1. Introduction
Operational regional forecasts at the Met Office make use of a cloud fraction (cf) parameterization, even at convection-permitting scales (Lean et al. 2008; Clark et al. 2016; Kain et al. 2017; Bush et al. 2020, hereafter B20). Parameterizing subgrid cloud variability, rather than assuming grid boxes to be fully cloudy or fully clear, is beneficial to simulations of surface radiation, fog, and cloud cover for grid spacings down to 100 m (Hughes et al. 2015; Boutle et al. 2016; B20). The midlatitude operational regional model configuration at the Met Office parameterizes cloud fraction based on Smith (1990). This scheme diagnoses cloud cover based on a unimodal, symmetrical subgrid distribution of the saturation departure. This approach allows clouds to form in a grid box that is subsaturated in a mean sense.
However, based on comparisons against aircraft observations, Wood and Field (2000) found that the Smith (1990) scheme underestimates the cloud cover and predicts too small a cloud cover for a given liquid water content. Therefore, operational regional forecasts use an empirical adjustment to the cloud cover, following Wood and Field (2000). This approach performs very well for conditions typical of the midlatitudes (B20). However, over the tropics the performance of the empirical adjustment to the cloud cover is rather poor (B20). This highlights the need for a more physically based and regime-dependent approach to improve cloud forecasts more generally.
Observations from lidar and aircraft (Wood and Field 2000; Wulfmeyer et al. 2010; Behrendt et al. 2015), as well as large-eddy simulations (Zhu and Zuidema 2009; Griewank et al. 2018), indicate that moisture and temperature variability in the atmosphere cannot typically be represented by a symmetric unimodal probability density function (PDF). It therefore seems plausible that the poor performance of the Smith (1990) scheme is related to its assumption of a unimodal symmetric PDF.
Many advanced parameterizations for boundary layer clouds have been developed, often allowing for skewed and multimodal distributions (Lewellen and Yoh 1993; Randall et al. 1992). Many of these approaches unify the boundary layer turbulence, shallow convection and the cloud scheme in a so-called assumed PDF approach (Golaz et al. 2002; Bogenschutz and Krueger 2013). Such schemes predict the subgrid vertical transport of moisture, heat and clouds and require a consistent underlying joint PDF of thermodynamical variables and vertical velocity. Averages and higher-order moments such as (co)variances of prognostic variables, are advected through the model domain and these moments are in turn used to select a member from a family of assumed PDF shapes. Other, slightly simpler schemes only predict tendencies of the moments of a bivariate or univariate PDF of temperature and/or moisture (Tompkins 2002) or of the cf and water contents directly (Tiedtke 1993; Wilson et al. 2008).
While such schemes are attractive mathematically, there are still a number of assumptions about the PDF parameters and the closure, while process rates for the moments of the PDF are difficult or impossible to observe (although they can be inferred from large-eddy simulations). Furthermore, in an operational context, these schemes can be less attractive given their computational cost, but also the difficulty to determine which parameters are most suitable for the inevitable tuning.
This paper seeks a more pragmatic and diagnostic approach to allow for skewed and bimodal subgrid saturation departure PDFs in cloud schemes. We emphasize that the scheme proposed here only determines the liquid cloud cover and water content, while the Lock et al. (2000) boundary layer parameterization still handles subgrid turbulent transport of moisture and temperature.
The scheme presented here identifies entrainment zones below any sharp inversion. Within the entrainment zone, air from the mixed layer (ML) is allowed to coexist with pockets of air entrained from above the inversion, giving rise to the often observed bimodal distribution of subgrid variability (Wulfmeyer et al. 2016). Weights are applied to the two modes so that the gridbox mean saturation departure is conserved.
Each individual mode is represented by a unimodal Gaussian distribution with variances obtained from turbulent theory, extending the parameterization for mixed-phase clouds in turbulent environments by Furtado et al. (2016) to all liquid clouds. Cloud fraction and liquid water content for each mode is calculated separately and combined using their respective weights.
This paper is the first of two and, after reviewing the operational cloud cover scheme, describes the fundamentals of the new scheme. Aircraft and ground-based observations from the Midlatitude Continental Convective Clouds Experiment (MC3E; Jensen et al. 2016) campaign are used to perform a process-based evaluation of the new scheme and compare it against the operational scheme. This paper is accompanied by a companion paper (Van Weverberg et al. 2021, hereafter Part II) which, using more observations, provides a thorough evaluation of the cloud properties and assesses the scale awareness of the bimodal and several other cloud schemes. Ultimately, the aim of the new diagnostic cloud scheme would be improved performance in different climate regimes, including the tropics, through a more physically based approach. However, the papers presented here focus on verification against observations in the midlatitudes to understand whether the new scheme can compete with existing operational approaches. Future studies will focus on other regions and climate conditions.
The remainder of this paper (Part I) is organized as follows. An overview of the observation campaign and the model setup is provided in section 2. The following section first revisits the basic principles of the Smith (1990) cloud scheme and explains operationally used additions (section 3). This section then outlines the structure of the bimodal cloud scheme. Three different cloud scheme configurations are evaluated against observed cloud properties in section 4. The main findings from Part I and the benefits and limitations of the bimodal scheme are highlighted in section 5.
2. Observations and model setup
a. MC3E intensive observation period
Simulations are performed for MC3E (Jensen et al. 2016), a measurement campaign conducted at the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) Central Facility site in Oklahoma from 22 April to 6 June 2011. A broad range of routinely operated state-of-the-art instruments are available at this site, complemented with more frequent sounding observations, aircraft data and a detailed ground-based water content retrieval product during the MC3E campaign.
b. Observations
1) ARSCL and microbase
Vertically distributed cloud locations and water contents are derived from the Active Remote Sensing of Clouds (ARSCL) and the Microbase ARM synergistic data products, respectively.
ARSCL provides a vertically distributed cloud mask, based on cloud radar, lidar, ceilometer, and radiometer data (Clothiaux et al. 2000), with high temporal (4 s) and vertical (30 m) resolution. Given observed wind speeds at each model level and a 1 km model grid length, a moving time window is applied to this cloud mask to establish the “observed” cf. This approach has been widely used in model evaluation studies (Illingworth et al. 2007; Morcrette et al. 2012; Van Weverberg et al. 2015) and gives reliable results for low-order moments and long time series (Grüetzun et al. 2013). Wu et al. (2014) compared 6 different estimates of cf near the SGP for 10 years and found ARSCL-derived cf to be typically about 8% larger than satellite retrievals.
The Microbase provides profiles of liquid and ice water content, based on ceilometer, micropulse lidar, microwave radiometer, balloon soundings, and vertically pointing Ka-band cloud radar (KAZR) data at the SGP (Dunn et al. 2011) with the same temporal and spatial resolution as ARSCL. As for the ARSCL-derived cf, the Microbase retrievals were time averaged using the observed wind speeds. Zhao et al. (2014) estimated the random uncertainties in the retrievals of liquid and ice water content (LWC and IWC) to be around 15% and 55%, respectively.
Since the observations become unreliable when the radar radomes are wet due to precipitation, a filtering procedure is applied to the observation data. For consistency, the same filtering procedure is applied to the simulations. Data are only retained when no precipitation is detected by the collocated rain gauges and when the first observation level of ARSCL and Microbase (~130 m above the surface) is clear of hydrometeors. It should be noted that if any observed or simulated profile at a particular time is flagged with this procedure, all observed and simulated profiles for that time are removed from the analysis. Hence, the observed and simulated sample sizes are identical. Furthermore, the data become less reliable in mixed-phase clouds. Hence, only data at temperatures warmer than 0°C were retained, using temperatures from the interpolated soundings product (Toto and Jensen 2016). Due to these observational limitations, the evaluation here focuses on liquid-only clouds. However, Part II provides an in-depth evaluation of all cloud phases.
2) Aircraft data
In situ cloud properties were sampled with the University of North Dakota Citation aircraft for a number of cases during the MC3E campaign. Horizontal flight legs through warm clouds (>0°C) from all 15 flights that took place during MC3E were selected. Noting the instrument sampling rate of 1 Hz and the aircraft speed, the aircraft data were divided into sections representative of a 1 km model grid length. Liquid water content from the Cloud Droplet Probe was used, while cf was determined as the fraction of time in each section that the cloud droplet number exceeded 5 cm−3, similar to Wood and Field (2000). Total relative humidity was calculated from temperature and pressure, and the dewpoint retrieved from a chilled-mirror dewpoint hygrometer. The random humidity error for a 1 Hz sampling rate is on the order of 5%.
c. Model description
Simulations have been performed with the Met Office Unified Model (UM; Brown et al. 2012) in hindcast mode for the entire 6-week period of the MC3E campaign.
Convection-permitting simulations with the UM are performed operationally over many regions in the world, including the United Kingdom, Australia, New Zealand, India, South Korea, parts of Southeast Asia, and South Africa. All simulations presented here are for a nested high-resolution configuration, close to that of the widely used RA2–Midlatitude (RA2-M) configuration (at UM version 11.4, B20). The three nested domains have grid spacings of 4, 2, and 1 km and time steps of 150, 100, and 60 s, respectively.
Domain sizes are 400 × 400 grid points for the 4 and 2 km domains and 600 × 600 grid points for the 1 km domain, all centered around the SGP Central Facility (at 36.6°N, 97.5°W). The global model (using the GA6 configuration (Walters et al. 2017) at a resolution of N512, ≃30 km grid spacing near the SGP) is used to provide lateral boundary conditions for the 4 km domain and all other nested domains are driven by the next resolution up. Global simulations were initialized from the operational global UM analysis and the boundaries of each regional domain were updated hourly. Hindcasts were initialized at 0000 UTC and run for 30 h, omitting the first 5 h as a spinup. All configurations have 70 stretched levels in the vertical with a top at 80 km for the global and 40 km for the regional domain. The analysis presented here focuses on the 1 km domain, which is closest to the operational grid spacing (B20). The other domains will be analyzed in Part II to investigate the scale-awareness of the cloud schemes.
The regional RA2-M configuration does not parameterize shallow or deep convection and uses the blended turbulence boundary layer parameterization described in Boutle et al. (2014a). This scale-aware scheme combines the nonlocal 1D Lock et al. (2000) scheme with a subgrid-turbulence scheme for large-eddy simulations. The nonlocal component of this scheme diagnoses one of seven boundary layer types from the thermodynamic profile before applying a type-specific nonlocal eddy diffusivity profile. The microphysics uses a single-moment bulk approach including cloud water, rain, graupel and a combined ice and snow hydrometeor type, based on Wilson and Ballard (1999). This model version includes the revised ice particle size distribution by Field et al. (2007) and improvements to warm-rain production (Boutle et al. 2014b) and the raindrop size distributions (Abel and Boutle 2012).
The radiation scheme is called every 15 min. The liquid cloud droplet number is calculated from aerosol climatologies, consistent with the microphysics, and is combined with the liquid water content to produce an effective radius for the shortwave (SW) and longwave (LW) radiation calculations. A heterogeneity factor of 0.7 is used to represent in-cloud liquid water content variability in the radiation scheme, following Cahalan et al. (1994).
The operational RA2M configuration uses the Smith (1990) cf parameterization, with two important modifications: the empirically adjusted cloud fraction (EACF) and an area-cloud fraction parameterization (ACF). These modifications are applied operationally to correct for climatological biases in the cloud cover, as will be further detailed in the next section.
Apart from the above RA2M configuration, two more configurations of the UM are integrated, differing only in their cf parameterization assumptions. The second set of simulations, referred to as NOEACF, uses the Smith (1990) parameterization without operational bias adjustments.
A third set of simulations (BM) uses the new diagnostic, bimodal cf parameterization that will be introduced in the next section, with otherwise identical settings to RA2M and NOEACF. For guidance, an overview of the three experiments is given in Table 1.
Experiment overview. Apart from the cloud scheme configuration, these experiments have identical settings as indicated in the text.
3. Description of cloud cover parameterizations
a. Original Smith scheme
Most liquid cloud fraction (cf) parameterizations rely on the principle of instantaneous condensation, which requires that any liquid supersaturation is removed to form cloud locally within a grid box. A further assumption is that, within each grid box, a distribution exists of the saturation departure (SD), the so-called s-distribution (Sommeria and Deardorff 1977). Note that we define SD in terms of total water qT = qυ + qliq and liquid temperature [Tliq = T − (Lυ/cp)qliq] throughout this paper, where qυ is the specific humidity and qliq is the liquid water content. It is useful in the context of this paper to revisit the underpinnings of the Smith (1990) cloud scheme.
To further demonstrate the underlying mechanics of the Smith (1990) scheme, and its variations that will be discussed next, Fig. 1 shows a profile through a fairly well-mixed stratocumulus-topped boundary layer (taken from the 27 April 2011 case during the MC3E campaign, at 1630 LT at the SGP). The gray lines in Fig. 1 denote the observed profiles obtained from the interpolated sounding ARM product. The observed cf and qliq associated with this profile are given in Fig. 2, showing near full cloud cover and a considerable amount of liquid water. Note that this profile was selected for its near-saturated state near the boundary layer top and the reasonable agreement between the model and the observations. The purpose of these figures is not to evaluate model performance, but to demonstrate how the diagnosed and the observed cloud properties compare for similar environmental conditions. A process-based evaluation of the online model simulations will follow in section 4 and in Part II of this paper.
(a) Example vertical profile of the liquid potential temperature and total water and (b) the associated profiles of total relative humidity for 1630 LT 27 Apr 2011. Shown are the simulations using the bimodal scheme (black) and the observations from the interpolated sounding ARM product (gray). Model levels are indicated by the horizontal gray lines and the inversion at the top of the boundary layer with a thick gray line. Levels encompassed by the entrainment zone, associated with this inversion, are indicated with the horizontal blue lines. The level that the free-tropospheric mode is taken from is indicated with the horizontal red line. The dotted horizontal black line is the k level that will be used to demonstrate the impact of various flavors of the cloud schemes on the subgrid PDF, cf, and qliq.
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
Vertical profiles of (a) cf and (b) qliq, associated with different cloud scheme configurations, for the thermodynamic profiles shown in Fig. 1. Shown are the observed profiles (gray), and the profiles for the original Smith scheme (smith, blue), the Smith scheme with the empirical bias adjustment (smith + EACF, orange), the Smith scheme with the empirical bias adjustment and grid refinement (smith + EACF + ACF, red), the Smith scheme using a symmetric Gaussian PDF with the same variance as the mixture of PDFs in the bimodal scheme (unimodal-bmvar, light gray), and the bimodal scheme (bimodal, black). Cloud fraction and water content are calculated offline using the thermodynamic profile shown in Fig. 1. Model levels are indicated by the horizontal gray lines and the inversion at the top of the boundary layer with a thick gray line. Levels encompassed by the entrainment zone, associated with this inversion, are indicated with the horizontal blue lines. The level representing the free-tropospheric mode is indicated with the horizontal red line. The dotted horizontal black line is the k level used to demonstrate the impact of various flavors of the cloud schemes on the subgrid PDF, cf, and qliq.
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
Figure 3 shows the distribution G(s) for the level k indicated by the dotted line in Fig. 1. This level resides just below the inversion (indicated by the thick gray horizontal line on the Figure) and is near gridbox-average total water saturation (indicated by the RHT profile in Fig. 1b). By solving the integrals in Eqs. (5) and (6), using the triangular distribution function, one can diagnose the cf and qliq for the level k. The integral to obtain the cf is indicated by the gray-shaded area in Fig. 3 and the values of cf and qliq at level k can be inferred from Fig. 2. Given a gridbox mean total relative humidity [
Assumed triangular distribution G(s) for the level k in Fig. 1. The solid vertical black line denotes the gridbox mean total saturation departure μk and the dashed vertical gray line denotes total saturation (total saturation departure = 0). These two lines are close to one another as the level k is close to grid-mean total saturation. The gray-shaded area is the integral from s = 0 to s = ∞. Also shown is one standard deviation (σs) to the left and right of the mean (vertical black dotted lines) and the half-width of the distribution (bs).
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
b. Empirically adjusted cloud fraction (EACF)
From the previous section, the Smith scheme is capable of producing some cloud in a subsaturated grid box. It can also be seen from Fig. 3 that for RHT = 100% (and hence μ = 0), the cf will be 50% by definition for a symmetric unimodal PDF.
However, from the vertical profiles in Figs. 1b and 2, and more generally from previous low-cloud aircraft observations (Wood and Field 2000), cf in environments with RHT = 100% is typically observed to be larger than 50%.
The cf and qliq associated with the thermodynamic profile shown in Fig. 1 and calculated using the EACF is shown in Fig. 2. Clearly, this modification to the Smith scheme largely improves cf, although the qliq remains underestimated, as it is unaffected by the EACF.
c. Area cloud fraction (ACF)
A further modification used in the RA2-M regional configuration is the implementation of an area cloud fraction (ACF) (Boutle and Morcrette 2010). The aim of this modification is to better represent thin stratocumulus clouds when the vertical resolution of the model is not sufficient to resolve them. This method consists of splitting each model layer into three sublayers, with interpolated values of qT and TL. The interpolation method depends on whether strong gradients in humidity exist across the model layers, suggesting the presence of a sharp inversion. The cf is then calculated in each of the sublayers, using the EACF, and averaged to produce a bulk cf for the layer in question. Furthermore, the cf passed to the radiation scheme is the maximum area cf from the three sublayers. The impact of the ACF on the cf and qliq for the profiles shown in Fig. 1 is shown to be minimal in this case (Fig. 2), since the cloud layer spans several model levels. A vertical grid refinement hence does not significantly modify cf in this case.
d. Bimodal cloud scheme
1) Entrainment zone definition
While experience has shown that the EACF is beneficial to midlatitude configurations of the UM, it is a one-size-fits-all bias correction, agnostic of the physical mechanisms leading to the observed cf versus RHT relation. Indeed, the EACF has been shown not to perform well over tropical regions (e.g., B20). Therefore, in this section an alternative, physically based approach is explored, offering a similar or greater benefit compared to the EACF over midlatitude regions.
We reiterate that by definition, a symmetrical subgrid SD PDF will produce a cf of 50% at RHT = 100%, regardless of the width of the PDF, and hence will always fail to reproduce the observed joint distribution of RHT and cf (Wood and Field 2000; Webb et al. 2001).
Lidar, aircraft and tethered balloon measurements often show skewed and multimodal temperature and humidity PDFs (Wood and Field 2000; Wulfmeyer et al. 2010; Turner et al. 2014; Behrendt et al. 2015; Osman et al. 2018; Price 2006).
These skewed thermodynamic distributions typically occur near the boundary layer top, in the so-called entrainment zone (EZ) (Wulfmeyer et al. 2016). Many studies have attempted to observe the properties of the EZ using lidar, suggesting that the EZ is associated with enhanced moisture and temperature variance and skewness (Kiemle et al. 1997; Wulfmeyer 1999; Turner et al. 2014). From the specific humidity variance profile, Turner et al. (2014) found the depth of the EZ to be 0.12–0.20 times the ML-top height, while larger values of about half the ML depth were found by Wulfmeyer (1999). The EZ is influenced by engulfments in the ML top, leading to frequent incursions of free-tropospheric air into the ML. As such, within the EZ, two distinct modes of variability would coexist in a volume typical of a model grid box, inducing enhanced variance and negative skewness as a generally moist environment experiences occasional dry, warm intrusions.
The coexistence of air masses with distinctly different thermodynamic properties in a model grid box can be represented by allowing for two separate modes of s variability. Skewness of this mixture of two PDFs would then be induced as their weights and means vary.
A method is presented here to reconstruct such a bimodal PDF by using information from the vertical thermodynamic profiles. It should be emphasized that the cloud scheme presented here only aims to diagnose liquid cf and qliq at each level, similar to the Smith (1990) scheme. There is no transport or actual mixing of moisture or heat and qT and Tliq in each grid box will remain conserved throughout this diagnosis.
First, the EZ is defined. In the UM, entrainment is parameterized following Lock et al. (2000), leading to a deepening of the ML and a homogeneous drying and warming of the ML top. From observations, entrainment is far from a homogeneous process, however, with pockets of dry, free-tropospheric air entrained deep into the ML and limited immediate mixing with the ML air. The top of the EZ is defined here as the level that resides just below an inversion, in turn defined as a local maximum in the liquid potential temperature gradient (Δθliq/Δz) exceeding 0.1 × Γdry, where Γdry is the dry adiabatic lapse rate. The threshold of 0.1 × Γdry is consistent with the definitions of inversions in the boundary layer scheme. Model levels below this inversion are included in the EZ as long as they have a Δθliq/Δz > 0.1 × Γdry, monotonically decreasing away from the inversion level. Indeed, this condition indicates a warming with increasing height within the ML and entrainment of warmer air from above the inversion. The bottom level of the EZ is constrained to be no lower than half the ML depth.
For the vertical profile shown in Fig. 1, the EZ (indicated by blue levels) identified as above, has a depth of approximately 30% of the ML depth, which is consistent with lidar observations.
For each level within the EZ, a mixture of PDFs is supposed to exist: one PDF originating from the free troposphere above the inversion, and another PDF from the bottom of the EZ. The free-tropospheric mode is taken from the driest level upward from the inversion, within a distance of 25% of the inversion height, as long as the levels are monotonically drying. This definition is based on vertical profiles of skewness and variance from Turner et al. (2014), showing an influence of the EZ up to an altitude of 1.25 times the inversion height. Note that a limit of 400 m is imposed on the distance between the dry mode level and the inversion level, to avoid drawing in air from too high above the inversion when the ML becomes very deep. A definition based on distance above the inversion, rather than just taking the first level above the inversion, avoids introducing sensitivity to the vertical level spacing. For the profile shown in Fig. 1, the dry mode is indicated by the red horizontal line.
EZs are not just identified at the planetary boundary layer (PBL) top, but also within residual or decoupled layers, if an inversion exists with a gradient of θliq > 0.1 × Γdry. Definitions for these EZs are identical to those in the PBL, although for residual layers a limit of 400 m is imposed on the EZ depth. Slight variations to the choice of the definitions of the top and bottom mode of the EZ do not substantially change the behavior of the bimodal cloud scheme.
In the remainder of this paper, the free-tropospheric mode will be referred to as the “top” mode, while the bottom of the EZ mode will be referred to as the “bottom” mode.
For any level k encompassed by an EZ, the top mode and bottom mode are adiabatically brought to that level k. Hence two modes of s-variability coexist for all levels k within the EZ.
2) PDF shape of individual modes
While the previous section described the subgrid variability within EZs as a mixture of two PDFs, the functional shape of the individual PDFs has yet to be determined. The Smith (1990) scheme assumes a triangular PDF with a fixed variance, linked to the constant profile of RHcrit [Eq. (8)]. This is an important limitation of this scheme, since variances typically vary significantly throughout the atmosphere. However, information about the subgrid variability can also be derived from the turbulence parameterization.
Field et al. (2014) and Furtado et al. (2016) developed a parameterization for the production and maintenance of supercooled liquid cloud in mixed-phase, turbulent environments. Following arguments laid out in Furtado et al. (2016), we generalize this parameterization here to describe turbulence-based saturation-departure variability in all liquid clouds.
First, we review the formulations by Furtado et al. (2016) for the turbulent generation of liquid at subzero temperatures in the presence of ice, before expanding their formulations to liquid-only and warm conditions.
For any grid box that is not encompassed by an EZ, a unimodal Gaussian subgrid PDF is assumed, using the local σturb and μturb as defined above.
Within EZs, however, we assume two modes of variability to be present, as outlined in the previous section. Hence, σt and σb and μt and μb in Eq. (11) will be the σturb and μturb from the respective levels of origin for the top and bottom modes, adiabatically brought to the level of interest. Note that we use the turbulent properties of the level of origin for the two modes, but the local phase-relaxation time scale τP at the level to calculate cloud for, for consistency with the local ice-phase characteristics.
To ensure that the mixture of PDFs represents a continuous distribution, the top mode is slightly adjusted. Physically, this represents minimal mixing of the top mode as it is dragged down into the ML. To do so, the moist tail of the top mode is extended so that there is a minimal overlap between top and the bottom modes. The appendix provides more detail on how this modification is performed, ensuring a full PDF integral of unity and conservation of the liquid SD. In the remainder of this section, the μt and σt refer to the modified values as described in the appendix.
4. Bimodal cloud cover and water content
As in Fig. 3, but for the bimodal scheme. The solid vertical black line denotes the gridbox mean total saturation departure μk and the dashed vertical gray line denotes total saturation (total saturation departure = 0). These two lines are close to one another as the level k is close to grid-mean total saturation. The red PDF represents the mode from just above the inversion, while the blue PDF represent the mode from the bottom of the entrainment zone (indicated by the blue horizontal lines in Fig. 1). The standard deviations (σ) and mean total saturation departure (μ) of both modes are represented by vertical dotted and solid lines, respectively, each in their respective colors. The adjusted first moment of the air from above the inversion (μt_adj) is indicated as well. The colored shading represents the integral from s = 0 to s = ∞ for each of the modes. The light gray PDF denotes a Gaussian symmetric distribution with the same first and second moments as the mixture of PDFs in the bimodal scheme, for reference (BMvar).
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
Figure 4 also shows the PDF of a unimodal, symmetric Gaussian distribution with the same mean and variance than the mixture of PDFs. This configuration will be further referred to as unimodal-bmvar and is provided to separately assess the impact of the variance and the skewness from the bimodal scheme on cloud properties.
From Fig. 2, the unimodal-bmvar has cf no larger than 50% for the subsaturated profile in Fig. 1, similar to Smith. However, the qliq (Fig. 2b) is much larger than in the Smith calculation. The difference between the Smith and unimodal-bmvar profiles in Fig. 2 is nearly entirely due to differences in the variance, rather than the different PDF shapes (Gaussian or triangular).
From Fig. 2, the bimodal scheme considerably improves the cf profile compared to Smith, and even outperforms Smith + EACF + ACF. Moreover, unlike Smith + EACF + ACF, the bimodal scheme also improves the qliq compared to Smith (Fig. 2b).
It is interesting to note that qliq using the bimodal scheme is not as tightly linked to the cf as in the Smith configurations. This allows qliq to get larger near the top of the stratocumulus layer, despite a somewhat smaller cf than the levels just below. This is consistent with the observed profile in Figs. 2a and 2b.
Vertical profiles of the (a) s-variance and (b) s-skewness as diagnosed in the bimodal Smith scheme for the profiles shown in Fig. 1. Both the local variance for the individual modes, diagnosed from the turbulent kinetic energy and mixing length following Furtado et al. (2016) (dashed), and the variance of the mixture of PDFs (solid) are shown. For levels that do not belong to an entrainment zone, both variances are identical. The gray dashed line in (a) denotes the variance of the triangular PDF, corresponding to the profile of RHcrit in the NOEACF configuration. Model levels are indicated by the horizontal gray lines and the inversion at the top of the boundary layer with a thick gray line. Levels encompassed by the entrainment zone, associated with this inversion, are indicated with the horizontal blue lines. The level that the free-tropospheric mode is taken from is indicated with the horizontal red line. The dotted horizontal black line is the k level used to demonstrate the impact of various flavors of the cloud schemes on the subgrid PDF, cf, and qliq.
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
5. Evaluation of different schemes
a. Observed cloud properties
Using aircraft and ground-based observations collected during the MC3E campaign, a process-based evaluation is performed of the online model simulations, described in section 2c. In all following analysis, observed and simulated data have been filtered for the occurrence of precipitation and temperatures below 0°C, as detailed in section 2b.
Joint density distributions of (a) total relative humidity and cf for the aircraft observations and (b)–(d) all simulations for warm clouds. The gray shading of the data points indicates the relative frequency (0.000 25 in gray to 0.0005 in black). The fitted dark blue line in (a) is the fit of the MC3E-aircraft observations, following Eq. (32) and the red line is the fit for the original Wood and Field (2000) data. The yellow lines in (b)–(d) are the fits for each of the respective model simulations, while the faint gray lines are the fits through the observations for reference. The dashed yellow line in (d) is the fit for cloudy points encompassed by an entrainment zone only. The horizontal gray dashed line in each panel denotes the cf at RHT = 100%. The histograms to the right and the bottom of each panel show the total distributions of cf and RHT, respectively. Again, faint gray lines in these histograms are for the observations for reference.
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
Fitting parameters for Eq. (32). Original parameters as proposed by Wood and Field (2000) are provided. Furthermore, fits using aircraft data from all flight legs during the MC3E campaign are provided, as well as for simulations with three permutations of the diagnostic cf scheme in the UM (NOEACF, RA2M, and BM). For BM, fitting parameters for grid points encompassed by entrainment zones only have been provided as well [BM (EZ)]. Also provided are the standard deviations of the observed and simulated cloud fraction around the mean fits (STD), as well as the mean absolute error of the simulated cloud fraction against the aircraft-observed fit (MAE). All data, apart from the data by Wood and Field (2000), are for an assumed grid length of 1000 m.
As in Fig. 6, but for the joint distribution of qliq and cf. (a) Observations include the Microbase/ARSCL derived data in black (for nonprecipitating, warm clouds during MC3E) and the aircraft-derived data in blue (for all available warm cloud flight legs). Fits are provided following Eq. (33) for each of the observational data sources in light gray and blue in (a) and for all simulations in yellow in (b)–(d). Faint gray lines in (b)–(d) are the observed fits for reference and the faint gray lines in the histograms are the Microbase-observed distributions for reference.
Citation: Monthly Weather Review 149, 3; 10.1175/MWR-D-20-0224.1
Fitting parameters for Eq. (33). Original parameters as proposed by Wood and Field (2000) are provided. Furthermore, fits using aircraft data from all flight legs during the MC3E campaign are provided, as well as for the entire MC3E period using ground-based Microbase/ARSCL observations, and simulations with three permutations of the diagnostic cf scheme in the UM (NOEACF, RA2M, and BM). Fits for the qliq vs cf relation [Eq. (33)] are provided for normalized and non-normalized qliq for the simulations and the aircraft data and only for non-normalized data for the ground-based observations. For BM, fitting parameters for grid points encompassed by entrainment zones only have been provided as well [BM (EZ)]. Also provided are the standard deviations of the observed and simulated cloud fraction around the mean fits (STD), as well as the mean absolute error of the simulated cloud fraction against the aircraft-observed fit (MAE). These two metrics are provided for the non-normalized data only. All data, apart from the data by Wood and Field (2000), are for an assumed grid length of 1000 m.
Histograms at the bottom and to the right of each panel in Figs. 6 and 7 show the histograms of RHT, qliq, and cf, respectively. From Fig. 7a, the distribution of qliq peaks around 10−3 kg kg−1, with a long tail toward smaller values. A fairly uniform distribution of cf exists for fractions between 0.1 and 0.9, but a peak near cf ≈ 1 exists in both observational datasets.
b. NOEACF
The NOEACF uses the Smith (1990) scheme as in the RA2-M configuration, but without the operational adjustments (ACF and EACF). This configuration fails to reproduce the observed RHT–cf (Fig. 6b) and qliq–cf (Fig. 7b) relations. The fit following Eq. (32) predicts a cf value of 47% for RHT of 100% (horizontal gray dashed line in Fig. 7), which is much lower than the observed value of 90%. For nearly the entire range of RHT values, the Smith scheme produces cf that is biased low.
Figure 7b reveals that also for a given qliq, the cf in Smith tends to be lower than the observations suggest. Moreover, since the qliq itself is biased low (histogram at the bottom of Fig. 7b), full cloud cover (cf > 90%) is far too infrequent, while the frequency of small cf is overestimated (histogram to the right of Fig. 7b).
The Smith scheme also produces a too tight relation between qliq, RHT, and cf, failing to show the considerable spread in these parameter spaces in the observations (Figs. 6a and 7a). This is also clear from the observed and simulated standard deviation (STD) in Tables 2 and 3. This is due to the space- and time-invariant profile of RHcrit and hence near-constant subgrid variability.
c. RA2M
The RA2M configuration includes the operational bias corrections (see sections 3b and c). As expected, this leads to an improved relation between RHT and cf (Fig. 6c), with improved fitted parameters compared to the NOEACF and a reduced mean absolute error (MAE; Table 2). The RA2M produces cf correctly exceeding 50% at RHT = 1, with the functional fit predicting a cf of 65%. This is closer to, albeit still lower than the observed values.
The relation between qliq and cf is also improved (Fig. 7c), although cf is still slightly underestimated for a given qliq. Furthermore, the qliq distribution is still biased toward smaller-than-observed values in the RA2M (histograms at the bottom of Fig. 7c), similar to the NOEACF. Overall, the RA2M exhibits an improved cf distribution (right panel in Fig. 7c), although full cloud cover is still too infrequent, while intermediate cf is too abundant.
Since the EACF is a one-size-fits-all adjustment, the very tight coupling between the qliq and cf still remains (STD in Table 3). Therefore, the RA2M also shows a lack of spread in the joint PDFs of RHT and cf and qliq and cf.
d. BM
The bimodal cloud scheme is used in the BM configuration. The BM is capable of producing cf above 50%, even for RHT < 100% (Fig. 6d). The functional fit [following Eq. (32)] predicts a value of 54% for a RHT = 1 for all grid points combined (solid yellow line), considerably lower than observed. However, Fig. 6d also shows the fit only for grid points encompassed by an EZ (dashed yellow line). This subset of grid points is capable of replicating the observed relation better than NOEACF and RA2M, showing a cf of 69% for RHT = 1 and a much smaller MAE (Table 2). Furthermore, unlike NOEACF and RA2M, BM produces many data points in the upper left quadrant of Fig. 6d, consistent with the observations (Fig. 6a).
The relation between qliq and RHT [following Eq. (33)] is slightly improved in BM compared to NOEACF, but not compared to the RA2M (Table 3 and Fig. 7).
Probably due to the larger variances in the EZ, the qliq distribution (histogram at the bottom of Fig. 7d) is better captured in BM than in the other configurations. Furthermore, the cf distribution (histogram to the right of Fig. 7d) is improved compared to the NOEACF, mainly for large cf.
Due to the much larger degree of freedom of the variance and skewness of the subgrid PDFs, the spread in the qliq–cf relation is much larger in BM than in RA2M and NOEACF, consistent with the observations (STD in Table 3).
6. Discussion and conclusions
Regional, operational forecasts with the Unified Model use a Smith (1990)-like diagnostic cloud fraction (cf) scheme, assuming unimodal, nonskewed subgrid saturation-departure variability. However, given the tendency for this scheme to underestimate cloud cover, an empirical adjustment is made operationally, following Wood and Field (2000).
While this bias adjustment results in much improved cloud cover, it has no knowledge of the underlying physical mechanisms responsible for this enhanced cloud cover. Hence, under certain conditions the performance of the bias adjustment is detrimental (B20), and a more physically based scheme would be preferable.
Aircraft and lidar observations typically show skewed thermodynamic variable distributions, mainly within the entrainment zone (EZ) at the top of the PBL (Wood and Field 2000; Turner et al. 2014; Behrendt et al. 2015). Frequent intrusions of dry free-tropospheric air into the upper mixed layer are responsible for this skewness (Wulfmeyer et al. 2016). Hence, in the EZ, air masses with distinctly different thermodynamic properties coexist in a volume typical of a model grid box. This suggests that cf parameterizations would struggle to obtain the correct cloud properties using a unimodal, symmetric PDF.
Advanced parameterizations exist that prognose higher-order moments of the subgrid variability PDF (Tompkins 2002; Golaz et al. 2002; Bogenschutz and Krueger 2013). While these schemes are attractive mathematically, they typically still have to make pragmatic assumptions to obtain closure for all the prognostic equations. Furthermore, these schemes can be less attractive in an operational context, given their higher computational cost and the difficulty to determine which parameters are most suitable for inevitable tuning.
This paper introduces an alternative, diagnostic liquid cf scheme. This is done by inferring a bimodal subgrid PDF of the saturation departure from the large-scale state at each time step.
The scheme identifies an EZ below any significant inversion, allowed to span several model levels and based on the liquid temperature gradient. For each level encompassed by the EZ, a bimodal PDF is reconstructed, consisting of a dry, warm free-tropospheric mode and a moist mixed-layer mode. Subsequently, weights are applied to the two modes so that the gridbox mean saturation departure remains conserved.
Hence, the subgrid variability for levels within the EZ is treated as a mixture of two PDFs. Outside the EZ, the scheme reverts to unimodal Gaussian PDFs. Each PDF (i.e., each of the two PDFs within the EZ, or the unimodal PDF outside the EZ) is assumed to be symmetric and Gaussian, although minimal mixing between the modes is allowed to skew the free-tropospheric mode. Moments for each individual PDF are determined from an extension of the turbulence-based mixed-phase cloud scheme by Furtado et al. (2016) to all liquid clouds. Subgrid variability is hence linked to the turbulent kinetic energy and a scale-aware mixing length, and accounts for water vapor competition from ice in mixed-phase conditions. Cloud fraction (cf) and liquid water content (qliq) are calculated for each PDF separately and gridbox mean values are derived from their weighted average.
The bimodal cloud scheme is capable of diagnosing a similar cloud cover enhancement compared to the operational configuration, but in a more physically meaningful way. An emergent property of this approach is that it implicitly yields skewness and variance profiles similar to observed profiles.
A process-based evaluation was performed of the Smith scheme, the Smith scheme with operational bias adjustment and the bimodal scheme, using aircraft observations and ground-based retrievals from the Midlatitude Continental Convective Clouds Experiment (MC3E; Jensen et al. 2016). The Smith (1990) scheme is shown to largely underestimate the cf for a given total relative humidity (RHT) or qliq. Moreover, the qliq itself is biased low, further exacerbating the lack of cloud in this scheme. The operational bias correction improves the RHT–cf relation, but still fails to capture the observed qliq distribution. While intermediate cf is too abundant in this configuration, there is still a lack of overcast conditions.
The RHT–cf relation is better captured in the bimodal scheme than in the Smith (1990) scheme as well, but in a more physically sensible way. Moreover, an important improvement is found in the distribution of qliq using the bimodal parameterization. Another important benefit of the bimodal cloud scheme is that the tight relation between qliq and cf in the Smith (1990) scheme is broken.
There are limitations to the bimodal framework. It appears that even with the bimodal scheme, the overall cloud cover is still somewhat underestimated. The simulated relation between RHT and cf is improved compared to the original Smith scheme, but still falls short of accurately reproducing the observations.
The definition of the EZ involves a number of ad hoc assumptions, although it should be said that slight changes to these assumptions did not substantially change the overall scheme behavior. Furthermore, the scheme assumes minimal mixing between the two modes of variability within EZs, which could be improved by linking the mixing between these modes to the turbulent properties or the large-scale environment. The bimodal scheme can infer skewness associated with EZs, but would not be able to infer skewness resulting from microphysical and precipitation processes. For instance, precipitation should disproportionally remove water from the most supersaturated part of the PDF, which will affect the shape of the PDF. Also, the scheme currently only treats liquid clouds, but could be further extended to incorporate the formation of ice cf as well, making improved assumptions about the overlap between liquid and ice cloud within a grid box. In principle the framework presented here could also be expanded to include the convective parcel mode in simulations that use a shallow or deep convection scheme (similar to e.g., Klein et al. 2005). However, we leave these further additions for future research. This paper serves as a proof-of-concept of a computationally feasible, diagnostic approach of inferring cf and qliq from asymmetric subgrid variability distributions in turbulent environments.
A more in-depth evaluation of the simulations for the MC3E campaign will be provided in Part II of this paper, using satellite and ground-based remote sensing, and surface observations. Given that the turbulence-based variance of the Gaussian PDFs of the individual modes scales with resolution, it is expected that the bimodal scheme should also become more scale aware. Part II of this paper will hence also investigate sensitivities of a range of cloud scheme configurations to changes in horizontal and vertical resolution.
Acknowledgments
The authors are grateful to Adrian Lock, Volker Wulfmeyer, and Vince Larson for stimulating discussions. The ground-based and aircraft observational data for this article were gratefully obtained from the U.S. Department of Energy ARM data archive (http://www.archive.arm.gov/armlogin/login.jsp), sponsored by the DOE Office of Science, Office of Biological and Environmental Research, Environmental Science Division. The work of K. Van Weverberg was supported by the Met Office Weather and Climate Science for Service Partnership (WCSSP) Southeast Asia as part of the Newton Fund.
APPENDIX
Variance Adjustment of Top Mode
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