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  • View in gallery

    Number concentration of ice crystals formed from ice nucleation Nin (L−1) as a function of temperature (°C) showing the Fletcher, Meyers, and Cooper curves taken from Thompson et al. (2004) with some modifications. The red dashed line indicates the curve used in the FA scheme.

  • View in gallery

    (a) Ice fall speed (m s−1) increase, VrimeF, as a function of RF (ordinate) and (abscissa). (b) Ice fall speeds (m s−1) vs mean particle diameter (mm) shown for unrimed ice (blue), the FER scheme (purple), and the FA scheme (red). An RF of 45 was assumed for the curves shown. Calculations are valid for P = 105 Pa, thus simplifying calculations here by making Γs = 1.

  • View in gallery

    (a) Schematic illustration of the drizzle parameterization for a single cloud layer in which drizzle forms from a low-level liquid water cloud at >0°C only when it is completely disconnected from rain formed from melting ice aloft. (b) The scatterplot from Westbrook et al. (2010) shows retrieved rain rate (R, mm h−1) vs the normalized rain intercept parameter (NL in m−4, where NL = Nor for exponential distributions) based on lidar observations of drizzle. The different values of Nor described in (7) are overlaid on the figure with the red line showing the variation of Nor as a function of rain rate for rain contents between 0.02 and 0.5 g m−3.

  • View in gallery

    NMMB 9-h forecast valid 0000 UTC 30 Jun 2012 of simulated maximum-in-column reflectivity (dBZ) (a) without and (b) with RF advection; and (c),(d) corresponding vertical cross-sections of reflectivity and temperature [°C, dashed (<0) and solid lines (≥0)] along line AB, respectively. Horizontal distance in km is plotted along the abscissa.

  • View in gallery

    (a),(b) Maximum-in-column reflectivity and (c),(d) vertical cross-sections of reflectivity valid along line AB at (a),(c) 2100 and (b),(d) 2200 UTC 29 Jun 2012. Horizontal distance in km is plotted along the abscissa.

  • View in gallery

    Vertical cross-sections of (a),(b) cloud ice + snow, and (c),(d) graupel in g kg−1 (color shaded), as well as (e),(f) RF for an NMMB run (left) without and (right) with RF advection. Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 0000 UTC 30 Jun 2012 (forecast hour 09). Horizontal distance in km is plotted along the abscissa. Snow and cloud ice are assumed to occur where RF ≤ 5 and graupel where RF > 5. The RF values are shown for mass contents ≥ 0.1 kg m−3.

  • View in gallery

    Histograms of hourly composite reflectivity (dBZ) aggregated over all 11 cases for reflectivity at 5-dBZ intervals starting with (a) 5 dBZ and (b) 45 dBZ. Histograms of 3-hourly accumulated precipitation aggregated over all 11 cases for amounts (c) starting with 0.01 in. and (d) starting with 1 in. Displayed in the figure are the observed counts (red) and counts from runs with the RF advection (blue) and without the RF advection (green). Rounding of the counts above the bars was done to make the plot more legible.

  • View in gallery

    Maximum-in-column reflectivity (dBZ) for runs with (a) n = 2 and (b) n = 1, as well as corresponding vertical cross-sections of (c),(d) reflectivity and (e),(f) precipitation ice fall speeds (m s−1). Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 0000 UTC 30 Jun 2012 (forecast hour 09). Horizontal distance in km is plotted along the abscissa.

  • View in gallery

    Histograms of 3-hourly accumulated precipitation aggregated over all 11 cases for amounts (a) starting with 0.01 in. and (b) starting with 1 in. Displayed in the figure are the observed counts (red), and counts from runs with n = 1 (blue) and with n = 2 (green). Rounding of the counts above the bars was done to make the plot more legible.

  • View in gallery

    Hybrid scan reflectivity (dBZ) and 1-km AGL reflectivity (dBZ) from (b) a Thompson run, (c) an FA run with a fixed rain intercept parameter Nor, and (d) an FA run with a variable Nor valid at 1500 UTC 1 Jul 2015 (forecast hour 15). The black solid line with the start and end points marked as A and B, respectively, represents the locations of vertical cross sections shown in Fig. 11.

  • View in gallery

    Vertical cross-sections of (a) reflectivity (dBZ) and (b) number-weighted, mean drop sizes (μm) of rain for the Thompson run and corresponding cross-sections for FA runs; (c),(d) using a fixed Nor and (e),(f) using a variable Nor. Refer to Fig. 10 for the location of the cross sections. Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 1500 UTC 1 Jul 2015 (forecast hour 15). Horizontal distance in km is plotted along the abscissa.

  • View in gallery

    (a) Observed composite reflectivity valid at 1300 UTC 24 May 2016 compared against 24-h forecasts from (b) a Thompson run, (c) an FA run using a fixed Nor, and (d) an FA run using a variable Nor valid at 1200 UTC 24 May 2016. No observations were available at 1200 UTC.

  • View in gallery

    (a) Composite reflectivity (dBZ) and (b) vertical cross-section of reflectivity for the Thompson run with the corresponding plots for the (c),(d) FER and (e),(f) FA runs valid at 0000 UTC 30 Jun 2012 (forecast hour 09). Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 0000 UTC 30 Jun 2012 (forecast hour 09). Horizontal distance in km is plotted along the abscissa.

  • View in gallery

    Vertical cross-section of snow number concentration (Ns) for the (a) Thompson, (b) FER, and (c) FA run at the same time and cross sections described in Fig. 13. Units are L−1.

  • View in gallery

    (a) Composite reflectivity (dBZ) and (b) vertical cross-section of reflectivity from observations with corresponding plots from (c),(d) an FER run, and (e),(f) an FA run, valid at 2200 UTC 20 May 2013 (forecast hour 22). Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 2200 UTC 20 May 2013 (forecast hour 22). Horizontal distance in km is plotted along the abscissa.

  • View in gallery

    Histograms of hourly composite reflectivity (dBZ) aggregated over all 11 cases for reflectivity at 5-dBZ intervals starting with (a) 5 and (b) 45 dBZ. Histograms of 3-hourly accumulated precipitation aggregated over all 11 cases for amounts (c) starting with 0.01 in. and (d) starting with 1 in. Displayed in the figure are the observed counts (red), and counts from FER runs (blue) and FA runs (green). Rounding of the counts above the bars was done to make the plot more legible.

  • View in gallery

    Flowchart of the FA microphysical scheme production terms listed in Table B1.

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Modified NAM Microphysics for Forecasts of Deep Convective Storms

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Abstract

The Ferrier–Aligo (FA) microphysics scheme has been running operationally in the National Centers for Environmental Prediction (NCEP) North American Mesoscale Forecast System (NAM) since August 2014. It was developed to improve forecasts of deep convection in the NAM contiguous United States (CONUS) nest, and it replaces previous versions of the NAM microphysics. The FA scheme is the culmination of extensive microphysical scheme sensitivity experiments made over nearly a dozen warm- and cool-season severe weather cases, as well as an extensive real-time testing in a full, system-wide developmental version of the NAM. While the FA scheme advects each hydrometeor species separately, it was the mass-weighted rime factor (RF) that allowed rimed ice to be advected to very cold temperatures aloft and improved the vertical structure of deep convection. Rimed ice fall speeds were reduced in order to offset an increase in bias of heavy precipitation as a consequence of the mass-weighted RF advection. The FA scheme also incorporated findings from 3-km model runs using the Thompson scheme, including 1) improved closure assumptions for large precipitating ice that targeted the convective and anvil regions of storms, 2) a new diagnostic calculation of radar reflectivity from rimed ice in association with intense convection, and 3) a variable rain intercept parameter that reduced widespread spurious weak reflectivity from shallow boundary layer clouds and increased stratiform rainfall.

Current affiliation: NOAA/NWS/NCEP/EMC, College Park, Maryland.

© 2018 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: Eric A. Aligo, eric.aligo@noaa.gov

Abstract

The Ferrier–Aligo (FA) microphysics scheme has been running operationally in the National Centers for Environmental Prediction (NCEP) North American Mesoscale Forecast System (NAM) since August 2014. It was developed to improve forecasts of deep convection in the NAM contiguous United States (CONUS) nest, and it replaces previous versions of the NAM microphysics. The FA scheme is the culmination of extensive microphysical scheme sensitivity experiments made over nearly a dozen warm- and cool-season severe weather cases, as well as an extensive real-time testing in a full, system-wide developmental version of the NAM. While the FA scheme advects each hydrometeor species separately, it was the mass-weighted rime factor (RF) that allowed rimed ice to be advected to very cold temperatures aloft and improved the vertical structure of deep convection. Rimed ice fall speeds were reduced in order to offset an increase in bias of heavy precipitation as a consequence of the mass-weighted RF advection. The FA scheme also incorporated findings from 3-km model runs using the Thompson scheme, including 1) improved closure assumptions for large precipitating ice that targeted the convective and anvil regions of storms, 2) a new diagnostic calculation of radar reflectivity from rimed ice in association with intense convection, and 3) a variable rain intercept parameter that reduced widespread spurious weak reflectivity from shallow boundary layer clouds and increased stratiform rainfall.

Current affiliation: NOAA/NWS/NCEP/EMC, College Park, Maryland.

© 2018 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: Eric A. Aligo, eric.aligo@noaa.gov

1. Introduction

The National Centers for Environmental Prediction (NCEP) North American Mesoscale Forecast System (NAM) contiguous United States (CONUS) nest is run at a horizontal grid spacing of 3 km (Rogers et al. 2017), the scale at which models are capable of resolving some circulations and physical processes associated with deep convective storms while also capturing convective mode with some degree of skill (e.g., Done et al. 2004; Kain et al. 2008). Correctly forecasting the mode of convection is important for predicting different types of severe weather. For example, Gallus et al. (2008) showed that quasi-linear mesoscale convective systems (MCSs) with trailing stratiform regions are most often associated with severe wind damage, and leading stratiform systems, while less common, are most often associated with hail and tornadoes. They also showed that tornadoes are most common in cellular systems within broken lines.

While the mode of a quasi-linear system can be initially governed by hydrometeor advection (Parker and Johnson 2000), microphysical processes can play an important role as well. The transport of buoyant air rearward from the convective region can result in further ascent in the stratiform region (Knupp and Cotton 1987) of trailing stratiform systems, with ascent being aided by condensation, freezing, and deposition (Houze 1982; Churchill and Houze 1984). Parker and Johnson (2004a,b) and Storm et al. (2007) showed that leading stratiform systems can be sustained by melting and evaporation, which can act to destabilize the atmosphere ahead of the convective line. Furthermore, microphysics impacts on supercells were shown in Gilmore et al. (2004), who illustrated the significance of assumed graupel density and intercept parameter on the areal coverage of rainfall in their idealized simulations. They found that slower-falling graupel particles, compared to faster-falling hail particles, remained suspended in the cloud for a longer period of time, allowing them to be advected outside of the updraft region and resulting in less total ground-accumulated rainfall. Initial near-surface downdrafts for the simulation with graupel particles were weaker, and outflow was warmer (compared to simulations using hail), as the slower-falling graupel particles took longer to reach the melting level and turn to rain, thus delaying the evaporation.

With forecasters regularly evaluating model-produced, storm-scale diagnostics, such as updraft helicity (Kain et al. 2008), hourly maximum 10-m winds (Kain et al. 2010), lightning (McCaul et al. 2009), and hail size (Adams-Selin and Ziegler 2016), forecasting the correct three-dimensional structure and mode of deep convective storms is critical. A motivation for developing the Ferrier–Aligo (FA) microphysics was in response to feedback from operational forecasters about the lack of structure and perceived realism in the NAM CONUS nest’s forecasts of deep convective storms, as seen from forecast reflectivity.1 This feedback, coupled with an ever-increasing focus on higher-resolution operational forecast model output (e.g., Clark et al. 2012), helped serve as motivation to begin taking an in-depth, careful look at the issue.

The FA scheme replaces previous versions of the Ferrier microphysics scheme (FER) (Ferrier et al. 2002, 2011) that had been run operationally in the NCEP operational NAM since 2001. Operational microphysical schemes outside the United States used in convection permitting model runs include the 3-ICE single-moment scheme used by Météo-France, with cloud ice, snow, and graupel being separated by the amount of riming (Seity et al. 2011). Nearly a dozen European countries use a variation of the 3-ICE scheme (Szintai et al. 2015; Bengtsson et al. 2017), while the Deutscher Wetterdienst (DWD; German Weather Service) uses a Lin et al. (1983)-type single-moment microphysics scheme (Baldauf et al. 2011). In the operational environment, computational efficiency is important, and the FA scheme continues to be a fast scheme, while more sophisticated microphysical schemes (double and triple moment) were too expensive to be implemented with the 2017 NAM upgrade. For example, Wolff et al. (2016) noted that Nonhydrostatic Multiscale Model on the B-grid (NMMB) runs with the Thompson scheme (Thompson et al. 2004, 2008), double moment in cloud ice and rain, take on average 50% more compute time than runs with the FA microphysics when using the operational NAM configuration. While the Thompson scheme could not be run operationally in the NAM, results from 3-km NMMB model runs using the Thompson scheme were used to develop the FA scheme.

Before the FA microphysics was implemented into the operational NAM, it underwent thorough pre-implementation testing spanning multiple seasons. The procedure for advancing this scheme into operations closely followed the steps described in Wolff et al. (2016), which included detailed case study analyses of some high-impact cases followed by extensive real-time testing, in combination with other model system changes. This paper describes and illustrates the modifications made to the FER scheme, based on microphysics sensitivity experiments that were carried out in the development of the FA scheme, to address concerns from various NCEP centers and NWS forecast offices.

While extensive evaluation of all cases described in Table 1 were used in development, in the interest of brevity, only illustrative results from a subset of representative cases are shown from the model microphysics sensitivity experiments, in addition to bias statistics summarizing the results over all of the cases. One of the cases shown in this paper includes the progressive derecho (Johns and Hirt 1987; Guastini and Bosart 2016) that affected an area from Iowa to Maryland on 29/30 June 2012. This progressive derecho produced widespread swaths of damage, resulted in numerous injuries and fatalities, and led to many indirect, heat-related fatalities following the event, when much of the affected region was without power during a heat wave (Vescio et al. 2013; Fierro et al. 2014; Guastini and Bosart 2016).

Table 1.

List of cases evaluated in NMMB runs with the initialization times and forecast length indicated. The first four cases were of cold-season events, and the last seven cases were of warm-season events. The cases with the superscripts a and b were of interest to SPC and WPC, respectively. The reports were provided by SPC and were refined based on the integration domain and forecast hours evaluated.

Table 1.

Section 2 describes the datasets that were used and the configuration of the model runs. Section 3 describes the general principles of the FA scheme and outlines the microphysics sensitivity experiments that were carried out in its development. Section 4 shows results of the microphysics sensitivity experiments and then provides a comparison between the FER and FA scheme results. A summary and conclusion can be found in section 5. The appendixes provide a detailed description of the FA microphysics, including definitions of microphysical scheme source and sink terms, as well as the extensive set of 12 lookup tables that are used, with modifications to the FER scheme clearly documented.

2. Data and methods

All model sensitivity runs were made with the Nonhydrostatic Multiscale Model on the B grid (Janjić 2005; Janjić and Black 2007; Janjić and Gall 2012) in full convection-allowing mode (i.e., no use of parameterized convection) at a horizontal grid spacing of 3 km using the following configuration: RRTMG radiation (Iacono et al. 2008), MYJ planetary boundary layer (Janjić 2001), MYJ surface layer (Janjić 1996), and Noah land surface model (Chen and Dudhia 2001; Ek et al. 2003). The cases evaluated with the NMMB are shown in Table 1 for both cool-season (first four) and warm-season (last seven) deep convection events. Some of the warm-season cases were of particular interest to the Storm Prediction Center (SPC) and the Weather Prediction Center (WPC) because they represented forecast failures of the operational 4-km NAM CONUS nest. The two 2015 cases in Table 1 were evaluated during the 2015 Flash Flood and Intense Rainfall Experiment (FFaIR; NOAA 2015) and identified by WPC as representative examples of the operational 4-km NAM CONUS nest’s high bias in heavy precipitation.

While the microphysics modifications went through vigorous pre-implementation testing in all seasons, the focus in this paper was on the deep convection events for which the modifications were intended. Not surprisingly, the microphysics modifications had only a minor impact on cold-season, nonconvection events due to the lack of supercooled liquid available for riming in the absence of intense updrafts (not shown). Components of the FA scheme that were modifications and additions to the FER scheme were activated only under more extreme conditions associated with warm-season deep convection. Eight of the cases in Table 1 used initial and lateral boundary conditions from the 12-km NAM interpolated to a 32-km grid, while the remaining three cases used an experimental, hourly updated version of the NAM run at grid spacing close to that of the operational NAM CONUS nest. One of the more unique aspects of this experimental system was its distinct data assimilation cycle for both the 12-km parent and CONUS nest domains. Initial NMMB runs of the 29 June 2012 derecho were initialized off of the 13-km Rapid Refresh version 2 (RAPv2) provided by the NOAA Earth System Research Laboratory (ESRL).

Precipitation histograms were constructed from 3-hourly model output and 3-hourly observations from the Climatology-Calibrated Precipitation Analysis (CCPA; Hou et al. 2014). Composite reflectivity histograms were constructed from hourly model output and hourly radar observations described in Liu et al. (2016). Both the precipitation and reflectivity data were interpolated onto a common 5-km grid using a neighbor interpolation option (Accadia et al. 2003).

3. The Ferrier–Aligo microphysics scheme

a. General principles

As with the FER scheme, the FA scheme is also a single-moment scheme that predicts the mixing ratios of cloud water (qc), rain (qr), cloud ice (qci), and snow–graupel (qs). A thorough description of the general microphysical scheme relationships is in appendix A, and a detailed description of the production terms is in appendix B. Also shared between the two schemes is the calculation of a diagnostic array called the “rime factor” (RF), which represents the degree of riming onto snow–graupel. The definition of the RF was changed to take into account the temperature of the ice particle, the impact velocity of the cloud droplet on the ice particle, and the size of the cloud droplet (see appendix C), and this was motivated by results that will be discussed in the subsections below. For all practical purposes, one can categorize precipitation ice as snow, graupel, or hail, similar to the ice species predicted in other microphysical schemes based on the value of the RF. For example, an RF = 1 represents unrimed snow; lightly rimed snow occurs when 1 < RF < 2; heavily rimed snow when 2 < RF ≤ 5; graupel when 5 < RF < 10; and frozen drops or hail when RF ≥ 10.2 In reality, the RF knows no arbitrary cutoff between different ice categories, and the categorizations above are somewhat subjective. Hereafter, we will also refer to mixing ratios of snow/graupel as precipitation ice. An advantage to having one precipitation ice category is the absence of an artificial boundary separating one precipitation ice species from another, which allows for a smoother evolution of particle properties (e.g., Morrison and Milbrandt 2015; Lin and Colle 2011). This approach is conceptually similar to that employed by Morrison and Milbrandt (2015), who used a single ice category and predicted its attributes, including rime volume and rime growth. Single-moment schemes that use a separate snow and graupel species include, but are not limited to, Weather Research and Forecasting (WRF) single-moment 6-class (WSM6; Hong et al. 2004; Hong and Lim 2006), which is used operationally in the NCEP High-Resolution Window (HIRESW) forecast system, and the Lin scheme (Lin et al. 1983).

Owing to operational computation constraints, the sedimentation process in both the FER and FA schemes does not use finite differencing of precipitation fluxes in the vertical in order to circumvent the requirement that small time steps be used in order to maintain numerical stability, particularly since the vertical resolution of the model increases dramatically near the ground. The algorithm is instead based upon a partitioning of precipitation already present in the grid box at the beginning of the time step and the precipitation entering the grid box from above at the end of the time step. A more detailed description of the sedimentation algorithm can be found in appendix D.

Throughout the rest of the model outside of the microphysics, the cloud ice (qci) and precipitation ice (qs) are combined into a total ice mixing ratio (qi = qci + qs). While there are no resolution restrictions to using the FA scheme, it is intended for convection-allowing resolutions where most of the benefits would more likely be noticed. The subsections below describe the components of the FA scheme developed through microphysical scheme sensitivity experiments that are modifications and additions to the FER scheme.

b. Modified ice nucleation

In the FER and FA microphysics, the mixing ratio of cloud ice is calculated as
e1
where Nci is the cloud ice number concentration given by
e2
Nin represents the number concentration of ice crystals formed from ice nucleation, ρa is the air density, and Mci is the mean mass of the cloud ice crystals, which is assumed to be
e3
where is a lookup table of the normalized mass of unrimed snow defined in (A22), and = 50 μm is the minimum mean diameter assumed for unrimed (RF = 1) snow. Several different relationships for estimating Nin were tested, in which Meyers et al. (1992) was used in FER but was replaced by a combination of Fletcher (1962; NinFL) and Cooper (1986; NinC) in the FA microphysics over concern that Meyers et al. (1992) could produce too much ice at higher temperatures. It was thought that delaying the onset of ice initiation at higher temperatures could enhance reflectivity in the poorly defined convective region seen in the FER scheme (shown later); however, sensitivity experiments indicated no noticeable impact on MCS convective region reflectivity between different ice nucleation relationships. Minimal impact on precipitation totals in winter nonconvection cases (not listed in Table 1) was found (not shown) using the different relationships. Figure 1, which is a modified figure from Thompson et al. (2004), compares the different Nin formulations as functions of temperature, and the caps on the Fletcher and Cooper curves were those presented in Thompson et al. (2004). To promote the presence of supercooled liquid water, Nin is defined as the minimum value of the two formulations noted above (red dashed line in Fig. 1), where
e4
If no ice is present, then ice nucleation occurs only if there is supersaturation with respect to water and the air temperature is ≤−12°C, following the constraints used in a version of the Thompson scheme available at the time. An additional constraint in the FA scheme caps the maximum cloud ice number concentration at 250 L−1, but the cap can also be larger by an amount equivalent to 10% of the total ice content. This latter criterion occurs only in targeted conditions at temperatures colder than −35°C where the total ice contents are high, such as at upper levels in areas of deep convection, where there are large uncertainties in the number concentrations and size distributions of ice crystals, noted in Yuan et al. (2010). Although experiments that increased the assumed fraction of cloud ice from 10% to 50% of the total ice content at temperatures colder than −35°C led to increased transport of ice into anvil areas adjacent to deep convection, fractions larger than 20% were found to degrade the vertical structure of radar reflectivity and led to spurious development of stratiform precipitation in areas that were not observed (not shown).
Fig. 1.
Fig. 1.

Number concentration of ice crystals formed from ice nucleation Nin (L−1) as a function of temperature (°C) showing the Fletcher, Meyers, and Cooper curves taken from Thompson et al. (2004) with some modifications. The red dashed line indicates the curve used in the FA scheme.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

c. Separate species and rime factor advection

While the FER scheme advects the total condensate mixing ratio, defined as
e5
the FA scheme advects each of the hydrometeor species on the right-hand side of (5) separately. Although initial experiments tested separately advecting the mixing ratios of cloud water, rain, and total ice, none of the experiments were able to reproduce key structures found in vertical cross-sections of observed radar reflectivity, particularly narrow and deep convective cores that penetrated well above the freezing level up into the upper troposphere. Because highly supercooled liquid water is not allowed to exist at temperatures colder than −40°C due to homogeneous freezing of cloud water, the only way to get graupel at these cold temperatures was through upward transport in strong updrafts. This was accomplished in the FA scheme by advecting the mass-weighted rime factor (qi × RF; this process will hereafter be referred to as RF advection). While the RF advection was an important first step in improving the structure of deep convection, it was clear that further modifications to the FA microphysics were necessary because the RF advection led to an ill-defined stratiform rain region and a smaller anvil that was in less agreement with observations, as will be shown in the results section. The new RF calculation noted in section 3a increased anvil size as expected and was the result of lower RFs contributing to lower rimed ice fall speeds; however, the impacts were only modest. The relationship between the RF and the rimed ice fall speeds is discussed next.

d. Fall speeds of rimed ice

The mass-weighted fall speeds for unrimed ice are described in appendix A. For rimed precipitation ice, the modified fall speeds are
e6
where VrimeF ≥ 1 is a term derived from best number and Reynold number calculations from Böhm (1989) that represents the increase in rimed ice fall speeds when RF > 1, n is a tunable exponent, and Γs is the effect of air resistance on ice fall speeds. Figure 2a shows the variation of VrimeF as functions of RF and , which are stored in a lookup table. Maximum values of VrimeF correspond to 3.8-times-higher fall speeds for the largest ( = ) and most heavily rimed ice (largest RF) over that for unrimed (RF = 1) snow. Although heavily rimed ice fall speeds in the FER scheme could exceed 14 m s−1 (purple line in Fig. 2b), forecast fall speeds were closer to the unrimed values shown in Fig. 2b because an RF > 1 occurred over limited areas. An RF > 1 was much more widespread when the RF was advected, resulting in higher rimed ice fall speeds, which led to poorly defined anvil/stratiform regions, as well as high precipitation and reflectivity biases in the convective region (shown later in the results section). Because of the uncertainties associated with the fall speeds of rimed ice particles (e.g., Böhm 1989; Potter 1991; Johnson et al. 2015), sensitivity experiments were made using different values of the tunable exponent (n) in (6), with the result being a reduction in n from a value of 2 in the FER scheme to a value of 1 in the FA scheme. The reduction in rimed ice fall speeds in the FA scheme reduced a high bias in heavy rainfall and improved (increased) reflectivity in the stratiform–anvil region, as shown in the results section, but it often led to low reflectivity biases in convective regions that tended to also be too broad. To maintain narrow, coherent reflectivity structures in the convective region that closely matched observations, larger graupel fall speeds using a relationship from the Thompson scheme were used only when all three of the following conditions were met: 1) precipitation ice mass contents larger than 1.57 g m−3, 2) RF ≥ 5, and 3) Ds = Dsmax.
Fig. 2.
Fig. 2.

(a) Ice fall speed (m s−1) increase, VrimeF, as a function of RF (ordinate) and (abscissa). (b) Ice fall speeds (m s−1) vs mean particle diameter (mm) shown for unrimed ice (blue), the FER scheme (purple), and the FA scheme (red). An RF of 45 was assumed for the curves shown. Calculations are valid for P = 105 Pa, thus simplifying calculations here by making Γs = 1.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

e. Variable rain intercept parameter (Nor)

In response to concerns from both WPC and SPC over sharp gradients in precipitation associated with summertime deep convective storms, and a lack of reflectivity–rainfall at lower levels behind convective lines in 4-km operational NAM CONUS nest forecasts, respectively, a variable Nor was introduced to improve stratiform rainfall using results from 3-km NMMB runs with the Thompson microphysics. Because the Thompson scheme predicts the mixing ratios and number concentrations of rain drops (i.e., double moment in rain), Thompson scheme runs tended to have more robust stratiform rain regions with larger mean drop sizes ( and smaller Nor values than those forecast in FER, which resulted in less loss of rain by evaporation. An algorithm was developed in FA that allows Nor to vary with height and the mean drop diameter to be fixed below melting layers. In FER, like other single-moment microphysics schemes (WSM6 and Lin), a constant value for Nor is assumed. The algorithm in the FA scheme, similar to what is done in the Thompson scheme (Thompson et al. 2008), assumes that a snow–graupel particle about to enter the melting layer from above has the same mean mass as a drop formed from melting below the melting layer. The mean drop diameter calculated below the melting layer acts as the lower limit for the mean drop sizes as the rain descends to lower levels. This algorithm is only active if 1) the snow–graupel density above the melting level (i.e., Tc < 0°C) is <225 kg m−3 (which corresponds to an RF = 10), 2) the rain content does not exceed 1 g m−3, and 3) there is vertical continuity of the rain at lower levels with the rain that formed from melting ice. The impact of the Nor modifications on stratiform precipitation will be shown in section 4.

Several changes were made to reduce spurious areas of light (<20 dBZ) radar echoes that developed at the top of moist boundary layers over the southeastern United States, within warm conveyor belts, and over ocean areas off the Pacific and Atlantic U.S. coasts. The spurious light rain–drizzle often became a distraction to forecasters who were evaluating forecast reflectivity from the NAM CONUS nest. While the lack of low-level light reflectivity observations may have been due in part to hydrometeors located below the radar beam, similar biases were seen close to radar sites. Additionally, satellite observations often revealed the NMMB forecast boundary layer was too cloudy (not shown), and a high light rainfall bias is evident from the rainfall histograms shown in the results section.3 The first attempt to eliminate the spurious drizzle was to increase the number concentration of cloud water Ncw from 200 to 300 cm−3 and introduce a threshold autoconversion as described in appendix B. These microphysics changes only reduced some of the spurious radar echoes and light rainfall. A larger impact was seen using a variable Nor following Westbrook et al. (2010), an approach conceptually similar to that described in Thompson et al. (2008) for drizzle. Figure 3a shows an example of drizzle forming in a single low-level liquid cloud layer above 0°C, in which the smaller, more numerous drizzle drops produce lower radar reflectivities, compared to rain, with Nor = 8 × 106 m−4 assumed in the FER scheme. For multiple cloud layers, drizzle from low clouds must be completely disconnected from rain formed aloft from melting ice, such that a rain-free layer must separate any stratiform rain layer aloft from drizzle formed within liquid clouds at lower levels. Supercooled drizzle is also allowed to form from warm-rain processes below 0°C. The quantity Nor is modified only when the rainwater content is <0.5 g m−3, such that Nor is assumed to vary (red line in Fig. 3b) with rain content (ρa × qr) as
e7
In the future, an empirical relation to better fit the observational data will be used. The impact of the Nor modifications on light rain/drizzle is presented in section 4.
Fig. 3.
Fig. 3.

(a) Schematic illustration of the drizzle parameterization for a single cloud layer in which drizzle forms from a low-level liquid water cloud at >0°C only when it is completely disconnected from rain formed from melting ice aloft. (b) The scatterplot from Westbrook et al. (2010) shows retrieved rain rate (R, mm h−1) vs the normalized rain intercept parameter (NL in m−4, where NL = Nor for exponential distributions) based on lidar observations of drizzle. The different values of Nor described in (7) are overlaid on the figure with the red line showing the variation of Nor as a function of rain rate for rain contents between 0.02 and 0.5 g m−3.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

f. Treatment of precipitation ice

This section describes the evolution of the size distribution assumptions from the FER scheme to the FA scheme, detailing the changes made to reduce high anvil reflectivity biases and the development of three RF regimes in order to improve convective region storm structure. The end of this section describes a diagnostic algorithm to increase reflectivity in the convective region under limited circumstances.

The initial estimate for the precipitation ice mixing ratio is calculated as the difference between the total ice mixing ratio and the cloud ice mixing ratio,
e8
This value is used to calculate the time-averaged mixing ratios, number concentrations (Ns), mean diameters (), and mass-weighted fall speeds () of the precipitation ice. The FER and FA schemes differ from some other single-moment schemes by assuming the intercept for the precipitation ice size distributions (Nos) varies as a function of temperature, mass content, and the degree of riming (RF). Recent studies have shown the advantages of using a variable snow intercept parameter from improving short-term forecasts of precipitation to providing more realistic simulations of warm-season convection (e.g., Pan et al. 2016; Wainwright et al. 2014). In the FER scheme, the initial mean diameter of the snow–graupel is
e9
where (9) is based on the mean particles sizes from Houze et al. (1979). The quantity = 50 μm is the minimum mean diameter, = 1 mm is the maximum mean diameter, and
e10
where Tc is the air temperature (°C). While the number concentration of precipitation ice (Ns) using (A21) is calculated to be
e11
an upper limit is sometimes enforced, depending on the version of the microphysics. In FER, the upper limit on Ns (Nsmax) was assumed to vary from 5 to 20 L−1, depending on the version of the scheme. However, after comparing with results from the Thompson microphysics, constraints on Ns were modified in the FA scheme to be a function of RF (or ice density). The assumptions for each of the three RF regimes were chosen by evaluating simulated storm characteristics and analyzing vertical cross-sections of cloud, precipitation, and thermodynamic fields for multiple cases. In the first RF regime where RF ≤ 2, little riming onto snow is consistent with assuming that stratiform conditions prevail, where is calculated from (9). This treatment produced a more stratified behavior in Ns and , with height and temperature that compared favorably with runs using the Thompson scheme. Only in this regime is Ns limited not to exceed Nsmax, which is assumed to be 250 L−1 in FA. The larger values of Ns that developed at lower temperatures produced smaller mean sizes for snow and reduced the high biases in radar reflectivity in the upper portions of stratiform anvil regions (see section 4). In the second regime, when 2 < RF ≤ 10, a transition from snow to graupel-like properties is assumed. For 2 < RF < 5, the mean diameter is first estimated to be a blend of (9) and the following mean diameter when the RF = 5:
e12
where a graupel density ρg and number concentration Ng are assumed to be 300 kg m−3 and 5 L−1, respectively. For 5 < RF < 10, is a blend as a function of RF between estimates from (12) and the in the third regime when RF ≥ 10 and hail-like properties are assumed:
e13
where a hail density (ρh) and number concentration (Nh) are assumed to be 500 kg m−3 and 1 L−1, respectively. If in any of the RF regimes, then is set to . Once is computed, Ns is calculated from (11) and then used to calculate the microphysical processes. For reference, the WSM6 and Thompson schemes assume a graupel density of 500 kg m−3. When run in hail mode, the WSM6 scheme assumes a hail density of 700 kg m−3.

In either FER or FA, if Ns > Nsmax, then Ns = Nsmax, and new estimates for are calculated from an updated mean mass with the requirement that . When precipitation ice contents are high and , then Ns > Nsmax can occur following (11) in order to preserve mass conservation. Because it is rare for Ns > Nsmax in the FA scheme, new estimates of are not usually calculated, except when lightly rimed (RF ≤ 2) snow is moderately high at cold temperatures (<−30°C).

Reducing the rimed ice fall speeds tended to reduce the magnitude of reflectivity in the convective region, as more of the precipitation ice particles were able to advect rearward into the stratiform region. To recover the higher reflectivity, a diagnostic algorithm was employed in FA in areas with high graupel/hail contents (RF > 5) exceeding 2.5 g m−3, for which large subgrid-scale variability is assumed with the observed reflectivity being dominated by returns from the fewest, largest particles. The number concentrations of graupel/hail were assumed to decrease proportionally with mass content to a value of Ns = 1 L−1 at 5 g m−3. Values of Ns were assumed to decrease further to a minimum value of Ns = 0.2 L−1, when graupel–hail contents reached 10 g m−3 and were assumed to be fixed at Ns = 0.2 L−1 for larger graupel–hail contents. This algorithm produced reflectivities of 56 and 69.1 dBZ when the graupel–hail mass contents reached 5 and 10 g m−3, respectively. This algorithm does not modify process rates, and the thresholds above were determined through iterative case study analyses.

4. Results

a. RF advection

When the mass-weighted RF was advected, convective region reflectivity (i.e., ≥45 dBZ) was often higher, and these higher values were often seen at temperatures colder than −40°C. An example of this can be seen from a 9-h (09h) forecast of the progressive derecho event of 29 June 2012, valid at 0000 UTC 30 June 2012, where only scattered areas of >50-dBZ composite reflectivity were present without the RF advection (Fig. 4a), compared to a larger area predicted in the RF advection run (Fig. 4b). Without the RF advection, vertical cross sections indicated >50-dBZ values were limited to temperatures warmer than −20°C (Fig. 4c), but with the RF advection, >50-dBZ echoes extended to higher levels at temperatures colder than −40°C (Fig. 4d). Since the observed derecho was more progressive than in any of the NMMB runs, observed composite reflectivity and vertical cross-sections of reflectivity from earlier times (valid at 2100 and 2200 UTC 29 June 2012) are shown in Fig. 5 for comparison. Intense convective cores with >50-dBZ reflectivities extended higher than 10 km above ground level (AGL) at both times. In the FA cross sections shown, −40°C is at approximately 250 hPa, indicating the run with the RF advection more correctly forecasted the vertical depth of the 50-dBZ reflectivity.

Fig. 4.
Fig. 4.

NMMB 9-h forecast valid 0000 UTC 30 Jun 2012 of simulated maximum-in-column reflectivity (dBZ) (a) without and (b) with RF advection; and (c),(d) corresponding vertical cross-sections of reflectivity and temperature [°C, dashed (<0) and solid lines (≥0)] along line AB, respectively. Horizontal distance in km is plotted along the abscissa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Fig. 5.
Fig. 5.

(a),(b) Maximum-in-column reflectivity and (c),(d) vertical cross-sections of reflectivity valid along line AB at (a),(c) 2100 and (b),(d) 2200 UTC 29 Jun 2012. Horizontal distance in km is plotted along the abscissa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Without the RF advection, most of the ice was in the form of cloud ice and snow (Fig. 6a), with only a small amount in the form of graupel (Fig. 6c), but with RF advection, cloud ice and snow mixing ratios were lower by a factor of 2 in the convective region (Figs. 6b,c), and graupel mixing ratios doubled in magnitude and extended to lower temperatures (Fig. 6d). From the reflectivity cross sections, it is clear the stratiform rain region suffered with the RF advection (Fig. 4d) as the result of higher RF values extending farther back into the stratiform region and to lower temperatures aloft (Figs. 6e,f). When RF increases, VrimeF and the resultant precipitation ice fall speeds (Fig. 2a) increase. With more precipitation ice in the form of faster-falling graupel, there was less of the lower-density, slower-falling snow that could be advected rearward into the stratiform region.

Fig. 6.
Fig. 6.

Vertical cross-sections of (a),(b) cloud ice + snow, and (c),(d) graupel in g kg−1 (color shaded), as well as (e),(f) RF for an NMMB run (left) without and (right) with RF advection. Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 0000 UTC 30 Jun 2012 (forecast hour 09). Horizontal distance in km is plotted along the abscissa. Snow and cloud ice are assumed to occur where RF ≤ 5 and graupel where RF > 5. The RF values are shown for mass contents ≥ 0.1 kg m−3.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Aggregating hourly composite reflectivity statistics over all 11 cases (Table 1) corroborated the subjective results that the RF advection reduced the size of the stratiform region, as seen by lower 15–40-dBZ reflectivity counts (Fig. 7a), and increased the instances of high (>45 dBZ) reflectivity (Fig. 7b). With the RF advection, there was a high bias in high reflectivity; however, without the RF advection, there was a low bias in the high reflectivity. A common complaint concerning the NAM CONUS nest prior to the implementation of the FA scheme in August 2014 was a low bias in high reflectivity, brought to our attention by SPC and NWS forecast offices; advecting the RF as is done currently in the operational NAM CONUS nest addressed those concerns. With the RF advection, instances of light precipitation (≤0.5 in.) were reduced (Fig. 7c), while instances of heavy precipitation (≥1 in.) were increased (Fig. 7d), compared to runs without the RF advection. This was not a surprising result, considering the partitioning of precipitation ice to be more in the form of higher-density ice when the RF is advected.

Fig. 7.
Fig. 7.

Histograms of hourly composite reflectivity (dBZ) aggregated over all 11 cases for reflectivity at 5-dBZ intervals starting with (a) 5 dBZ and (b) 45 dBZ. Histograms of 3-hourly accumulated precipitation aggregated over all 11 cases for amounts (c) starting with 0.01 in. and (d) starting with 1 in. Displayed in the figure are the observed counts (red) and counts from runs with the RF advection (blue) and without the RF advection (green). Rounding of the counts above the bars was done to make the plot more legible.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

b. Fall speeds of rimed ice

Sensitivity experiments were made using different values of the tunable exponent (n) in (6) that controls the increase in the rimed ice fall speeds to reduce a high bias in heavy precipitation, which was a common complaint especially from WPC due to the RF advection. After an extensive evaluation using the cases in Table 1, it was determined that a value of n = 1 reduced the high precipitation biases while not degrading the reflectivity structure of MCSs. Figure 8 shows that the area of 20-dBZ composite reflectivity was too small in spatial extent with n = 2 (Fig. 8a) and in much better agreement with observations (Fig. 5) when n = 1 in the derecho case (Fig. 8b). Very little stratiform reflectivity/rainfall existed in the run with n = 2 (Fig. 8c), while a more robust stratiform region could be seen with n = 1 (Fig. 8d) that more closely matched observations (Fig. 5). When n = 2, the precipitation ice fall speeds in the convective region were as high as 15 m s−1 (Fig. 8e), but with n = 1, they generally did not exceed 7 m s−1 (Fig. 8f). Slowing the rimed ice fall speeds allowed more of the heavily rimed ice to be advected rearward, which resulted in a better-defined stratiform region reflectivity. Not surprisingly, the instances of light precipitation (≤0.5 in.) were higher with n = 1 (Fig. 9a), while the instances of heavy precipitation (≥1 in.) were lower, compared to the n = 2 runs (Fig. 9b). Although a high bias in precipitation amounts ≥2 in. might still be present with n = 1 (the current setting used in the operational NAM), it was an improvement over the higher biases when n = 2.

Fig. 8.
Fig. 8.

Maximum-in-column reflectivity (dBZ) for runs with (a) n = 2 and (b) n = 1, as well as corresponding vertical cross-sections of (c),(d) reflectivity and (e),(f) precipitation ice fall speeds (m s−1). Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 0000 UTC 30 Jun 2012 (forecast hour 09). Horizontal distance in km is plotted along the abscissa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Fig. 9.
Fig. 9.

Histograms of 3-hourly accumulated precipitation aggregated over all 11 cases for amounts (a) starting with 0.01 in. and (b) starting with 1 in. Displayed in the figure are the observed counts (red), and counts from runs with n = 1 (blue) and with n = 2 (green). Rounding of the counts above the bars was done to make the plot more legible.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

c. Variable Nor

Implementation of a variable Nor improved stratiform reflectivity and rainfall at low levels with forecasts more comparable to observations. For example, a comparison of observed hybrid scan radar reflectivity valid at 1500 UTC 1 July 2015 (Fig. 10a; Table 1) and 1-km AGL reflectivity for a 15-h forecast from a Thompson scheme run (Fig. 10b) indicates a larger stratiform rain region in the FA run using a variable Nor instead of a constant Nor (Figs. 10c,d), closer in spatial extent to what was observed and similar to what was seen in the Thompson run. Similar patterns were seen at other forecast times (not shown). While one should be careful making direct comparisons of 1-km AGL reflectivity from the model output to the hybrid scan reflectivity as different parts of the storm are being sampled, the areal coverage can be gleaned from the comparison.

Fig. 10.
Fig. 10.

Hybrid scan reflectivity (dBZ) and 1-km AGL reflectivity (dBZ) from (b) a Thompson run, (c) an FA run with a fixed rain intercept parameter Nor, and (d) an FA run with a variable Nor valid at 1500 UTC 1 Jul 2015 (forecast hour 15). The black solid line with the start and end points marked as A and B, respectively, represents the locations of vertical cross sections shown in Fig. 11.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

The more robust stratiform rain region in the Thompson microphysics run, also depicted in the vertical cross-sections of reflectivity (Fig. 11a), was associated with number-weighted mean drop sizes exceeding 400 μm, which remained nearly constant with height down to the surface (Fig. 11b). Stratiform region reflectivity from rain struggled to reach the surface in the FA run with a constant Nor (Fig. 11c), as drop sizes were often less than 250 μm (Fig. 11d) and decreased dramatically in size approaching the surface. Larger drops surviving to the surface were more evident in the stratiform region of the FA run with a variable Nor (Figs. 11e,f) and were a result of drop sizes assumed to be fixed with height below the melting layer, allowing the rain intercept parameter to vary with height.

Fig. 11.
Fig. 11.

Vertical cross-sections of (a) reflectivity (dBZ) and (b) number-weighted, mean drop sizes (μm) of rain for the Thompson run and corresponding cross-sections for FA runs; (c),(d) using a fixed Nor and (e),(f) using a variable Nor. Refer to Fig. 10 for the location of the cross sections. Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 1500 UTC 1 Jul 2015 (forecast hour 15). Horizontal distance in km is plotted along the abscissa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Introducing a variable Nor also removed some of the spurious widespread light reflectivity often forecast in FER runs. At 1300 UTC 23 May 2003 (observations were not available at 1200 UTC), observations showed convection dissipating in Oklahoma (Fig. 12a), with a similar system forecast an hour earlier in the Thompson, FER, and FA runs (Figs. 12b–d). The FA run using a constant Nor had widespread scattered radar echoes <20 dBZ throughout central Texas, while the area and intensity of the scattered light reflectivity was reduced and agreed better with observations in the FA run using a variable Nor. These differences were present at other forecast times and in almost all of the cases were evaluated especially around sunrise (not shown).

Fig. 12.
Fig. 12.

(a) Observed composite reflectivity valid at 1300 UTC 24 May 2016 compared against 24-h forecasts from (b) a Thompson run, (c) an FA run using a fixed Nor, and (d) an FA run using a variable Nor valid at 1200 UTC 24 May 2016. No observations were available at 1200 UTC.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

d. FER versus FA

Several factors that contributed to improvements in storm structure in the FA scheme include the RF advection, the faster Thompson-based graupel fall speeds used for large RFs, the diagnostic enhancement of reflectivity, and the forcing of smaller Ns values in the convective region. Additionally, a high bias in anvil reflectivity, noted via the analysis of many vertical cross sections, was reduced when Nsmax was relaxed to be more consistent with results from Thompson scheme runs. The examples below highlight the differences between the FER and FA runs for two representative cases.

One of the features that stood out in the observed reflectivity of the 29 June 2012 derecho (Figs. 5c,d) was the narrow, nearly solid line of deep intense convection. Slightly less organized, but with similar intensity, the Thompson scheme run produced a well-defined, deep convective core with a robust stratiform region (Figs. 13a,b), as can be seen at 09 h, valid 0000 UTC 30 June 2012. However, the FER run had a more diffusive convective region (Fig. 13c), with composite reflectivity values that barely reached 50 dBZ and with the vertical extent of the 50-dBZ echoes restricted to below the melting level (Fig. 13d). The FA run was an improvement over the FER run, with higher convective region reflectivity (Figs. 13e,f), more intense echoes in the convective region, and 50-dBZ echoes extending to lower temperatures (below −40°C). None of the runs simulated the well-organized, solid line of convection that was observed (Figs. 5a,b).

Fig. 13.
Fig. 13.

(a) Composite reflectivity (dBZ) and (b) vertical cross-section of reflectivity for the Thompson run with the corresponding plots for the (c),(d) FER and (e),(f) FA runs valid at 0000 UTC 30 Jun 2012 (forecast hour 09). Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 0000 UTC 30 Jun 2012 (forecast hour 09). Horizontal distance in km is plotted along the abscissa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

The 09-h forecast reflectivity cross sections behind the convection from the Thompson scheme run (Fig. 13b) and from observations (Figs. 5c,d) indicated that the FER run produced anvil reflectivities that were too high by as much as 15 dBZ (Fig. 13d), compared with more reasonable values in the FA run (Fig. 13f). Anvil reflectivity at temperatures near −30°C approached 30 dBZ in the FER run versus 15 dBZ in the Thompson run and in the FA run. In the Thompson run, Ns values exceeded 250 L−1 at very cold temperatures (Fig. 14a) and were generally <30 L−1 in the FER run (Fig. 14b) and as high as 250 L−1 in the FA run (Fig. 14c).

Fig. 14.
Fig. 14.

Vertical cross-section of snow number concentration (Ns) for the (a) Thompson, (b) FER, and (c) FA run at the same time and cross sections described in Fig. 13. Units are L−1.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

The FER and FA schemes were also compared for the Moore, Oklahoma, tornado outbreak on 20 May 2013. At 2200 UTC 20 May 2013, large, isolated storms extended southwest to northeast across Oklahoma (Fig. 15a). A cross section through one of the convective storms showed a strong cell with 50-dBZ reflectivity extending up to 11 km AGL (Fig. 15b). A run with the FER microphysics produced two lines of storms that were oriented correctly, but with a mode that was too linear (Fig. 15c). A vertical cross section through the primary (strongest) line of storms showed reflectivity barely exceeding 45 dBZ (Fig. 15d). In the FA run, the mode of convection was more discrete and in better agreement with observations with higher reflectivity in the convective cores (Fig. 15e), a feature seen at other times and in other cases. A vertical cross section through one of the storms in the FA run showed the 50-dBZ reflectivity extended to temperatures near −40°C (~225 hPa), which was closer to the observed height of the 50-dBZ echo (Fig. 15f). As in the derecho case above, the anvil reflectivity in the FA run was in better agreement with observations than the anvil reflectivity in the FER run.

Fig. 15.
Fig. 15.

(a) Composite reflectivity (dBZ) and (b) vertical cross-section of reflectivity from observations with corresponding plots from (c),(d) an FER run, and (e),(f) an FA run, valid at 2200 UTC 20 May 2013 (forecast hour 22). Each cross section contains temperature [°C, dashed (<0) and solid lines (≥0)] with all fields valid along line AB and at 2200 UTC 20 May 2013 (forecast hour 22). Horizontal distance in km is plotted along the abscissa.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Aggregating the composite reflectivity statistics over all 11 cases indicated that for reflectivity ≤25 dBZ, a high bias existed in both the FER and FA runs; however, the high bias was reduced in the FA run, likely in areas associated with lower anvil reflectivity, and by representing rain as drizzle within shallow clouds that form at the top of moist PBLs (Fig. 16a). While a forecast high bias in the reflectivity ≤25 dBZ might exist, an underestimate of the observed counts at these lower thresholds is possible because the lowest 0.5° elevation angle used at most radar sites will not capture widespread light precipitation often seen at low levels (Zhang et al. 2005). Although a high bias can be seen in both the FER and FA schemes for the 45–50-dBZ reflectivity bin (Fig. 16b), there was a pronounced low bias in >55-dBZ reflectivity in the FER scheme that was eliminated with the FA scheme. Precipitation histograms were also aggregated over all 11 cases. The biases were reduced for precipitation amounts ≤0.5 in. (Fig. 16c), which was the result of the RF advection reducing the stratiform precipitation. The reduced bias was an improvement for the 0.01–0.1 in. precipitation bin, but resulted in a slight low bias in precipitation in the FA scheme for the 0.1–0.5 in. bins. For the 1 in.+ amounts (Fig. 16d), the FER scheme had a low bias, with 11% fewer counts than observed, while the FA scheme had a slight high bias, with 3% more counts than observed. The higher precipitation bias in the FA scheme is the result of having higher-density, faster-falling precipitation ice in the convective region.

Fig. 16.
Fig. 16.

Histograms of hourly composite reflectivity (dBZ) aggregated over all 11 cases for reflectivity at 5-dBZ intervals starting with (a) 5 and (b) 45 dBZ. Histograms of 3-hourly accumulated precipitation aggregated over all 11 cases for amounts (c) starting with 0.01 in. and (d) starting with 1 in. Displayed in the figure are the observed counts (red), and counts from FER runs (blue) and FA runs (green). Rounding of the counts above the bars was done to make the plot more legible.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

5. Summary and conclusions

The flagship microphysics of the NAM for more than a decade was updated with the FA microphysics scheme in order to improve the vertical structure of convective storms in the NAM CONUS nest. Running with a more sophisticated microphysics scheme like Thompson was not viable at the time; however, results from Thompson scheme runs proved useful in developing the fast FA scheme and addressing the concerns from forecast centers and NWS forecast offices. Notable components of the FA scheme include the advection of separate species in addition to the advection of the mass-weighted RF, which resulted in graupel reaching very cold temperatures. The RF advection, as well as the reduced number concentrations of precipitation ice (Ns) assumed in areas of convection, helped eliminate the low reflectivity biases seen in FER runs in areas associated with deep convection. The RF advection resulted in a high bias in heavy precipitation that was noted by WPC, and to counter this effect, the rimed ice fall speeds were reduced. The use of graupel fall speeds from the Thompson scheme in intense convection allowed for more coherent, narrow convective regions, while a diagnostic method was added to boost reflectivity in the most intense convection. A high reflectivity bias in the upper portions of stratiform anvils at temperatures colder than −10°C was reduced, allowing the number concentrations of snow to reach 250 L−1, which is more consistent with results seen in Thompson microphysics runs.

A variable rain intercept parameter was introduced into the FA scheme to address concerns regarding rainfall outside the convective region. Larger rain number concentrations and smaller drop sizes reduced or eliminated spurious, widespread radar echoes that developed in shallow liquid water clouds that form at the top of the PBL, as was often noted in the operational 4-km NAM CONUS nest. In areas associated with light-to-moderate precipitation rates away from areas of active convection, the mean drop diameter was conserved rather than the rain intercept, reducing the loss of rain by evaporation and allowing more rain produced from melting ice to reach lower levels, which reduced (improved) the low bias in light rainfall and low-level reflectivity.

Development of the FA scheme initially addressed concerns raised by SPC and NWS forecast offices regarding the proper representation of deep convective storms. Further modifications to the FA scheme addressed concerns from WPC regarding a high heavy precipitation bias and large gradients in forecast precipitation. Currently, the majority of modeling systems employing convection-permitting grid spacing exist primarily in the regional domain. However, in the coming years, it is likely that convection-permitting applications will extend to the global domain, where the challenges are vast and the impacts more far-reaching (e.g., tropical convection and the global circulation). Efficient yet sophisticated microphysics schemes, such as FA, will play a significant role in the development toward such a high-resolution, global modeling paradigm.

Acknowledgments

The authors thank SPC, particularly Steven Weiss and Israel Jirak. Matthew Pyle and Eric Rogers are thanked for technical support and valuable comments. The authors would also like to thank ESRL for providing RAPv2 grid data and Greg Thompson for incorporating his microphysics into the NMMB. Comments and suggestions from the anonymous reviewers led to substantial improvements to the original manuscript.

APPENDIX A

General Microphysics Relationships

a. Mass–diameter relationships

All liquid and ice species follow the mass–diameter relationship described by
ea1
where mx(Dx) is the mass associated with a particle of diameter Dx. Liquid drops in the form of small cloud droplets (x = w) and larger rain drops (x = r) are assumed to be spherical in shape, where
ea2
and ρw = 1000 kg m−3 is the density of liquid water. The mass of unrimed ice particles as a function of their maximum diameter [mus(Ds)] is assumed to be an equal mix of three different ice habits (bullets, columns, and plates) for crystals sizes <1.5 mm and an equal mix of three different types of aggregates for larger size particles, where
ea3
with values of ai and bi for i = 1–6 listed in Table A1. The mass–diameter for rimed ice particles (RF > 1) is assumed to be
ea4
Table A1.

Values of ai and bi used in (A3) for bullet rosettes (i = 1), columns (i = 2), and plates (i = 3) from Heymsfield (1972) and Starr and Cox (1985), while those for aggregates of unrimed radiating assemblages of dendrites or dendrites (Agg1, i = 4), aggregates of unrimed radiating assemblages of plates, side planes, bullets, and columns (Agg2, i = 5), and aggregates of unrimed side planes (Agg3, i = 6) are from Locatelli and Hobbs (1974). The mass of the ice are in units of mg and the diameters are in mm, which are converted into mks units in the final lookup table calculations.

Table A1.

b. Fall speed relationships

The fall speeds for cloud droplets are assumed to be negligible. The fall speeds for other hydrometeors are described by
ea5
where Vy(Dy) is the terminal fall speed associated with a particle of diameter Dy, Γy is the air resistance effect, and ∝y and βy are the fall speed coefficients for hydrometeors in the form of rain (y = r), snow/graupel (y = s), and cloud ice (y = i). For rain drops (y = r) ignoring air resistance effects [i.e., Γr = 1 in (A5)],
ea6
where Vr is in m s−1 and Dr is in cm. The formulation combines drop fall speeds from Rutledge and Hobbs (1983) for the smallest drops (≤0.42 cm) and Gunn and Kinzer (1949) for larger drops. The air resistance effects for drops (Beard 1985) is given by
ea7
where ρa is the air density. As noted above, the fall speeds for unrimed ice are assumed to be an equal mix of three different ice habits (bullets, columns, and plates) for crystals sizes <1.5 mm and an equal mix of three different types of aggregates for larger size particles, where
ea8
Note that multiple ice crystal fall speed relationships are used for different size intervals. The values for αi,j and βi,j listed in Table A2 correspond to the different ice habits (i = 1–6) and for different particle size intervals (j = 1, 6), where D1, D2, D3, D4, and D5 in (A8) correspond to 200, 400, 600, 800 μm, and 1.5 mm, respectively. The air resistance effects for ice particles (Starr and Cox 1985) is given by
ea9
where p is the pressure (Pa). To assist with increasing the area coverage of anvils, the fall speeds of ice were reduced below 300 hPa by enforcing an upper limit of Γs = 1.5 in the FA microphysics. The adjustment for rimed ice when RF > 1 is discussed in section 3d.
Table A2.

Values of αi,j and β i,j used in (A8) for the same set of ice habits listed in Table A1, but for different size intervals. Ice fall speeds are in units of m s−1. The fall speed relationships for the first five size categories of ice crystals (j ≤ 5) are from Starr and Cox (1985) and Heymsfield (1972), in which the particle diameters are in units of μm. The fall speed relationships for the largest size category (j = 6), which involve different types of aggregates, are from Locatelli and Hobbs (1974).

Table A2.

c. Microphysics moments

This subsection describes the various moments of the particle size distribution function, such as the number concentration, mean diameter, mass content, mass-weighted fall speeds, and reflectivity. An exponential distribution is assumed for the rain (z = r) and ice (z = s) hydrometeors, where
ea10
Noz is the intercept parameter, and λz is the slope parameter. The total number concentration is obtained by integrating (A10) over all particle sizes,
ea11
and rearranging terms yields
ea12
The mean diameter () is defined as
ea13
After integrating the numerator, using (A12) to represent the denominator, and rearranging terms, this yields
ea14
A general expression for the mass content of rain (z = r) and ice (z = s) is
ea15
For rain mass contents (i.e., z = r), substituting (A10) for Nr(Dr) in (A15), using (A2), then rearranging terms yields
ea16
where
ea17
and Nor = 8 × 106 m−4 (Marshall and Palmer 1948) is assumed as a nominal value (note that different values of Nor are used for drizzle and stratiform rain described in section 3e). Lookup tables, which store values of MASSr() as a function of the mean drop size () ranging from = 50 to = 1000 μm (1 mm) at 1-μm intervals, are solved by numerically integrating (A17) at 1-μm size intervals over drop sizes ranging from Drmin = 50 μm to Drmax = 1 cm. When rain contents are below the threshold value of , then Nor is adjusted such that
ea18
For precipitation ice (z = s), a similar procedure is followed where (A4) is substituted in (A15), which yields, after rearranging terms,
ea19
with
ea20
Because the intercept for precipitation ice (Nos) is not constant, and an iterative algorithm is used to calculate and Ns, (A19) is revised slightly to produce
ea21
with
ea22
Lookup tables store values of as a function of the unrimed mean ice-particle ice size ranging from = 50 to = 1000 μm (1 mm) at 1-μm intervals. For each value of , (A20) is numerically integrated at 1-μm size intervals over particle sizes ranging from Dsmin = 20 μm to Dsmax = 2 cm.
The rain rate is defined as
ea23
where is the mass-weighted fall speed for rain, which is a function of the mean drop size and Vr(Dr) is defined by (A6). Substituting (A10) into (A23) and using (A12) and (A14) yields
ea24
where
ea25
Ignoring air resistance effects (Γr = 1), the mass-weighted drop fall speeds are found by rearranging (A24) and using (A16) to produce
ea26
As with lookup tables store values of and as functions of the mean drop size ranging from = 50 to = 1000 μm (1 mm) at 1-μm intervals. For each value of , they are solved by numerically integrating (A25) over drop sizes ranging from Drmin = 50 μm to Drmax = 1 cm at 1-μm intervals.
The precipitation rate for total ice is a little more complicated because it combines the small sedimentation rates from small cloud ice crystals, along with the precipitation rates from larger ice. The precipitation rate for unrimed ice (RF = 1) is
ea27
where
ea28
Solving for in (A27) using (A21) yields the mass-weighted fall speeds for unrimed ice:
ea29
As with MASSs, lookup tables store values of IRATE and as functions of the mean ice particle ranging from = 50 to = 1000 μm (1 mm) at 1-μm intervals. For each value of , (A28) is numerically integrated over ice-particle sizes ranging from Dsmin = 20 μm to Dsmax = 2 cm at 1-μm intervals. The IRATE and tables include a mix of different ice crystal habits and types of aggregates by using (A3) and Table A1 for the mass–diameter relationships and (A8) and Table A2 for the velocity–diameter relationships. Since the mass-weighted fall speeds for small cloud ice crystals () are assumed to be the same as for unrimed snow (RF = 1), with = 50 μm,
ea30
The final precipitation rate for total ice (cloud ice and precipitation ice), which includes the contributions from small ice crystals and precipitation ice, is
ea31
The radar equation for rainwater is
ea32
and for precipitation ice, it is
ea33
following Ferrier (1994).

APPENDIX B

FER and FA Microphysics Production Terms

A schematic illustration of the microphysical scheme production terms is shown in Fig. B1. The terms are defined in Table B1, along with a list of symbols in Table B2. All units are in mks unless otherwise specified. The budget equations listed below represent the total changes integrated over the physics time step (Δt) in water vapor (DELV), cloud water (DELW), total ice (DELI), rain (DELR), and temperature (DELT):
eb1
eb2
eb3
eb4
eb5
Fig. B1.
Fig. B1.

Flowchart of the FA microphysical scheme production terms listed in Table B1.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-17-0277.1

Table B1.

List of microphysical processes, their description, and the equations in appendix B where they are defined. All processes are in units of kg kg−1.

Table B1.
Table B2.

List of symbols.

Table B2.

Each of the microphysical source sink terms referenced above will be discussed in more detail below.

a. Production terms for ice (DELI)

Cloud water is assumed to freeze rapidly by homogeneous freezing when Tc < −40°C,
eb6
such that during these conditions, ice deposition is assumed to adjust conditions to ice saturation (wsi),
eb7
where PIDEP = DEPt, w is the water vapor mixing ratio, T is the air temperature (K), Ls is the latent heat of sublimation, Rυ is the specific gas constant for water vapor, and cp is the specific heat of air at constant pressure (see also Table B2). During conditions when PIHOM > 0, a dummy temperature (T*) is used in (B7) that accounts for the rapid freezing of cloud water:
eb8
The quantity DEPt represents the upper limit for vapor deposition onto ice that results in adjusting conditions to ice saturation.
When the air is supersaturated with respect to ice (w > wsi) between 0° and −40°C, vapor deposition onto existing small ice crystals (DEPci) and large precipitation ice (DEPs) particles is
eb9
eb10
eb11
eb12
where DEPt is from (B7), Ls is the latent heat of sublimation, the thermal conductivity of water vapor is
eb13
the diffusivity of water vapor is
eb14
Nci is the number concentration of cloud ice crystals calculated from (2), Ns and are the number concentration and mean diameter of precipitation ice (respectively), and = 50 μm. The terms in brackets in (B10) and (B11) represent a combination of the capacitance and ventilation effects associated with falling cloud ice and precipitation ice particles (respectively), where
eb15
eb16
are calculated offline and stored in lookup tables that are functions of the mean precipitation ice diameters ranging from = 50 μm to = 1 mm at 1-μm intervals. The values for venti1 and venti2 as functions of ice-particle diameter (d in mm) are
eb17
eb18
where the different formulations used to calculate (B17) and (B18) are listed in Table B3. For small ice crystals where d ≤ 0.2 mm, a simplified approximation is used in (B17) that assumes a linear increase in the ventilation coefficient from a value of 1 for d ≤ 0.05 mm to 1.1 at d = 0.2 mm. The coefficients used for venti1 and venti2 are from Hall and Pruppacher (1976) for the larger ice crystals (0.2 < d < 1.5 mm), while the coefficients from Rutledge and Hobbs (1983) are used for d > 1.5-mm aggregates. For every value of , the integrations over all particle sizes assume an equal mix of three different ice-crystal habits for sizes ranging from Dsmin = 20 μm to D = 1.6 mm, and an equal mix of three different types of aggregates are assumed for larger particles ranging from D > 1.6 mm to Dsmax = 2 cm. The terms SFci and SFs in (B10) and (B11) represent the increase in terminal fall speeds of cloud ice and precipitation ice (respectively), which are assumed to be independent of particle size following (6) and contribute to enhanced ventilation,
eb19
eb20
where the dynamic viscosity is
eb21
The effects of riming that lead to increased fall speeds of precipitation ice and increased ventilation are accounted for by the VrimeFn term in (B20).
Table B3.

List of equations used to calculate venti1 and venti2, where the variable d represents the maximum ice-particle dimension in mm. For ice crystals <1.5 mm in length, the various relationships for bullets are from Heymsfield (1972, 1975), whereas for plates and columns, they are from Young (1993). For the larger aggregates whose maximum dimensions are >1.5 mm, the various relationships assume spherical particles. The fall speed relationships for the different ice habits listed in Table A2 are used to calculate Xagg and Xavg.

Table B3.
Although small ice crystals can be initiated when there is little if any ice present and the air is supersaturated with respect to ice (w > wsi) between 0° and −40°C, it is a process that is still not well understood. As noted in association with (4), ice nucleation occurs when the air reaches water saturation at temperatures colder than −12°C, when there are fewer ice cloud ice crystals present than the number of ice nuclei (Nci < Nin), such that
eb22
MY600(Tc) are tabulated values of ice crystal growth (in grams, g) from Miller and Young (1979) as a function of temperature over a period of 600 s in water-saturated conditions, the 10−3 factor is to convert the calculations from g to kg, and the (Δt/600)1.5 term normalizes the ice crystal growth calculations by the physics time step. Values of MY600(Tc) are assumed to be constant for temperatures colder than −27°C. The code includes an extra check that prevents too much ice deposition from occurring, such that
eb23
When the air is subsaturated with respect to ice (w < wsi), sublimation will occur:
eb24
where PIDEP < 0 since all terms in (B24) are <0. Because large physics time steps have been used in the NAM, PILOSS represents the largest sink that results in the loss of all ice falling into the grid box from above () and the existing ice within the layer (qi),
eb25
and Δη = ρ × Δz is the mass thickness of the model layer.
When Tc > −40°C, the collection of cloud water by precipitation ice (snow–graupel) is
eb26
eb27
where FRsw is the fraction of cloud water collected by precipitation ice,
eb28
For simplicity, the efficiency that cloud water is collected by snow is assumed to be Esw = 0.5, and
eb29
are calculated offline and stored in lookup tables that are functions of the mean precipitation ice diameters ranging from = 50 μm to = 1 mm at 1-μm intervals. For every value of , the integrations over all particle sizes use the Vs(Ds) relationships in Table A2 and assume an equal mix of the three different crystal habits for sizes ranging from Dsmin = 20 μm to D = 1.6 mm and an equal mix of the three different types of aggregates for D > 1.6 mm to Dsmax = 2 cm.
The freezing of rain drops can occur through probabilistic freezing following Bigg (1953) and by collisions with other precipitation ice particles. The probabilistic freezing of rain is
eb30
where Bfr = 100 m−3 s−1. The fraction of rain that freezes due to collisions with precipitation ice particles was derived by rearranging similar formulas from other schemes (e.g., Lin et al. 1983; Rutledge and Hobbs 1984):
eb31
where
eb32
is an approximation (Murakami 1990), and Esr = 1 is assumed, which resulted in the freezing of rain due to collisions with precipitation ice (snow–graupel) as
eb33
Both contributions to drop freezing are combined and limited to be less than the maximum amount of rain falling into the grid box from above plus the existing rain within the layer (qr), where
eb34
eb35
When Tc > 0°C, the melting of precipitation ice is described by
eb36
where and are from the lookup tables, as discussed in association with (B15) and (B16). During melting, if the air is subsaturated with respect to water and there is no cloud water present, then the vapor exchange between the melting ice and the environment is
eb37
where ws0 is the saturation mixing ratio at 0°C because it is assumed that the temperature of the liquid water coating the ice during melting is fixed at 0°C. Depending on the sign of w − ws0,
eb38
represents the loss of melting precipitation ice by evaporation (PIEVP < 0), and
eb39
represents vapor condensation onto melting precipitation ice (PICND > 0), which will be discussed in more detail in section c of this appendix. Since processes (B36)(B39) are calculated when Tc > 0°C, they are included in the final budget equations in (B1)(B5) only when the combined effects from all of the other microphysical processes do not cause Tc ≤ 0°C. The total loss of precipitation ice by melting and evaporation cannot exceed PILOSS:
eb40

b. Production terms for cloud water (DELW)

When Tc > −40°C, and the air is supersaturated with respect to water (w > ws) or cloud water is present (qc > 0) and the air is subsaturated (w < ws), the net cloud water condensation (>0) or evaporation (<0) is calculated using the algorithm of Asai (1965):
eb41
in which T, w, qc, ws, and PCOND are updated in an iterative manner until the final relative humidity is within 0.1% of the saturated value (RHgrd) or all of the cloud water is evaporated. Note that RHgrd = 100% in the 3-km NAM nests and 98% in the coarser 12-km parent domain. For example, when condensation occurs in the nests, the final relative humidity is between 99.9% and 100.1%. Although the algorithm is allowed to have as many as 10 iterations, it almost always converges within one or two iterations.

The collection of cloud water by precipitation ice (PIACW) is given by (B26).

The autoconversion or self-collection of cloud droplets to form rain drops was revised from a Kessler-type formulation in FER to the parameterization proposed by Liu and Daum (2004) and Liu et al. (2006) in FA, in which a generalized gamma distribution is assumed for cloud droplets, where
eb42
eb43
eb44
Ncw = 200 × 106 m−3 (200 cm−3) is the assumed cloud droplet number concentration, and RDIS = 0.5 is the assumed relative dispersion of the droplet distribution. Although the advantage of this approach is that it needs no threshold cloud water mixing ratio (or content), too much rain–drizzle formed in association with low-level clouds, so it was decided to reintroduce a threshold cloud water mixing ratio (qco) in FA, where
eb45
eb46
Dca = 20 × 10−6 m (20 μm) is the threshold cloud droplet diameter for autoconversion to occur (Manton and Cotton 1977; Banta and Hanson 1987), and Ncw was increased to 300 cm−3 in FA in order to further delay the onset of rain–drizzle.
The collection of cloud water by rain is
eb47
where FRrw is the fraction of cloud water collected by rain, given by
eb48
For simplicity, the efficiency that cloud water is collected by rain is assumed to be Erw = 1, Γr is the change in rain fall speeds due to air resistance effects (Beard 1985), and
eb49
are calculated offline and stored in lookup tables that are functions of the mean rain drop diameters ranging from = 50 μm to = 1 mm at 1-μm intervals. For every value of , the integrations over all drop sizes use the Vr(Dr) relationships in (A6) from Drmin = 50 μm to Drmax = 1 cm.

c. Production terms for rainwater (DELR)

Evaporation of rain will occur when it falls into subsaturated (w < ws) air with no cloud water present (qc = 0), such that
eb50
eb51
eb52
and Lυ is the latent heat of vaporization. The terms in brackets in (B51) represent a combination of the capacitance and ventilation effects associated with falling rain, but unlike ice, which has very complicated shapes, the treatment for rain is much simpler, where
eb53
eb54
are calculated offline and stored in lookup tables that are functions of the mean drop diameters ranging from = 50 μm to = 1 mm at 1-μm intervals. For every value of , the integrations over all drop sizes use the Vr(Dr) relationships in (A6) from Drmin = 50 μm to Drmax = 1 cm. The SFr in (B51) represents the increase in terminal fall speeds of rain that are independent of particle size and contribute to enhanced ventilation,
eb55
The PRLOSS term in (B50), which is calculated in (B34), is the maximum loss that removes all rain in the grid box, plus the rain that falls into the grid box from above.
For the remaining terms that have not been defined above,
eb56
represents the shedding of cloud water collecting by melting ice to form rain. It does not contribute to temperature changes because no change in phase has occurred. The condensation of vapor onto melting ice (PICND), defined in (B39), is assumed to be shed to form rain. Because a change in phase has occurred in going from vapor to liquid, PICND acts to heat the air slightly.

APPENDIX C

New Rime Factor Calculation

Among the steps that were taken to increase the size of the anvil–stratiform region in FA was modifying how RF is calculated in the microphysics. The time-averaged values of RF for the FER and FA schemes are
ec1
which is a weighted average of the RF associated with the precipitation ice in the grid box (first term in the numerator of C1) and the RF associated with precipitation ice falling in the grid box from above (second term in the numerator of C1). In (C1), K is the vertical index, with K = 1 being the first model level at the top of the domain, the symbol , and represent the mixing ratio of total ice (qi), and the RF, respectively, at model level K and time step N; while and represent the fall of precipitation in the form of total ice, and the RF, respectively, that has fallen into the grid box from above during one physics time step With ΔηK = ρa,K × ∆ZK representing the mass thickness of the grid box, the denominator in (C1) represents the total mass of precipitation ice available in the grid box before the microphysics updates:
ec2
Within FER, the updated RF after the microphysics was
ec3
in which it was assumed that the riming of cloud water onto ice (PIACWI) and the freezing of rain drops to produce precipitation ice (PIACR) produced high-density rimed ice, while the denominator was assumed to represent an estimate of the growth of ice by vapor deposition only. Values of the updated RF from (C3) were also restricted to be ≥1 and ≤RFmax (described below). Other ice processes, such as homogeneous freezing of cloud water (PIHOM), cloud ice initiation (PINIT), sublimation of ice (PIDEP < 0), melting (PIMLT), and evaporation (PIEVP) or condensation (PICND) from melting ice were assumed not to impact the RF.
However, because the density of rime collected on ice is actually a function of the surface temperature of the ice particle, the impact velocity of the droplet onto the ice, and the size of the cloud droplets (Macklin 1962; Pflaum and Pruppacher 1979; Heymsfield and Pflaum 1985), a more refined calculation was done in FA that accounts for these effects following Straka and Mansell (2005). The definition of the rime factor was also revised to represent the ratio of the average density of rimed ice to the average density of unrimed ice :
ec4
The new calculation of RF involves a mass-weighted average of the growth of unrimed ice (RF = 1) associated with vapor deposition onto precipitation ice [DEPs > 0; see Eq. (B11)], the freezing of drops assuming an average density of 900 kg m−3, and the density of rime (frozen cloud water) collected on precipitation ice (Heymsfield and Pflaum 1985; Straka 2009) given by
ec5
where rcw is the mean droplet radius in μm, the impact velocity was approximated by the mass-weighted fall speed of the precipitation ice , and the surface temperature of the ice was approximated by the air temperature (Tc). An average particle density for rimed ice is calculated from the following relationship:
ec6
which can be integrated analytically by assuming Dsmin = 0 and Dsmax = ∞, where Dsmin and Dsmax are the minimum and maximum precipitation ice diameters. Rearranging terms, making use of the Gamma function, and combining (A12) and (A14), (C6) becomes
ec7
where the average density of the rimed ice is
ec8
Using (A21) to represent the precipitation ice content and rearranging terms yields
ec9
where MASSs is a lookup table of the normalized mass of unrimed snow defined in (A22). The maximum RF associated with the freezing of drops (RFmax) is calculated assuming = 900 kg m−3 in (C9), while the RF of rime associated with the riming (and freezing) of cloud water onto precipitation ice (RFrime) is calculated by assuming = ρrime and using (C5) in (C9). The final, updated RF calculation in FA becomes
ec10
The new calculation of RF is a mass-weighted average that considers the growth of unrimed ice (RF = 1) associated with deposition onto precipitation ice [DEPs > 0; see Eq. (B11)], the maximum RF associated with the freezing of drops that yields an ice density of 900 kg m−3 is given by
ec11
and the RF associated with the density of rime collected on precipitation ice (Heymsfield and Pflaum 1985; Straka 2009) is given by
ec12
where rcw is the mean droplet radius in μm.

APPENDIX D

Sedimentation

The historical foundation for the sedimentation process for all versions of the FER microphysics, and subsequently FA, was to improve upon the absence of sedimentation in Zhao and Carr (1997) and to keep the microphysics fast and efficient in its execution to meet operational computational requirements. Explicit calculation of the vertical flux convergence of precipitation was considered too expensive in order to maintain numerical stability (e.g., a 50-s physics time step is used in the 12-km operational NAM parent domain). In the remaining discussion within this subsection, the symbol represents the mixing ratio of precipitation in the form of rain (q = qr) or total ice (q = qi = qci + qs) at model level K and time step N, while represents the fall of precipitation in the form of rain or total ice that has fallen into the grid box from above during the time step. Note that the NMMB employs a top-down vertical index, with K = 1 being the first model level at the top of the domain. With ΔnK = ρa,K × ΔZK representing the mass thickness of the Kth grid box, the total mass of precipitation available for calculating microphysical processes (TOTx, x = i for total ice and x = r for rain) is
ed1
where the first term on the right side is the mass in the grid box, and the second term is the mass of precipitation falling into the grid box from above during the time step. This configuration reflects the fundamental partitioning algorithm of the sedimentation scheme discussed in section 3a. A time-averaged, first-guess estimate of precipitation mixing ratio () is calculated from
ed2
where is the mass remaining in the grid box, is the precipitation that falls through the bottom of the Kth grid box during the time step, and is the mass-weighted fall speed of precipitation. The first-guess, time-averaged precipitation mixing ratio is solved by rearranging (D2) and incorporating (D1):
ed3
The first-guess mixing ratio (D3) is then used as input for calculating the different microphysical sources and sinks described in appendix B. The microphysical sources and sinks provide a new mixing ratio in the grid box, , that can be expressed as
ed4
where DEL can be the total change in total ice mixing ratio (B3) or the total change in rain mixing ratio (B4) due to microphysical processes. To ensure the rainfall exiting the box from the bottom is accurate (using consistent values of and ), the drop size and the first-guess estimate of rain fall speed, in (D4), are updated using the new mixing ratio from (D4). The drop size and rain mixing ratio are recalculated up to three times until the difference in drop size is less than 2 microns (the solution converges).

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1

At the time, the NAM CONUS nest was run at 4-km horizontal grid spacing.

2

These values of RF are approximate and are most applicable to conditions where Tc > −10°C.

3

Spurious light rainfall was also noted in NMMB runs using the WSM6 and Thompson schemes (not shown).

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