, which have a nearly constant value of C1 ≈ 0.05 throughout the boundary layer. Note that the Na75_nx576 bin simulation has very low rain fraction on average (Fig. 5), perhaps explaining the volatility of the C1 profile for this case. In cloud (z/zi ≳ 0.5), qr is substantially more intermittent than below cloud for the Na35 and Na75 cases. Below cloud, bin rain C1 is generally consistent with the observed range (roughly 0.13 < C1 < 0.2). Both the Na35 and Na55 simulations have a peak in C1 near cloud base at (z/zi ≈ 0.5, not shown), perhaps corresponding to a region of enhanced asymmetries of updrafts and downdrafts. It is not clear why Na55 exhibits a different vertical structure with lower C1 in cloud than either Na35 or Na75.

6. Discussion

Modeled cloud water generally has similar multifractal structure to observed clouds (Figs. 6 and 12), whether using simple saturation adjustment (bulk) or a more detailed activation/condensation scheme (bin). It is expected from spectral analysis that cloud water should bear a close resemblance to passive scalars in turbulence (i.e., the spectral slope of qc is close to −5/3), from which we may infer that the spatial structure of PBL cloud water primarily responds to that of the fluctuations of vertical velocity that induce supersaturation.

The increased smoothness of bin results (for which H1 is routinely greater than the theoretical prediction of 0.33 for passive scalars) may result from numerical errors that cause the cloud field to be overly diffusive (Morrison et al. 2018; Lee et al. 2021). In particular, Lee et al. (2021) posit that erroneous DSD broadening due to condensation is especially pronounced when collision–coalescence is active, and all bin simulations exhibit nontrivial surface accumulation. The thermodynamic nudging and lack of aerosol processing are likely the main contributors to the sustained, unrealistically high surface precipitation rates in the Na35 and Na55 simulations (approximately 100 mm day−1, Fig. 9), from which we infer that precipitation appears to be a dominant factor in the increase in bin cloud C1 with decreasing Nc (Fig. 12). The nudging approach (taken to maintain PBL thermodynamic state and a constant inversion height) may also have dampened the development of cellular organization by preventing mesoscale variability in cloud-top height. Evidence for this interpretation can be found in Fig. 9, from which it can be seen that cloud cover was not appreciably below 90% for any of the simulations and the inversion height zinv was maintained between 780 and 795 m for the entirety of all simulations.

Overall, both the bulk and bin microphysics frameworks produce precipitation with markedly different (H1, C1) than observed drizzle. Bulk rain closely resembles cloud water, while bin rain exhibits improved intermittency but is far too smooth. The consistency of bin rain C1 as a function of Na is encouraging, although the proximity of rain and cloud points for the Na35_nx576 case shows that the bin scheme, too, can produce “cloud-like rain” or “rain-like cloud,” depending on one’s perspective. Additionally, the fact that bin rain C1 is usually higher than the observed range is concerning. The highest C1 observational data point was derived from radar time series. Radar reflectivity is more responsive to the rarest (largest) drops than qr, so the intermittency of reflectivity should be an upper bound on the C1 value a lower moment like qr can take. This could be tested in the future by applying a radar simulator to model output (as in Rémillard et al. 2017; Lee et al. 2019, 2021) and comparing with vertically pointing or scanning radar observations such as those analyzed by Lamer et al. (2019).

Given the poor performance of bulk and bin (in one word, Eulerian) microphysics parameterizations in matching observed precipitation structure, a major outstanding question is whether a Lagrangian microphysics scheme (e.g., Shima et al. 2009; Riechelmann et al. 2012; Dziekan et al. 2019) can produce improved agreement. Lagrangian schemes are not subject to many of the numerical issues inherent to Eulerian approaches, but they are not without drawbacks. With present-day computational resources, Lagrangian microphysics faces a trade-off of computational domain size (i.e., number of grid points) and superdroplet density, with at least a few hundred superdroplets per grid cell needed to produce robust statistics (Grabowski et al. 2019). There are several simplifications that can be made to concentrate available resources on precipitation formation, e.g., via Twomey CCN activation (as in Grabowski et al. 2018), which may make this approach more computationally tractable for large domains (several tens of kilometers in each horizontal direction) in the near term, and this will be explored in future work.

Beyond Lagrangian schemes, how else might an Eulerian approach to precipitation formation be improved? If bin model numerics are indeed a key factor limiting the ability of these models to represent observed rain bifractal properties, refining vertical grid spacing (Morrison et al. 2018) or bin mass grid spacing (Lee et al. 2021) can reduce numerical diffusion in condensational growth. For bulk schemes, allowing the rain DSD shape to evolve independently following the three-moment approach (e.g., Paukert et al. 2019) would add an additional degree of freedom to represent the rain DSD. Another possible approach is to directly introduce stochasticity to the microphysical process rates (viz., autoconversion and accretion). This is notably different from previous approaches, such as Stanford et al. (2019) who stochastically perturbed microphysical parameters (describing the mass–diameter relationship and fall speed of ice particles). In terms of “tuning” such a stochastic parameterization, the multifractal framework represents a potentially powerful diagnostic tool because of the orthogonality of additive and multiplicative noise in the bifractal plane: additive noise roughens a field (i.e., decreases H1) whereas multiplicative noise increases intermittency. Whether modification of collision–coalescence rates translates directly to improvement in the bifractal properties of the rain mixing ratio field is an open question.

7. Conclusions

A drizzling stratocumulus field based on the well-characterized DYCOMS-II RF02 case study (Ackerman et al. 2009) is simulated with a range of domain sizes and prescribed cloud drop/aerosol concentrations using bulk and bin microphysics to study the bifractal characteristics of clouds and rain. Domain-average cloud and rain statistics (condensate mixing ratio, area fraction, LWP) behave as expected as a function of cloud/aerosol concentration (Figs. 4, 5, 9), but the bifractal spatial statistics are more variable as a function of number concentration than domain size (Figs. 68, 1214). In particular, bifractal properties appear to be robust for nx576 and nx1152 while domains with nx < 288 are difficult to analyze because the range of scales is small and confidence is low in the structure function fitting algorithm.

The LES results presented in this study adequately simulate cloud liquid water structure for nearly all cases, but the model struggles to reproduce the observed structure of precipitation using either bulk or bin microphysics. We infer that this may represent a fundamental limitation of bulk and bin microphysics schemes. In general, we found that the bulk simulations more closely match observed multifractal properties for cloud water than rain. Previous analyses (Schertzer and Lovejoy 2011) have proposed that cloud water (broadly writ) is roughly equivalent to a passive scalar in turbulence; our results suggest that nonprecipitating cloud simulated by the bulk schemes is typically somewhat rougher and less intermittent than a passive scalar. However, the bin scheme, especially at low concentrations, produced cloud water that was somewhat too smooth (H1 ≈ 0.4) and far too intermittent (for the N35_nx576 simulation, C1 = 0.26—even higher than the corresponding C1 point for rain). Dwelling further on rain, the bulk simulations produced precipitation fields that were decidedly too “cloud-like,” with both smoothness and intermittency in the vicinity of the observed cloud water (H1, C1) range. Bin rain has much higher intermittency (0.2 < C1 < 0.24) but, unrealistically, is as smooth as the cloud water field.

Previous observational studies have placed minimal focus on the vertical structure of cloud multifractal properties, with the notable exception of Ma et al. (2017), who explored cloud water (H1, C1) in the entrainment interface layer near cloud top. Profiles of bulk cloud (H1, C1) (Fig. 7) show some similarities across all large-domain simulations near cloud base, but contrasting properties aloft as a function of the prescribed Nc, where the precipitating simulations (N10) exhibit both higher H1 and C1. Notably, the monotonically increasing relationship of (H1, C1) with height inferred by Ma et al. (2017) does not agree well with the simulation results presented here, although there is very limited evidence of increasing (H1, C1) for nonprecipitating cases above z/zi = 0.9. Profiles of bulk rain (H1, C1) (Fig. 8) were only obtained for the N10 and SAM_KK simulations due to the high sensitivity to aerosol of the SB scheme. Both multifractal parameters for rain are nearly constant with height. Cloud H1 increases descending from cloud top to z/zi = 0.8 and is constant below; cloud C1 is approximately constant throughout the profile. We speculate that this lack of vertical variability may be caused by the strong constraints on rain DSD evolution from imposing a fixed functional form for the DSD, and we will further explore this idea in future work.

Bulk and bin simulations display considerably different macro- and microphysical evolution in terms of LWP and surface rain accumulation, respectively, likely leading to different bifractal properties. Only the bulk simulations with Nc = 10 mg−1 produce significant precipitation, while the bin Na35 and Na55 cases have higher LWP and are strongly precipitating (R ∼ 100 mm day−1). The bin Na75 case exhibits low rain fraction but relatively high intensity surface precipitation (R ∼ 1 mm day−1), comparable to the bulk N10 simulations. The shape of the cloud (H1, C1) profiles for the bin Na55 and Na75 cases qualitatively match the bulk profiles, albeit shifted to higher values for both H1 and C1 compared to the bulk simulations. The vertical structure of (H1, C1) for the bin Na35 case is unique, with a nearly monotonic decrease in H1 descending from cloud top and a marked maximum in C1 at z/zi = 0.7 absent from all other simulations.

To return to the question posed in the introduction of whether LES or comparable fine-scale model output can be used to guide development of scale-aware PDF-based parameterizations, our results indicate that cloud water structure is indeed accurately represented. This appears to result from the fact that the multifractal behavior of cloud water mostly follows that of a passive scalar in turbulence, which LES is expected to capture well. However, the multifractal behavior of simulated rain does not agree with observations, being too smooth in the bin simulations and both too smooth and lacking intermittency in the bulk simulations, regardless of domain size. Thus, the LES cannot realistically represent higher-order statistics of the observed rain structure. This likely indicates problems in the representation of rain processes, which in general cannot be directly observed, particularly those processes that depend on the qr and qc distribution tails (e.g., autoconversion). Moreover, multifractal characteristics depend on the cloud and rain physical structure, providing much more information than PDFs alone. In this way, the multifractal analysis provides additional clues about process-level realism (or lack thereof) in the model. For example, it may be possible for a model to represent the qr PDF well but fail to capture the observed rain structure and hence its multifractal behavior, thereby revealing a process-level deficiency. This would suggest the model is “getting the right answer for the wrong reason” (in terms of the rain PDF). Future work will focus more on implications for model representations of cloud and rain processes. We also advocate using scanning radar observations both to characterize PDFs of cloud and rain properties as well as for higher-order multifractal analysis. For the latter, a wavelet technique for obtaining multifractal parameters may be desirable to improve computational efficiency over the structure function analysis used here.

Looking forward, several model development options have been proposed to test whether Eulerian microphysics parameterizations can be improved to better match the observed multifractal properties of rain. Some involve modifications of the modeling configuration, e.g., refining vertical grid spacing or using more complex surface forcing and radiation schemes to allow more degrees of freedom in terms of microphysical interactions with the thermodynamics and dynamics. Others are more targeted at microphysics: refining drop size spacing in bin schemes, allowing the shape of the rain DSD to evolve in bulk microphysics, or introducing stochasticity to the bulk collision–coalescence process rate formulations. Another approach is to use a Lagrangian microphysics scheme, which avoids many of the numerical problems of Eulerian bin microphysics (Grabowski et al. 2019) and is a promising avenue for anticipated future study of cloud and rain bifractal properties.

Acknowledgments.

The majority of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). MKW was supported in part by the U.S. Department of Energy’s Atmospheric System Research (ASR), an Office of Science Biological and Environmental Research program, under Grant DE-SC0020332. MKW and JT were also supported by the U.S. Department of Energy under Grant DE-SC0019242 and the National Science Foundation under Grant 1916619. HM was supported by the U.S. Department of Energy’s ASR under Grant DE-SC0020118. MKW thanks the NCAR Computational and Information Systems Laboratory for a postdoctoral computing allocation in 2019. ABD is supported by JPL’s SRTD program (PBL observation study). Finally, the authors thank Graham Feingold for providing SAM output and the three anonymous reviewers for their constructive feedback.

Data availability statement.

The UCLA-LES code (https://github.com/uclales/) and the bin microphysics code (https://www.esrl.noaa.gov/csd/staff/graham.feingold/code/) are available online. From the VOCALS experiment, raw rain drop data can be obtained at https://doi.org/10.5065/D6VH5M40, optical array probe processing software from https://github.com/abansemer/soda2, and 25 Hz Particulate Volume Monitor (PVM) cloud liquid water content data are available from the National Center for Atmospheric Research Earth Observing Laboratory (NCAR EOL) at https://doi.org/10.5065/D69K48JK. Sample-rate (500 Hz) PVM data are available on request from NCAR EOL (email: eol-archive@ucar.edu).

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