Due to their vast areal coverage (Warren et al., 1986, 1988; Hahn and Warren, 2007) and strong radiative cooling effect (Hartmann et al., 1992; Chen et al., 2000), small changes in the coverage or thickness of marine boundary layer (MBL) clouds could change the radiative energy budget significantly (Hartmann and Short, 1980; Randall et al., 1984) or even offset the radiative effects produced by increasing greenhouse gases (Slingo, 1990). The lifetime of MBL clouds remains an issue in climate models (Yoo and Li, 2012; Jiang et al., 2012; Yoo et al., 2013; Stanfield et al., 2014) and represents one of the largest uncertainties in predicting future climate (Wielicki et al., 1995; Houghton et al., 2001; Bony and Dufresne, 2005).

MBL clouds frequently produce precipitation, mostly in the form of drizzle (Austin et al., 1995; Wood, 2005a, 2012; Leon et al., 2008). A significant amount of drizzle evaporates before reaching the surface, for example, about ∼76 % over the Azores region in the northeast Atlantic (Wu et al., 2015), which provides a water vapor source for MBL clouds. Due to their pristine environment and their proximity to the surface, MBL clouds and precipitation are especially sensitive to aerosol perturbations (Platnick and Twomey, 1994). Thus, accurate prediction of precipitation is essential in simulating the global energy budget and in constraining aerosol indirect effects in climate projections.

Due to the coarse spatial resolutions of the general circulation model (GCM) grid, many cloud processes cannot be adequately resolved and must be parameterized. For example, warm-rain parameterizations in most GCMs treat the condensed water as either clouds or rain from the collision–coalescence process that is partitioned into autoconversion and accretion subprocesses in model parameterizations (Kessler, 1969; Tripoli and Cotton, 1980; Beheng, 1994; Khairoutdinov and Kogan, 2000; Liu and Daum, 2004). Autoconversion represents the process of drizzle drops being formed through the self-collection of cloud droplets and accretion represents the process where raindrops grow by collecting cloud droplets. Autoconversion mainly accounts for precipitation initiation, while accretion primarily contributes to precipitation intensity. Autoconversion is often parameterized as functions of cloud droplet number concentration (Nc) and cloud water mixing ratio (qc), while accretion depends on both cloud and rain water mixing ratios (qc and qr) (Kessler, 1969; Tripoli and Cotton, 1980; Beheng, 1994; Khairoutdinov and Kogan, 2000; Liu and Daum, 2004; Wood, 2005b). The majority of previous studies suggested that these two processes can be represented by power–law functions of cloud and precipitation properties (see Sect. 2 for details).

In conventional GCMs, the lack of information on the subgrid variance of clouds and precipitation leads to the unavoidable use of the grid-mean quantities (Nc‾, qc‾, and qr‾, where the overbar henceforth denotes grid mean) in calculating autoconversion and accretion rates. MBL cloud liquid water path (CLWP) distributions are often positively skewed (Wood and Hartmann, 2006; Dong et al., 2014a, b); that is, the mean value is greater than the mode value. Thus, the mean value only represents a relatively small portion of samples. Also, due to the nonlinear nature of the relationships, the two processes depend significantly on the subgrid variability and co-variability of cloud and precipitation microphysical properties (Weber and Quass, 2012; Boutle et al., 2014). In some GCMs, subgrid-scale variability is often ignored or hard coded using constants to represent the variabilities under all meteorological conditions and across the entire globe (Pincus and Klein, 2000; Morrison and Gettleman, 2008; Lebsock et al., 2013). This could lead to systematic errors in precipitation rate simulations (Wood et al., 2002; Larson et al., 2011; Lebsock et al., 2013; Boutle et al., 2014; Song et al., 2018), where GCMs are found to produce too-frequent but too-light precipitation compared to observations (Zhang et al., 2002; Jess, 2010; Stephens et al., 2010; Nam and Quaas, 2012; Song et al., 2018). The bias is found to be smaller when using a probability density function (PDF) of cloud water to represent the subgrid-scale variability in autoconversion parameterization (Beheng, 1994; Zhang et al., 2002; Jess, 2010), or more complexly, by integrating the autoconversion rate over a joint PDF of liquid water potential temperature and total water mixing ratio (Cheng and Xu, 2009).

Process rate enhancement factors (E) are introduced when considering subgrid-scale variability in parameterizing grid-mean processes and they should be parameterized as functions of the PDFs of cloud and precipitation properties within a grid box (Morrison and Gettleman, 2008; Lebsock et al., 2013; Boutle et al., 2014). However, these values in some GCM parameterization schemes are prescribed as constants regardless of surface or meteorological conditions (Xie and Zhang, 2015). Boutle et al. (2014) used aircraft in situ measurements and remote sensing techniques to develop a parameterization for clouds and rain, in which they not only consider the subgrid variabilities under different grid scales but also consider the variation of cloud and rain fractions. The parameterization was found to reduce the precipitation estimation bias significantly. Hill et al. (2015) modified this parameterization and developed a regime- and cloud-type-dependent subgrid parameterization, which was implemented to the Met Office Unified Model by Walters et al. (2017), who found that the radiation bias is reduced when using the modified parameterization. Using ground-based observations and retrievals, Xie and Zhang (2015) proposed a scale-aware cloud inhomogeneity parameterization that they applied to the Community Earth System Model (CESM) and found that it can recognize spatial scales without manual tuning and can be applied to the entire globe. The inhomogeneity parameter is essential in calculating enhancement factors, since they affect the conversion rate from clouds to rain liquid. Xie and Zhang (2015), however, did not evaluate the validity of CESM simulations from their parameterization; the effect of Nc variability or the effect of covariance of clouds and rain on the accretion process was not assessed. Most recently, Zhang et al. (2018) derived the subgrid distribution of CLWP and Nc from the MODIS cloud product. They also studied the implication of subgrid cloud property variations for the simulation of autoconversion, in particular the enhancement factor, in GCMs. For the first time, the enhancement factor due to the subgrid variation of Nc was derived from satellite observation, and results reveal several regions downwind of biomass burning aerosols (e.g., Gulf of Guinea, east coast of South Africa), air pollution (i.e., East China Sea), and active volcanos (e.g., Kilauea in Hawaii and Ambae in Vanuatu) where the enhancement factor due to Nc is comparable or even larger than that due to CLWP. However, one limitation of Zhang et al. (2018) is the use of passive remote sensing data only, which cannot distinguish cloud and rain water.

Dong et al. (2014a, b) and Wu et al. (2015) reported MBL cloud and rain properties over the Azores and provided the possibility of calculating the enhancement factors using ground-based observations and retrievals. In this study, a joint retrieval method to estimate qc and qr profiles is proposed based on existing studies (Appendix A). Most of the calculations and analyses in this study are based on the Morrison and Gettelman (2008, MG08 hereafter) scheme. The enhancement factors in several other schemes are also discussed and compared with the observational results, and the approach in this study can be repeated for other microphysics schemes in GCMs. This paper is organized as follows: Sect. 2 includes a summary of the mathematical formulas from previous studies that can be used to calculate enhancement factors. Ground-based observations and retrievals are introduced in Sect. 3. Section 4 presents results and discussions, followed by a summary and conclusions in Sect. 5. The retrieval method used in this study is in Appendix A.

Autoconversion and accretion rates in GCMs are usually parameterized as power–law equations (Tripoli and Cotton, 1980; Beheng, 1994; Khairoutdinov and Kogan, 2000; Liu and Daum, 2004): ∂qr∂tauto=Aqc‾a1Nc‾a2, ∂qr∂taccr=Bqc‾qr‾b, where A, a1, a2, B, and b are coefficients in different schemes listed in Table 1. qc‾, qr‾, and Nc‾ are grid-mean cloud water mixing ratio, rain water mixing ratio, and droplet number concentration, respectively. Because it is widely used in model parameterizations, the detailed results from the Khairoutdinov and Kogan (2000) parameterization that has been used in the MG08 scheme will be shown in Sect. 4, while a summary will be given for other schemes.

Ideally, the covariance between physical quantities should be considered in the calculation of both processes. However, qc‾ and Nc‾ in Eq. () are arguably not independently retrieved in our retrieval method, which will be introduced in this section and Appendix A. Thus, we only assess the individual roles of qc and Nc subgrid variations in determining the autoconversion rate. qc and qr, on the other hand, are retrieved from two independent algorithms, as shown in Dong et al. (2014a, b), Wu et al. (2015), and Appendix A. The effect of the covariance of qc and qr on accretion rate will be assessed.

At the subgrid scale, the PDFs of qc and Nc are assumed to follow a gamma distribution based on observational studies of optical depth in MBL clouds (Barker et al., 1996; Pincus et al., 1999; Wood and Hartmann, 2006): Px=ανΓνxν-1e-αx, where x represents qc or Nc with grid-mean quantity qc‾ or Nc‾, represented by μ, α=ν/μ is the scale parameter, σ2 is the relative variance of x (= variance divided by μ2), and ν=1/σ2 is the shape parameter. ν is an indicator of cloud field homogeneity, with large values representing homogeneous and small values indicating inhomogeneous cloud fields.

By integrating autoconversion rate, Eq. (), over the grid-mean rate, Eq. (), with respect to subgrid-scale variation of qc and Nc, the autoconversion rate can be expressed as ∂qr∂tauto=Aμqca1μNca2Γν+aΓννa, where a=a1 or a2. Comparing Eq. () to (), the autoconversion enhancement factor (Eauto) can be given with respect to qc and Nc: Eauto=Γν+aΓννa. In addition to fitting the distributions of qc and Nc, we also tried two other methods to calculate Eauto. The first is to integrate Eq. () over the actual PDFs from observed or retrieved parameters and the second is to fit a lognormal distribution for subgrid variability as has been done in other studies (e.g., Lebsock et al., 2013; Larson and Griffin, 2013). It is found that all three methods provide similar results. In this study, we use a gamma distribution that is consistent with MG08. Also, note that, in the calculation of Eauto from Nc‾, the negative exponent (-1.79) may cause singularity problems in Eq. (). When this situation occurs, we perform direct calculations by integrating the PDF of Nc‾ rather than using Eq. ().

To account for the covariance of microphysical quantities in a model grid, it is difficult to apply a bivariate gamma distribution due to its complex nature. In this study, the bivariate lognormal distribution of qc and qr is used (Lebsock et al., 2013; Boutle et al., 2014) and can be written as Pqc‾,qr‾=12πqc‾qr‾σqcσqr1-ρ2exp⁡-1211-ρ2ln⁡qc‾-μqcσqc2-2ρln⁡qc‾-μqcσqcln⁡qr‾-μqrσqr+ln⁡qr‾-μqrσqr2, where σ is standard deviation and ρ is the correlation coefficient of qc and qr.

Similarly, by integrating the accretion rate in Eq. () from Eq. (), we get the accretion enhancement factor (Eaccr) of Eaccr=1+1νqc1.152-1.1521+1νqr1.152-1.152exp(ρ1.152ln⁡1+1νqcln(1+1νqr)).

The datasets used in this study were collected at the Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) Mobile Facility (AMF), which was deployed on the northern coast of Graciosa Island (39.09∘ N, 28.03∘ W) from June 2009 to December 2010 (for more details, please refer to Rémillard et al., 2012, Dong et al., 2014a, and Wood et al., 2015). The detailed operational status of the remote sensing instruments at AMF is summarized in Fig. 1 of Rémillard et al. (2012) and discussed in Wood et al. (2015). The ARM Eastern North Atlantic (ENA) site was established on the same island in 2013 and provides long-term continuous observations.

The cloud-top heights (Ztop) were determined from the W-band ARM cloud radar (WACR) reflectivity and only single-layered and overcast low-level clouds with Ztop≤3 km were selected (the detailed selection criteria can be found in Dong et al., 2014a, b). Cloud-base heights (Zbase) were detected by a laser ceilometer (CEIL) and the cloud thickness was simply the difference between cloud-top and cloud-base heights. The cloud liquid water path (CLWP) was retrieved from microwave radiometer (MWR) brightness temperatures measured at 23.8 and 31.4 GHz using a statistical retrieval method with an uncertainty of 20 g m-2 for CLWP<200 g m-2 and 10 % for CLWP>200 g m-2 (Liljegren et al., 2001; Dong et al., 2000). Precipitating status is identified through a combination of WACR reflectivity and Zbase. As in Wu et al. (2015), we labeled the status of a specific time as “precipitating” if the WACR reflectivity below the cloud base exceeded -37 dBZ. Note the differences of the reflectivity thresholds used here and in other studies. For example, these were -15 dBZ in Sauvageot and Omar (1987), -17 dBZ in Frisch et al. (1995), -19 to -16 dBZ in Wang and Geerts (2003), and -30 dBZ or lower in Kollias et al. (2011). The threshold used in this study is set at the cloud base rather than for the entire cloud layer as in the abovementioned studies. The -37 dBZ threshold is a statistical value from WACR observations over the Azores presented by Wu et al. (2015, Fig. 2a), in which it is found that using a higher threshold will miss a significant number of drizzling events, especially the clouds with virga.

The ARM merged sounding data have a 1 min temporal and 20 m vertical resolution below 3 km (Troyan, 2012). In this study, the merged sounding profiles are averaged to 5 min resolution. Pressure and temperature profiles are used to calculate air density (ρair) profiles and to infer adiabatic cloud water content.

Cloud droplet number concentration (Nc) is retrieved using the methods presented in Dong et al. (1998, 2014a, b) and is assumed to be constant with height. Vertical profiles of cloud and rain liquid water content (CLWC and RLWC) are retrieved by combining WACR reflectivity and CEIL attenuated backscatter, and by assuming adiabatic growth of cloud water. A detailed description is presented in Appendix A with the results from a selected case. The CLWC and RLWC values are transformed to qc and qr by dividing by air density (e.g., qc(z)=CLWC(z)/ρair(z)).

The estimated uncertainties for the retrieved qc and qr are 30 % and 18 %, respectively (see Appendix A). We used the estimated uncertainties of qr and qc as inputs of Eqs. () and () to assess the uncertainties of Eauto and Eaccr. For instance, 1±0.3qc is used in Eq. () and the mean differences are then used as the uncertainty of Eauto. The same method is used to estimate the uncertainty for Eaccr.

The autoconversion and accretion parameterizations dominate at different levels in a cloud layer. Autoconversion dominates around the cloud top, where drizzle drops form by the self-collection of cloud droplets, and accretion is dominant in the middle and lower parts of the cloud where raindrops grow by collecting cloud droplets. In accordance with the physical processes, we estimate autoconversion and accretion rates at different levels of a cloud layer in this study. The averaged qc values within the top five range gates (∼215 m thick) are used to calculate Eauto. To calculate Eaccr, we use the averaged qc and qr within five range gates around the maximum radar reflectivity. If the maximum radar reflectivity appears at the cloud base, then five range gates above the cloud base are used.

The ARM merged sounding data are also used to calculate lower tropospheric stability (LTS=θ700hPa-θ1000hPa), which is used to infer the boundary layer stability. In this study, unstable and stable boundary layers are defined as LTS less than 13.5 K and greater than 18 K, respectively, and an environment with an LTS between 13.5 and 18 K is defined as mid-stable (Wang et al., 2012; Bai et al., 2018). Enhancement factors in different boundary layers are summarized in Sect. 4.2 and may be used as reference for model simulations. Further, two regimes are classified: CLWP greater than 75 g m-2 as precipitating and CLWP less than 75 g m-2 as non-precipitating (Rémillard et al., 2012).

To evaluate the dependence of autoconversion and accretion rates on subgrid variabilities for different model spatial resolutions, an average wind speed within a cloud layer was extracted from merged sounding and used in sampling observations over certain periods to mimic different grid sizes in GCMs. For example, 2 h of observations correspond to a 72 km horizontal equivalent grid box if mean horizontal in-cloud wind speed is 10 m s-1, and if the wind speed is 5 m s-1, 4 h of observations are needed to mimic the same horizontal equivalent grid. We used six horizontal equivalent grid sizes (30, 60, 90, 120, 150, and 180 km) and mainly show the results from 60 and 180 km horizontal equivalent grid sizes in Sect. 4. For convenience, we use “equivalent size” to imply “horizontal equivalent grid size” from now on.

In this section, we first show the data and methods using a selected case, followed by statistical analysis based on 19 months of data and multiple time intervals.

To better understand the influence of subgrid cloud variations on the warm-rain process simulations in GCMs, we investigated the warm-rain parameterizations of autoconversion (Eauto) and accretion (Eaccr) enhancement factors in MG08. These two factors represent the effects of subgrid cloud and precipitation variabilities when parameterizing autoconversion and accretion rates as functions of grid-mean quantities. Eauto and Eaccr are prescribed as 3.2 and 1.07, respectively, in the widely used MG08 scheme. To assess the dependence of the two parameters on subgrid-scale variabilities, we used ground-based observations and retrievals collected at the DOE ARM Azores site to reconstruct the two enhancement factors in different equivalent sizes.

From the retrieved qc and qr profiles, the averaged qc values within the top five range gates are used to calculate Eauto, and the averaged qc and qr within five range gates around maximum reflectivity are used to calculate Eaccr. The calculated Eauto values from observations and retrievals increase from 1.96 at an equivalent size of 30 km to 3.18 at an equivalent size of 150 km. These values are 38 % and 0.625 % lower than the prescribed value of 3.2. The prescribed value in MG08 represents well the large grid sizes in GCMs (e.g., 1802 km2 grid). On the other hand, the Eaccr values increase from 1.53 at an equivalent size of 30 km to 1.76 at an equivalent size of 180 km, which are 43 % and 64 % higher than the prescribed value (1.07). The higher Eauto and lower Eaccr prescribed in GCMs help to explain the issue of too-frequent precipitation events with too-light precipitation intensity. The ratios of rain to cloud liquid water increase with increasing Eaccr from 0 to 2 and then decrease thereafter; the values at Eaccr=1.07 and Eaccr=2.0 are 0.063 and 0.142, further underscoring that the prescribed value of Eaccr=1.07 is too small to simulate correct precipitation intensity in models.

To further investigate what conditions and parameters can significantly influence the enhancement factors, we classified low-level clouds according to their boundary layer conditions and CLWPs. Both Eauto and Eaccr increase when the boundary layer conditions become less stable, and the values are larger in precipitating clouds (CLWP>75 g m-2) than those in non-precipitating clouds (CLWP<75 g m-2). The Eauto values in both stable and mid-stable boundary layer conditions are smaller than the prescribed value of 3.2, while those in unstable boundary layer conditions are significantly larger than 3.2 regardless of whether or not the cloud is precipitating (Table 2). All Eaccr values are greater than the prescribed value of 1.07. Therefore, the selection of Eauto and Eaccr values in GCMs should be regime-dependent, which also has been suggested by Hill et al. (2015) and Walters et al. (2017).

This study, however, did not include the effect of uncertainties in GCM simulated cloud and precipitation properties on subgrid-scale variations. For example, we did not consider the behavior of the two enhancement factors under different aerosol regimes, a condition which may affect the precipitation formation process. The effect of aerosol–cloud–precipitation interactions on cloud and precipitation subgrid variabilities may be of comparable importance to meteorological regimes and precipitation status and deserves further study. Other than the large-scale dynamics, e.g., LTS in this study, upward/downward motion at subgrid scale may also modify cloud and precipitation development and affect the calculations of enhancement factors. The investigation of the dependence of Eauto and Eaccr on aerosol type and concentration as well as on vertical velocity would be a natural extension and complement the current study. In addition, other factors may also affect precipitation frequency and intensity even under the same aerosol regimes and even if the clouds have similar cloud water content. Wind shear, for example, as presented in Wu et al. (2017), is an external variable that can affect precipitation formation. Further studies are needed to evaluate the role of the covariance of qc and Nc at subgrid scales on Eauto, which is beyond the scope of this study and requires independent retrieval techniques.