Abstract

Observations of the vertical structure of rainfall, surface rain rates, and drop size distributions (DSDs) in the southern Appalachians were analyzed with a focus on the diurnal cycle of rainfall. In the inner mountain region, a 5-yr high-elevation rain gauge dataset shows that light rainfall, described here as rainfall intensity less than 3 mm h−1 over a time scale of 5 min, accounts for 30%–50% of annual accumulations. The data also reveal warm-season events characterized by heavy surface rainfall in valleys and along ridgelines inconsistent with radar observations of the vertical structure of precipitation. Next, a stochastic column model of advection–coalescence–breakup of warm rain DSDs was used to investigate three illustrative events. The integrated analysis of observations and model simulations suggests that seeder–feeder interactions (i.e., Bergeron processes) between incoming rainfall systems and local fog and/or low-level clouds with very high number concentrations of small drops (<0.2 mm) govern surface rainfall intensity through driving significant increases in coalescence rates and efficiency. Specifically, the model shows how accelerated growth of small- and moderate-size raindrops (<2 mm) via Bergeron processes can enhance surface rainfall rates by one order of magnitude for durations up to 1 h as in the observations. An examination of the fingerprints of seeder–feeder processes on DSD statistics conducted by tracking the temporal evolution of mass spectrum parameters points to the critical need for improved characterization of hydrometeor microstructure evolution, from mist formation to fog and from drizzle development to rainfall.

1. Introduction

Quantitative estimation and modeling of precipitation remains elusive, partly owing to the lack of complete understanding of microphysical processes governing the evolution of hydrometeor distributions in the troposphere from droplet formation to rainfall (Testik and Barros 2007; Rennó et al. 2013). Current rainfall-monitoring systems generally rely on indirect estimation from microwave radiometer (passive) and radar (active) measurements (Hou et al. 2001; Tapiador et al. 2011). Satellite-based radars have significant difficulty in the detection of light rainfall, particularly at high temporal resolution (Huffman et al. 2007) and in complex terrain (Prat and Barros 2010a; Duan and Barros 2013). Rainfall estimation is an inverse and ill-posed problem that requires fundamental knowledge of the space–time distribution of hydrometeors in the atmosphere in addition to the indirect microwave measurement proper, which is typically unavailable or only partially available. Therefore, simplifying assumptions are required. For warm rain, such assumptions include steady state, height independence of the raindrop size distribution (DSD), and limited, if any, dependence of the DSD on storm regime (Iguchi et al. 2000; Seto et al. 2013). Simulations from numerical weather prediction models in a data assimilation framework can address some of these assumptions, but severe limitations remain because of time-scale disparities between observations and model simulations and model errors, including the incomplete representation of microphysical processes (Hou and Zhang 2007).

In the case of radar observations, the reflectivity–rain rate (ZR) relation varies significantly in space and time with the DSD [Battan (1973) and Raghavan (2003) list over 100 empirical relationships], but for practical reasons and owing to the lack of confidence in their general applicability, estimation algorithms tend to rely on just a few (Tokay et al. 2009). The application of these ZR relations also assumes steady-state DSDs constant with height, although there is strong evidence that (i) differences exist in functional form between the DSD measured at the surface and that measured aloft (Uijlenhoet 1999) and (ii) the DSD time to equilibrium is often longer than the evolution of rainfall systems and the measurement time scale (Prat and Barros 2009). One objective of this research is to illuminate the physical mechanisms behind the temporal and vertical evolution of warm rain DSDs in complex terrain, leading to improved estimation algorithms.

In mountainous regions globally, light rainfall and fog are critical for freshwater sustainability (Bruijnzeel et al. 2011; Klemm et al. 2012). In the southern Appalachians, which experience a persistent fog regime, observations made by a high-density rain gauge network, represented by the purple circles in Fig. 1, indicate that light rainfall (rain rates less than 3 mm h−1 at 5-min time scales) at ridgelines represents 30%–50% of total annual precipitation based on 2007–13 records. Average monthly accumulations from light and heavy rainfall are plotted in Figs. 2a and 2b, respectively. There is higher variability in the amounts received from heavy rainfall, while the amounts from light rainfall and its diurnal cycle (not shown) remain stable throughout the year and indeed from year to year.

Fig. 1.

(top) Map of the study region shown in context of the southeastern United States; (bottom left) MRR locations during the period October 2011–November 2012, along with the rain gauge network in the Pigeon River basin (Haywood County); and (bottom right) zoomed-in schematic of the R1 supersite (distances not to scale), including RGs [RG-Tower (RGT)], P1 (PARSIVEL-1 disdrometer), P2s (PARSIVEL-2 disdrometers), a WXT (Vaisala weather station, version 510), and MRRs. The entire suite of instrumentation in the supersite was recording observations from May to June 2012. See Table 2 for a description of the instrument and location abbreviations.

Fig. 1.

(top) Map of the study region shown in context of the southeastern United States; (bottom left) MRR locations during the period October 2011–November 2012, along with the rain gauge network in the Pigeon River basin (Haywood County); and (bottom right) zoomed-in schematic of the R1 supersite (distances not to scale), including RGs [RG-Tower (RGT)], P1 (PARSIVEL-1 disdrometer), P2s (PARSIVEL-2 disdrometers), a WXT (Vaisala weather station, version 510), and MRRs. The entire suite of instrumentation in the supersite was recording observations from May to June 2012. See Table 2 for a description of the instrument and location abbreviations.

Fig. 2.

Average monthly inputs from (a) rain rates less than 3 mm h−1 and (b) rain rates equal to or exceeding 3 mm h−1 as observed by the PMM rain gauge network during the observation period 2007–13. Rain rates were computed at 5-min time scales. Each bar represents one gauge during the month, and the gauges move from east to west (left to right).

Fig. 2.

Average monthly inputs from (a) rain rates less than 3 mm h−1 and (b) rain rates equal to or exceeding 3 mm h−1 as observed by the PMM rain gauge network during the observation period 2007–13. Rain rates were computed at 5-min time scales. Each bar represents one gauge during the month, and the gauges move from east to west (left to right).

Generally, two types of fog occur in the southern Appalachians: valley fog, a type of radiation fog that can reach middle/high elevations, and upslope fog, a type of advection fog often present in mountainous regions that can happen at night or during the day depending on mountain–valley winds. Surface observations indicate that the observed diurnal cycle of light rainfall is intertwined with the observed diurnal cycle of fog occurrence, with midday peaks concurrent with valley fog and evening peaks concurrent with radiation fog.

Valley fog forms when air at high elevations begins to cool and sink after sunset; it then becomes saturated and forms dense fog or cap clouds that dissipate with solar forcing after sunrise. Upslope fog is caused by warm, moist air being pushed upward over land surfaces where it condenses. It can be differentiated from radiation fog by time of occurrence and by its movement. Daily midday upslope fog occurs along ridges in the inner region of the southern Appalachians from late fall through early spring, and its presence is consistent with the diurnal peak of rainfall in the region. Detailed analysis of the time series of rainfall data shows that in valley and sheltered ridge locations, there are intermittent periods of very intense rainfall, especially in the warm season, concurrent with dense fog and/or cap clouds, that cannot be explained based on the prevailing storm regime. Furthermore, Prat and Barros (2010a) showed that the right-hand side of observed valley DSDs is “heavier” than that of DSDs in adjacent ridges for the same event, suggesting enhanced drop coalescence between ridge and valley locations. On the other hand, during such events, there are no significant differences among the reflectivity profiles obtained from paired Micro Rain Radars, suggesting that the rainfall enhancement is taking place primarily in the vertical direction and is not an artifact of lateral advection of heavier rainfall. In this study, we investigate the hypothesis that interactions between seeder stratiform rainfall and local feeder fog and low-level clouds anchored to landform explain the observed enhancement of rainfall intensity and duration. We build on previous research pointing to the importance of coalescence processes in nonconvective rainfall in this region (Barros et al. 2008; Prat and Barros 2010b) and in mountainous regions with persistent near-surface cloudiness generally (e.g., Bergeron 1960).

A short review of fog and rainfall microphysics relevant for this research is presented in section 2, followed by model description in section 3. Section 4 describes the study region and observations. Model implementation is discussed in section 5, and model simulations are analyzed in section 6. Finally, conclusions and future work planned to build on these results will be discussed in section 7.

2. Microphysics overview

Raindrop populations are described by statistical DSDs fit to histograms of observed drops organized by class size. Two of the DSD functional forms most often used are the negative exponential:

 
formula

which can be characterized by two parameters (slope Λ and intercept N0), and the gamma:

 
formula

where μ represents the shape. Negative exponential distributions are a subset of gamma distributions (i.e., μ = 0). The Marshall–Palmer distribution, historically the most commonly used DSD, is a negative exponential distribution with a fixed intercept value and a slope that is dependent upon rain intensity, based on drops collected for a limited range of rain rates (1–23 mm h−1; Marshall and Palmer 1948). Because the negative exponential distribution does not resolve completely either very large or very small drops, Ulbrich (1983) proposed the gamma distribution, which is used widely in operational retrieval algorithms such as the Tropical Rainfall Measurement Mission Precipitation Radar (TRMM PR) (Iguchi et al. 2000). Other researchers found the gamma distribution to be the best fit in terms of how it represented coalescence growth, drop evaporation, and the number concentration (Willis 1984), as well as its skill in replicating observed rain rates and other DSD moments (Atlas et al. 1984; Tokay and Short 1996; Mallet and Barthes 2009), though exhibiting regime dependence (Cho et al. 2004).

a. Warm rain microphysics

After cloud droplets are large enough to be stable, and depending on drop number concentration, collisional drop–drop interactions are the primary mechanism of DSD evolution in space and time (Srivastava 1978). Collision outcomes are determined by the magnitude of collision kinetic energy related to drop diameter (Testik and Barros 2007; Barros et al. 2008; Testik et al. 2011). Other mechanisms affecting the DSD include aerodynamic breakup, advection and redistribution by air motion, and phase changes (evaporation/condensation). Low and List (1982a,b) parameterizations, updated by McFarquhar (2004), have been the primary source for collision-based interactions between drops in models of raindrop population dynamics. Testik et al. (2011) and Straub et al. (2010) investigated the sensitivity of coalescence efficiency to the collision Weber number. Furthermore, Testik et al. (2011) analyzed the data from Barros et al. (2008) and established the physical basis for a new parameterization of breakup type and bounce described by Prat et al. (2012). They showed that explicit consideration of bounce (collisions where both drops retain the same size before and after the collision) can significantly influence the DSD especially in the small diameter range of the drop spectrum.

The DSD dynamics model (Prat and Barros 2007; Prat et al. 2012) used in this research has been used successfully in the past to independently reproduce DSD observations and integral parameters (e.g., Prat et al. 2008; Prat and Barros 2010b). Previous modeling studies indicating that coalescence moving very small drops to the midsize drop region is the dominant microphysical mechanism driving DSD evolution in light rainfall are particularly relevant (Prat and Barros 2009). Specifically, collisional interactions among drop populations in different regions of the size spectrum—that is, seeder (right) and feeder (left) populations—can act to significantly enhance coalescence at very fast time scales. These seeder–feeder interactions, also referred to as Bergeron processes, were first proposed by Bergeron (1960) to explain orographic enhancement of rainfall in Sweden. Subsequently, a large number of studies in humid mountainous regions and the midlatitudes (northern Europe, United Kingdom, Japan, etc.) attributed orographic rainfall enhancements of one to two orders of magnitude to Bergeron processes [see Barros and Lettenmaier (1994) and Cotton and Anthes (1989) for detailed literature reviews].

b. Fog microphysics

In the literature, fog is often quantitatively described by bulk measures such as total accumulation over relatively long time scales (ecology) or visibility (aviation, air quality). When fog droplets are measured directly, the most common instrument used is some version of a particle spectrometer (e.g., Gonser et al. 2012; Frumau et al. 2011; Hering et al. 1987). These instruments have measurement limitations; the smallest drop size that can be detected is generally around 5 μm, and the largest ranges from 40 to 200 μm. Other instruments used include photoelectric particle counters (Eldridge 1960), high-speed cameras (Okuda et al. 2009), special collectors (Straub and Collett 2002), and forward-scattering spectrometers (García-García et al. 2002; Gultepe et al. 2009).

Here, we are particularly interested in observational studies that report on fog DSD properties. A literature review including studies in the Swiss Alps (Müller et al. 2010), eastern Canada (Gultepe et al. 2009; Walmsley et al. 1996), Japan (Okuda et al. 2008, 2009), Germany (Bendix et al. 2005), the United States (Eldridge 1960; Hering et al. 1987), Taiwan (Gonser et al. 2012), Costa Rica (Frumau et al. 2011), Mexico (García-García et al. 2002), and in the laboratory (Straub and Collett 2002) is summarized in Table 1. Please note that, following conventions in the literature, the units of fog DSD properties describe the droplets in terms of micrometers and cubic centimeters, which is in contrast with raindrop DSDs described using millimeters and cubic meters. In general, the average diameter reported by these studies is between 10 and 20 μm. Fog droplets as large as 200 μm were observed (Okuda et al. 2008), but this is rare; usually the maximum diameter lies is in the 50–100-μm range. There is large variability in the number concentration of the smallest drop sizes (10–103 cm−3 μm−1), and the order of magnitude for the total number concentration varies between 102 and 103 cm−3. The reported liquid water content (LWC) is generally between 0.03 and 0.4 g kg−1. This indicates that studies reporting the N0 value to be as high as 103 cm−3 μm−1 also reported fog DSDs composed almost entirely of very small droplets. Other critical information to understand fog dynamics (e.g., duration and temporal evolution) is not reported. Nevertheless, because of the lack of fog observations beyond visibility in the southern Appalachians, we rely on these previous studies to specify physically reasonable fog DSDs in the model simulations that are consistent with low-level reflectivity observations from vertically pointing radars (VPRs).

Table 1.

Summary of fog characteristics from the literature.

Summary of fog characteristics from the literature.
Summary of fog characteristics from the literature.

3. Model description

This study uses a column (rainshaft) model to simulate drop–drop interactions in the liquid phase. A brief description of the number and mass conservative explicit bin model follows, but an exhaustive description can be found in Prat and Barros (2007) and Prat et al. (2012). The general continuous transport equation [stochastic collection equation–stochastic breakup equation (SCE–SBE)] is discretized using a grid independent number and mass conservative scheme (Kumar and Ramkrishna 1996). The resulting discrete equation over the ith discrete interval is given by

 
formula

where Ni(z, t) is the total number density of drops in the ith class size (cm−3):

 
formula

The terms on the right side of Eq. (3) describe the coalescence–breakup dynamics. From left to right, the terms represent the rate of change of gains resulting from coalescence, losses from coalescence, gains from breakup, and losses due to breakup. The representative volume of the ith size range is noted by xi. The collection kernel Ci,j is the product of the coalescence efficiency (Ecoali,j: Low and List 1982a,b) and the gravitational collision kernel Ki,j (Pruppacher and Klett 1978) incorporating results from Barros et al. (2008) and Testik et al. (2011). The breakup Bi,j and collection kernels are complementary (i.e., Bi,j = 1 − Ci,j). The parameters η (coalescence contribution) and κi,j,k (breakup contribution) are derived from the number-mass conservative discretization scheme. Specifically, η is the contribution to the drop population located at the ith interval due to coalescence of drops of volume xj and xk. When a drop of volume υ = xj + xk is created by coalescence and its size falls between two grid points xi and xi+1, the newly created drop is distributed proportionally to both adjacent grid points (xi and xi+1) so that both number and mass are preserved (Prat and Barros 2007). Similarly, the term κi,j,k is the contribution to drop population located at the ith interval due to collisional breakup of drops of volume xj and xk. The parameter κi,j,k is a function of the fragment distribution function P(υ, xj, xk), where P(υ, xj, xk) is the number of drops with a volume in the range υ to υ + dυ obtained from the collisional breakup of two drops of volume xj and xk. The overall fragment distribution function includes the three types of breakup (filament, sheet, and disc) identified by McTaggart-Cowan and List (1975). The occurrence of each type of breakup and the number of drops created by a collisional event follows Low and List (1982a,b), modified with results from Testik et al. (2011) and Prat et al. (2012). The fragment distribution function for each type of breakup is derived from the parameterization proposed by McFarquhar (2004) that provides expressions for the parameters (height, standard deviation, and modal diameter) of the Gaussian and lognormal functions constituting the fragment distribution for each type of breakup. The term κi,j,k is integrated over every bin category via routines specifically written for generalized Gaussian and lognormal distributions (Prat and Barros 2007). The overall fragment distribution is obtained from the weighted contribution of each type of breakup.

The irregularly discretized grid used in this study covers a diameter range from 0.01 to 6.2 mm over 60 bins and combines a geometric grid (d ≤ 0.1 cm) with a regular grid (spacing = 0.02 cm, d > 0.1 cm) in order to accurately capture the location of peaks in the small drop diameter range, the tail of the DSD in the large drop diameter range, and minimize numerical diffusion (Prat and Barros 2007). The moments of the DSD are expressed by

 
formula

where mi is the characteristic mass (g) of the ith class category. The term CC is a conversion coefficient dependent on the integral properties considered. Moments of interest include the drop number concentration M0(z, t) (k = 0; CC = 1; cm−3), the liquid water content [LWC(z, t) (k = 1; CC = 106; g m−3)], and the radar reflectivity factor {k = 2; CC = 1012[6/(πρ)]2 with ρ = 1 g cm−3; mm6 m−3 or dBZ} with . The rain rate is given by

 
formula

where Vi(m s−1) is the drop fall velocity for a drop of the ith bin category (Atlas et al. 1973) and Vair(z) is the mean air velocity at the height z that includes updrafts–downdrafts with appropriate sign conventions (positive toward the ground).

The numerical approximation of the SCE–SBE is achieved using a semi-Lagrangian formulation with a vertical resolution of 10 m and a time step of 1 s. At each time step, the distribution of drops inside the rainshaft is updated using the vertical positions for each drop size class and the DSD spectra is then modified locally under the influence of combined coalescence–breakup mechanisms.

4. Observations

Intensive observation periods (IOPs) were conducted on the Cataloochee Ridge in the Pigeon River basin [2011/12, ridge 1 (R1)] and in Madison County at a ridge [2011/12, ridge 2 (R2)] and in a valley [2012, valley 2 (V2)]. Figure 1 shows sensor locations during the observing periods, including a schematic of R1 during a period of cross calibration in late spring 2012. A list of the instrumentation with deployment dates is presented in Table 2. Instruments include tipping-bucket rain gauges (RGs) and a Vaisala automatic weather station (WXT) with an acoustic rainfall sensor that measures rainfall intensity at 0.1-mm resolution and Particle Size Velocity (PARSIVEL) optical disdrometers (P1 and P2) and Micro Rain Radars (MRRs) with sensitivity down to 0.01 mm h−1. MRRs observe 30 vertical levels in the atmosphere with range-gate resolution ranging from 35 to 150 m; 150-m resolution was used here. MRRs operate at 24 GHz (K band) using the frequency modulated continuous wave principle. The fall velocity of hydrometeors leads to a Doppler shift in the received frequency in addition to the range frequency shift. The fall velocity is the first moment of the Doppler spectra and the DSD is determined using the relationship between spectral reflectivity and single-particle-backscattering cross section (METEK 2012). The main limitation in this retrieval is that it does not account for wind (horizontal or vertical) at any scale.

Table 2.

Instruments deployed during 2011/12 field seasons organized by time period and location of installation, including abbreviations used throughout the paper.

Instruments deployed during 2011/12 field seasons organized by time period and location of installation, including abbreviations used throughout the paper.
Instruments deployed during 2011/12 field seasons organized by time period and location of installation, including abbreviations used throughout the paper.

Averaged and maximum vertical profiles for each MRR location are presented in Fig. 3, computed for each gate. The lower bound for the observed reflectivity was 15 dBZ. The largest differences are between the two ridge locations R1 and R2. R1 is located in the inner mountain region, while R2 and V2 are located in an open mountain pass. The reflectivity profile observed at R1 is characterized by a sharp decrease in reflectivity at altitudes higher than approximately 2 km AGL. The reflectivity profile in the mountain pass ridge (R2) has a greater vertical extent with consistently high reflectivity values at all levels. The differences between the two high-elevation ridges are higher than the differences between the valley and ridge location in the mountain pass. In the latter, as expected, the valley location (V2) observes the bright band at a higher-altitude AGL than the ridge, corresponding to the elevation difference between R2 and V2. This effect is clearer in the average profile than in the maximum. The valley also exhibits a steeper change of reflectivity with height than the ridge, and the ridge values are consistently higher. The maximum reflectivity profiles are similar, with the ridge showing slightly higher reflectivity values at low to midlevels than the valley.

Fig. 3.

Average vertical profiles are shown here for the MRRs in R1, R2, and V2 in (top) 2011 and (bottom) 2012. Note the difference in scale for average values vs maximums.

Fig. 3.

Average vertical profiles are shown here for the MRRs in R1, R2, and V2 in (top) 2011 and (bottom) 2012. Note the difference in scale for average values vs maximums.

Contoured frequency with altitude diagrams (CFADs), created by computing percentages of occurrence for reflectivity values over 15 dBZ at each gate, show the same differences between R1 and R2 in 2011 (Fig. 4). R2 CFADs show higher reflectivity values throughout the column while R1 shows a steeper gradient from the lower- to higher-elevation values and a clearer, lower bright-band signature. Low-level processes dominate at R1, and reflectivity values generally do not exceed 20 dBZ above 2 km AGL. In 2012, observations are available from R1 during May and June and from R2 and V2 during June and July. At R1, the observations from 2012 are similar to those in 2011. The values observed during 2012 are slightly higher than those in 2011, but the general pattern is still apparent. Reflectivity values are higher in the lower levels of the atmosphere, and above about 2 km AGL, they drop off significantly. Incidentally, in this region, satellite rainfall products often significantly underestimate precipitation (Prat and Barros 2010b). In the R2 and V2 plots, the shift in bright-band location about 500 m higher in the atmosphere at the valley location is shown; the milder decline with reflectivity and height at the ridge corresponds to the vertical profiles in Fig. 3.

Fig. 4.

CFADs for (top) May 2011 at R1 and R2, (middle) May 2012 at R1, and (bottom) summer 2012 [June–August (JJA); see Table 2] at V2 and R2. Percentages were computed for each height level (150-m resolution). There are 10 bin edges per height, represented on the x axis. Contour line spacing is at 1%. White lines indicate the average values for the profile during the CFAD period; black lines indicate the maximum observed values.

Fig. 4.

CFADs for (top) May 2011 at R1 and R2, (middle) May 2012 at R1, and (bottom) summer 2012 [June–August (JJA); see Table 2] at V2 and R2. Percentages were computed for each height level (150-m resolution). There are 10 bin edges per height, represented on the x axis. Contour line spacing is at 1%. White lines indicate the average values for the profile during the CFAD period; black lines indicate the maximum observed values.

The microphysical structure of DSDs observed at the surface by optical disdrometers during 2012 was also explored. Time series of DSDs observed by the disdrometer are plotted in three dimensions in Fig. 5 to show the variability in distributions with time, in addition to diurnal patterns. A diurnal cycle is observed with some droplets recorded each day. Longer, more intense events exhibit broader distributions.

Fig. 5.

Time series of drop size distribution observed by P2s at (top) R1, (middle left) R2, and (middle right) V2. The y axis is the drop diameters. The color fill represents the log of the number concentration. Time series of drop size distribution for (bottom left) R2 and (bottom right) V2 during a rain-free day. Note the difference in y-axis scale between the top and middle panels and the bottom panels.

Fig. 5.

Time series of drop size distribution observed by P2s at (top) R1, (middle left) R2, and (middle right) V2. The y axis is the drop diameters. The color fill represents the log of the number concentration. Time series of drop size distribution for (bottom left) R2 and (bottom right) V2 during a rain-free day. Note the difference in y-axis scale between the top and middle panels and the bottom panels.

Note the differences between the DSDs observed by the PARSIVEL disdrometer at R2 and V2. V2 shows a diurnal pattern of small-drop observations in significant numbers. We hypothesize that this indicates the presence of fog in the valley, whereas at R2, fog is not a dominant feature. Interestingly, the observations at R1 in the inner mountain region show similar characteristics to those at V2 despite the elevation difference. One significant event that occurred in R2 and V2 during this time period was enhanced at the valley location by the presence of fog, as observed by the VPR.

Tokay et al. (2013) conducted a careful measurement characterization study that revealed strong artifacts associated with the underestimation of the number of small drops by PARSIVEL disdrometers. This underestimation for drops smaller than 0.76 mm had a pronounced effect on the parameters of gamma-fit DSDs. Evidence of overestimation of midsize and large drops is also presented. We will refer to these findings again in section 7. At this point, it is important to emphasize that the study by Tokay et al. suggests that the PARSIVEL observations presented here are underestimates of actual fog DSDs, and therefore the number concentrations of midday fog detected at V2 should be higher, corresponding to higher LWC.

Two microphysical parameters of particular interest are mass spectrum parameters: mean mass-weighted diameter Dm and its standard deviation σm, respectively, the center of mass for LWC and the width of the distribution’s shape. These are more physically meaningful variables than the slope, shape, and intercept parameters, which are mathematical in nature (Haddad et al. 1996). However, they are still related to those mathematical parameters. The relationship between Dm and σm follows a power law:

 
formula

Dm and σm values corresponding to disdrometer DSDs observed each minute are presented in Fig. 6. There were about 2000 (1000) min of drop size observations available for each instrument at R1 (R2 and V2). All minutes that surpassed the minimum detectable rain rate of the disdrometers were included in the plots. The first fit shown was found using over 29 000 min of two-dimensional video disdrometers (2DVD) measurements during several different National Aeronautics and Space Administration (NASA) Ground Validation (GV) experiments (Williams et al. 2014). The remaining fits were found using corresponding data points from each disdrometer. At R1 and V2, the fit is very similar to what was found in Williams et al. despite the difference in geographic location and in instrumentation, suggesting the robustness of this fit. The α and β parameters found in that paper, as well as in the R1, R2, and V2 observations, are reported in Table 3.

Fig. 6.

Plots of Dm vs σm for each instrument and location. (top) For R1, the number of available minutes (restricted to rain rates greater than 0 that fall within the bounds described below) is as follows for each instrument: P1: 2281 min, P2(2): 3872 min. (bottom) For R2 and V2, the number of available minutes with the same constraints is as follows for each instrument: P2-R2: 1056, P2-V2: 1099. The R1, R2, and V2 fits use the data in the plot, with bounds where Dm does not exceed 3 mm and σm does not exceed 1.5 mm. For equations, see Table 3.

Fig. 6.

Plots of Dm vs σm for each instrument and location. (top) For R1, the number of available minutes (restricted to rain rates greater than 0 that fall within the bounds described below) is as follows for each instrument: P1: 2281 min, P2(2): 3872 min. (bottom) For R2 and V2, the number of available minutes with the same constraints is as follows for each instrument: P2-R2: 1056, P2-V2: 1099. The R1, R2, and V2 fits use the data in the plot, with bounds where Dm does not exceed 3 mm and σm does not exceed 1.5 mm. For equations, see Table 3.

Table 3.

Fitted parameters for the Dm and σm relationship found with different DSD observations, presented with 95% confidence intervals when available. Selected segments of the diurnal cycle are presented when enough observations during the given time period were available. See Table 2 for abbreviations.

Fitted parameters for the Dm and σm relationship found with different DSD observations, presented with 95% confidence intervals when available. Selected segments of the diurnal cycle are presented when enough observations during the given time period were available. See Table 2 for abbreviations.
Fitted parameters for the Dm and σm relationship found with different DSD observations, presented with 95% confidence intervals when available. Selected segments of the diurnal cycle are presented when enough observations during the given time period were available. See Table 2 for abbreviations.

At R2, the fit diverges from that found in past GV experiments and at R1 and V2. These observations highlight the spatial variability in this region and, in particular, the large differences between ridges in the inner mountain region and in the mountain pass. In the R2–V2 plot, note the difference between the P2 at the ridge elevation of about 1200 m and the P2 at the valley elevation of about 500 m. The R2 values exhibit higher variance than V2, and indeed any of the other field experiments, for a given drop diameter. The sample size for this period was limited to just a few significant precipitation events, so a caveat is that this difference may be related to particular properties of the events observed.

There are a much larger number of smaller Dm values reported at R1 than at either R2 or V2. The density of the Dm occurrences in the R2–V2 plot stays consistent up to about 2 mm, compared to 1.5 mm at R1. It is not clear whether part of this difference is reflected in the different seasons (late spring versus early summer) in which the disdrometers were deployed, or if the difference is primarily the result of spatial variability and hydrometeorological regime. In addition, there is more spread in the points observed at R1 than at R2 and V2. This is probably the result of the limited sample size of precipitation events recorded at R2 and V2 during the observation period. The presence of more different types of rain events would likely increase the variability in the Dmσm relationship.

The Dmσm observations were organized in several different ways in order to highlight the variations, including a separation based on rain intensity (3 and 5 mm h−1 were used as thresholds) and a separation based on time of the observation. Figure 7 presents plots for selected 3-hourly periods at R2 and V2. A 3-h period was included if it observed at least 100 rainy minutes. This figure shows that the highest Dm values, and the smallest variation with location, occur during 1200–1500 LT. This is consistent with midday peaks in fog that translate into a concurrent diurnal cycle of rainfall. The range of Dm is largest during 0000–0300 and 1500–1800 LT. The early morning hours at R2 show the highest variation with increasing Dm. Despite the limited number of precipitation events that occurred during this period, the observations contain information about the timing and frequency of rainfall occurrence and the shape of the DSD at different times of day.

Fig. 7.

Plots of Dm vs σm for selected 3-h segments of the day for the period June–July 2012. The red curve corresponds to the equation in Fig. 6. Parameters for the fit at each time of the day, and number of points available, can be found in Table 3.

Fig. 7.

Plots of Dm vs σm for selected 3-h segments of the day for the period June–July 2012. The red curve corresponds to the equation in Fig. 6. Parameters for the fit at each time of the day, and number of points available, can be found in Table 3.

5. Model setup: Boundary conditions and the seeder–feeder mechanism

The MRR observations were used to provide initial and boundary conditions for three case studies chosen from the observation periods described in section 4. At the initial time, DSDs estimated from MRR reflectivity observations were imposed at 150-m resolution in the vertical column. The top boundary condition changed with time as in real rain events and was updated every 60 s. The model was run as described in section 3, with temporal resolution of 1 s and vertical spatial resolution of 10 m.

a. Top boundary condition

The preprocessing done on the reflectivity field observed by the MRR proceeds as follows. First, a top boundary condition (TBC) is chosen. This is straightforward in stratiform precipitation, when a clear bright band is present. We follow Fabry and Zawadzki (1995) to determine the height of the bottom of the bright band from the MRR measurements as the level at which the reflectivity is at least 20% lower than the maximum reflectivity. Simulations are conducted below the bright band to ensure that ice microphysics will not play a significant role. When a clear bright band cannot be identified, the TBC reflectivity threshold is changed to 10%, and the TBC height is constrained to be above 1 km AGL. In both cases, the boundary condition moves when the algorithm finds a new TBC that is stable for at least 5 min.

Next, the TBC reflectivity is used to derive a DSD. There is not a unique mapping between the reflectivity value obtained by the MRR and the DSDs used for initial and boundary conditions; indeed, this is an intrinsic part of the problem that this research is trying to address. To use the reflectivity values observed by the MRR, the initial DSD is fit with a negative exponential distribution for simplicity. After this initialization, the model explicitly determines how the DSD evolves. The intercept parameter [see Eq. (1), section 2] is chosen based on the reflectivity value and is allowed to vary between 8000 and 80 000 m−3 mm−1, with higher values corresponding to higher reflectivity. The slope parameter is then computed and the resulting DSD is evaluated against the input reflectivity. This operation is repeated iteratively until the reflectivity values converge, with an error threshold of 0.001%. This procedure is repeated each time the TBC is reinitialized (every 60 s) to capture the impact of observed reflectivity changes according to the best fit.

Since analysis of the events observed by the MRRs and disdrometers indicated that the seeder–feeder mechanism, here understood as the layer of fog or cap cloud between the ground surface and a specified height representing its depth, can represent an important contribution to rainfall intensity and overall accumulation, a methodology was developed to include a representation of this mechanism in the model.

b. Seeder–feeder mechanism (SFM)

It is important to note that quantitative observations of fog intermittency and turbulence in our study region are not available. Therefore, we do not have specific values observed in the field to include in the simulation of realistic precipitation events, and instead typical values from the literature are used to guide the simulations. The authors recognize that these fields are highly significant in terms of their influence on the results, and with that in mind, future field campaigns have been planned in order to record these observations.

First, sensitivity studies were conducted to determine the most appropriate N0 value for the fog distributions in the model; based on the literature, the choice was made in the range 0.0004 and 0.04 cm−3 μm−1. These values are orders of magnitude below the higher N0 values reported in the literature, because the lowest diameter bin edge represented by the model grid is 10 μm. This is generally the average diameter of observed fog (see Table 1), and so the N0 in the model must correspond to the approximate number concentration of that size droplet. The fog distribution is allowed to reach a maximum diameter of either 130 or 50 μm, depending on time of day and fog duration. LWC is used as a model constraint as either a percentage of the seeder rainfall or by itself using the values found in the literature for guidance (ranging between 0.006 and 0.06 g kg−1 throughout the fog column to represent different stages of fog development as per the reflectivity measurements). As expected, this constraint has a much greater effect than N0 on the observed change in reflectivity and rain rate, when the fog evolution was modeled by itself (see Fig. 8b and Table 4). This is because of the very small size of the drops associated with larger N0. Higher-order moments are much more sensitive to large drops and do not change appreciably with changes in small drop concentration. Idealized steady-state simulations with the collection equation assuming a collision efficiency of 100% indicate that the relative increase in mass of small raindrops (diameter < 0.3 mm) falling through homogeneous fog layers (fog drop diameter ~ 30 μm) can be as high as 50% for a fog LWC of 0.06 g kg−1 (not shown). Although this is an upper bound given that breakup effects are neglected, the sensitivity analysis indicates that for the range of drop sizes at the number concentrations shown in Fig. 8b, significant increases in rainfall intensity and accumulation can be expected even with fogs of relatively low LWC.

Fig. 8.

(a) Sensitivity study showing the difference between the top boundary condition and low-level reflectivity values reported by the model for various top boundary conditions and percentages of seeder rainfall liquid water content incorporated. (b) Equilibrium DSD curves for various initial and boundary conditions simulating fog only. Equilibrium is defined as when the N(D) values for that minute and the minute directly before it are different by less than 1% for every bin. Parameters used in the simulations can be found by number in Table 4.

Fig. 8.

(a) Sensitivity study showing the difference between the top boundary condition and low-level reflectivity values reported by the model for various top boundary conditions and percentages of seeder rainfall liquid water content incorporated. (b) Equilibrium DSD curves for various initial and boundary conditions simulating fog only. Equilibrium is defined as when the N(D) values for that minute and the minute directly before it are different by less than 1% for every bin. Parameters used in the simulations can be found by number in Table 4.

Table 4.

Parameters used in idealized fog simulations (with Fig. 8b).

Parameters used in idealized fog simulations (with Fig. 8b).
Parameters used in idealized fog simulations (with Fig. 8b).

Simulations were also conducted to investigate how a static distribution of 300-m depth of fog (or cap cloud) evolves in the presence of a constant-reflectivity precipitating cloud above as boundary condition. Nonlinear changes in reflectivity were found in the lower levels (see Fig. 8a) with increases in values of up to 20% with boundary condition reflectivity up to 30 dBZ and then decreasing for higher values. This corresponds with the reflectivity values observed by the MRR and yields rain rates and accumulations on the same order as the observations made by ground instrumentation (RGs and disdrometers). Thus, whereas nonraining fog or cap clouds may not be detectable in isolation by radars, when interacting with an upper-level source, they act as the Bergeron feeder population to accelerate coalescence, leading to enhanced precipitation and thus explaining the observed reflectivity increase at low levels. Note that seeder–feeder interactions become dominated by breakup when seeder rainfall rates are high (reflectivities > 30 dBZ, convective systems), and thus the orographic enhancement of rainfall by Bergeron processes is regime dependent.

6. Case studies

In this section, the output of dynamic simulations of three distinct rain events is compared in space and time with integral parameters observed by the MRRs, and in time with integral parameters observed at the surface by the RGs, disdrometers, and WXT. The discretized DSDs produced by the model are compared with those observed by optical disdrometers through analysis of their mass spectrum parameters. Maximum accumulations, rain rates, duration, and other parameters from each event are catalogued in Table 5. Please note that the fog DSD characteristics are reported in units of micrometers and cubic centimeters as opposed to millimeters and cubic meters, following the literature conventions when discussing fog.

Table 5.

Selected characteristics describing the case studies in this paper. See Table 2 for abbreviations.

Selected characteristics describing the case studies in this paper. See Table 2 for abbreviations.
Selected characteristics describing the case studies in this paper. See Table 2 for abbreviations.

a. Case 1

The first case examined occurred on 26 May 2011, during a frontal passage along which convection arose in the early evening. The model was used to simulate the event at R1 and R2. The squall line reached R1 first. Although rainfall intensity was higher at R1, more intense reflectivity values were recorded at R2, consistent with the passage of embedded convective cells at that location. The ground instruments available for this case were MRRs (R1 and R2), a WXT (R2), and RGs (R1). At R1, the active depth of the atmosphere during the storm was shallower than at R2 by about 500 m. At both locations, this event lasted for about 4 h, with just under an hour delay in event onset at R2. R2 did not experience any additional forcing at low levels.

The reflectivity profile for the 26 May 2011 event observed at R1 shows a strong reflectivity gradient with decreasing altitude, corresponding with rainfall measurements at the surface. Therefore, it is hypothesized that the SFM is critical, and it was included in the model simulations. The event was simulated at R1 for a 4-h period, from 1717 to 2117 LT. The SFM was included with timing and intensity corresponding to the reflectivity profiles. A depth of 300 m was assumed to be constant throughout the event. During the period 1727–1752 LT, the characteristics of the DSD used for the SFM were LWC = 0.002 g kg−1, N0 = 0.004 cm−3 μm−1 (where N0 corresponds to the lowest bin in the model grid for a diameter of 10 μm), and Dmax = 50 μm. From 1925 to 2005 LT, the characteristics of the SFM DSD were the same, except LWC = 0.0007 g kg−1 based on reflectivity. For the remainder of the simulation, LWC = 0.0002 g kg−1 while the other parameters again remained the same. The top boundary condition was 1500 m AGL. The three different specified fog DSDs each remained constant throughout the period that they were, in effect, to simulate the raindrops falling through a relatively stable layer of fog. The fog layer was adjusted to the LWC described above during peaks in low-level reflectivity, based upon the idealized fog simulations presented in the previous section and in Fig. 8.

Results for R2 (Fig. 9) show the best agreement with the rain accumulation reported by the MRR at the range gate 600 m AGL. The WXT was the only surface instrument available during this event. There is reason to question the validity of the WXT’s reports in this instance, since it underreports the daily total as compared to nearby Community Collaborative Rain, Hail and Snow Network (CoCoRaHS) gauges also at high elevations; in addition, generally the WXT overestimates rainfall as compared to other surface instrumentation (case 2 in this study; Braten et al. 2013; Lanza et al. 2010). For this reason, we will accept in this case the MRR rain rate estimates as closer to the ground truth with which to compare model output. Nonetheless, both MRR and WXT data are included in the plots.

Fig. 9.

Observations and model results for case 1 at (left) R1 and (right) R2. (top) Reflectivity observations from the MRR are shown with contour levels at 1 dBZ. A black line over the observations indicates where the top boundary conditions were taken. (middle) Reflectivity output by the model, again with contour intervals at 1 dBZ. “Model-SFM” indicates model simulations including the presence of fog, and thus with seeder–feeder mechanism active. “Model” indicates model simulations without including fog. (bottom) Time series of cumulative rainfall over the simulated period, with observations as solid lines and model results as dashed lines.

Fig. 9.

Observations and model results for case 1 at (left) R1 and (right) R2. (top) Reflectivity observations from the MRR are shown with contour levels at 1 dBZ. A black line over the observations indicates where the top boundary conditions were taken. (middle) Reflectivity output by the model, again with contour intervals at 1 dBZ. “Model-SFM” indicates model simulations including the presence of fog, and thus with seeder–feeder mechanism active. “Model” indicates model simulations without including fog. (bottom) Time series of cumulative rainfall over the simulated period, with observations as solid lines and model results as dashed lines.

The R1 simulation including the seeder–feeder mechanism (model-SFM) at the initial time corresponding to the increase in reflectivity at lower levels significantly improved the model output as compared to the ground truth, considered in this case to be the ensemble of surface observations from rain gauges (see Fig. 1 for the distance between the MRR and the RGs). Results (Fig. 9) show good agreement between the simulated and observed reflectivity throughout the 4-h period, with almost all of the error concentrated in the duration of the first low-level peak. The rain rates show this behavior as well. Note the dramatic change in rainfall intensity between the simulations with and without the SFM. The total accumulation reported by model-SFM is between the totals reported by the RGs and within the instrument uncertainty. When the simulation was run without SFM, errors were on the order of 80%.

b. Case 2

To assess how robust the SFM is at the R1 location, the same methodology was applied to a similar event that occurred the following year on 29 May 2012. This event was meteorologically similar to the 2011 event. It happened during the same time of the day (early evening), and the overall synoptic structure of the storm environment was similar, including the prior day’s temperature, relative humidity, and wind profile (not shown). Both cases exhibit the same type of reflectivity signature in terms of the slope and magnitude of the vertical profile. The 2012 event had a slightly smaller total accumulation of rainfall and shorter duration (~25 mm and 1.5 h) than the 2011 event (~35 mm and 4 h), although its maximum rain intensity was higher than case 1.

Instrumentation available for ground validation during case 2 included three optical disdrometers, one WXT, and two RGs (see Fig. 1 for a schematic). The PARSIVELS observed lower rain rates and accumulations than the RGs, particularly during the period 1900–1915 LT. The highest intensities for all sensors correspond to the maximum low-level reflectivity observed by the MRR just before 1930 LT. The PARSIVELS are used as the most reliable measurement for rain rate and accumulation in this case. Most interestingly for our purposes, the presence of fog for over 1 h before the rainfall event onset was confirmed with an onsite webcam. Based on the available data, fog with N0 = 400 cm−3 μm−1, Dmax = 50 μm, and LWC = 0.0002 g kg−1 was specified. During the peaks of low-level reflectivity, the fog characteristics changed; from 1859 to 1907 and from 1951 to 1956 LT, the fog LWC was 0.0007 g kg−1 and from 1908 to 1925 LT, the LWC was 0.002 g kg−1 based on the reflectivity profiles. The simulation was run from 1845 to 2015 LT, and the SFM was activated in the lowest 300 m.

As in the 2011 case, results for this case show significant increase (and improvement compared to observations) in rainfall accumulation (greater than 50%) when the SFM is included with appropriate timing during the event (see Fig. 10, model-SFM). The dynamic evolution of the fog DSD improves the accuracy of the simulation during peak rainfall observations at the surface. Indeed, the time-varying LWC of the fog DSD is essential to capture the temporal variability in observed rainfall intensity. However, it is important to recall the actual dynamics of fog formation and advection, including turbulence, which would explain the temporal variability in LWC (here inferred indirectly from the reflectivity profile) and fog DSD proper (not done here), are not directly represented in the model. It is essential to begin taking observations of fog DSDs to improve our understanding of these processes and how they are organized by the terrain in order to generalize model applicability. In the case of generic remote sensing retrievals, it will be necessary to understand and map the regimes of the SFM without benefit of low-level reflectivity information.

Fig. 10.

Observations and model results for case 2. (left) Reflectivity plots are shown for the (top) MRR observations and (bottom) model results. (right) (top) Rain rates and (bottom) accumulations are shown for the available observations (solid lines) and model results (dashed lines). Legends, lines, and computations follow Fig. 9.

Fig. 10.

Observations and model results for case 2. (left) Reflectivity plots are shown for the (top) MRR observations and (bottom) model results. (right) (top) Rain rates and (bottom) accumulations are shown for the available observations (solid lines) and model results (dashed lines). Legends, lines, and computations follow Fig. 9.

c. Case 3

The 2012 event simulated at R2 and V2 occurred in the early afternoon on 11 July 2012 and was part of a larger, longer-duration precipitation event due to a frontal passage with enhanced convection on 10–11 July 2012. The reflectivity plots in Fig. 11 indicate high reflectivity values throughout the atmospheric column at the ridge. In the valley, the reflectivity gradient is steeper, and stronger values are observed near the surface. The bright band at the valley is located 500 m above the bright band at the ridge (3500 and 3000 m AGL, respectively), and its magnitude is higher at the ridge. The event was simulated during the period 1300–1600 LT at both ridge and valley locations. Event onset in the valley was about 45 min prior to that at the ridge. The highest reflectivity values and rain accumulations in the valley occur around 1400 LT and at the ridge closer to 1430 LT. At each location, disdrometers, MRRs, and RGs were available. Accumulations at the valley exceed those at the ridge. In both locations, the rain gauge observes higher accumulations than the disdrometers. The simulation at V2 includes low-level forcing throughout the depth of the valley (500 m), with N0 = 400 cm−3 μm−1, Dmax = 50 μm, and LWC = 0.0002 g kg−1. The LWC was 0.0007 g kg−1 during the periods 1330–1400 and 1530–1545 LT. During the highest peak, 1401–1430 LT, the LWC increased to 0.002 g kg−1, again inferred from low-level reflectivity. No SFM was observed at R2.

Fig. 11.

Observations and model results for case 1 at (left) V2 and (right) R2. Legends, lines, and computations follow Fig. 9.

Fig. 11.

Observations and model results for case 1 at (left) V2 and (right) R2. Legends, lines, and computations follow Fig. 9.

Model results at the ridge (R2) show good agreement between modeled and observed rainfall (Fig. 11). The model reports a rainfall total between the accumulations of the RG and the disdrometer. There is significant spread among rainfall estimates from various surface instruments (>10 mm over the 3-h period), with the differences seen throughout the event. Model results at the valley show good agreement for overall accumulation during the event; again, note the difference in peak rainfall time and rainfall accumulations between model and model-SFM simulations. The improvement against simulations without the SFM was greater than 50%.

Regarding the MRR observations, cases where the SFM was included show the greatest agreement with the first MRR gate at 150 m AGL. This gate is normally considered to contain a great deal of attenuation error, but in the case studies here, when the SFM is important, it correlates not only with model results but also with other surface observations. In cases where the SFM is not an important contributor to the rainfall event, the MRR-based estimates are correlated with surface rainfall at the 600-m gate, corresponding to previous findings (Prat and Barros 2010b).

7. Conclusions and discussion

Integrated analysis of model simulations and observations in the southern Appalachians demonstrate how feeder–seeder interactions can play an important role in the orographic enhancement of rainfall in landscapes characterized by persistent fog and cap clouds locked to topography. The high frequency of light rainfall throughout the year, in particular between 1100 and 1600 LT during the cold season on the ridgelines of the inner mountain region even in the absence of mesoscale forcing, suggest that interactions among advection fog and low-level clouds are a persistent regional feature. Warm-season observations during the passage of precipitating weather systems show much higher rainfall rates in the inner ridgelines and in valleys than what would be expected from an inspection of collocated vertical reflectivity profiles. Illustrative case studies examined here occurred during early evening and midday hours in May and July of 2011 and 2012.

A look at the results in terms of the relative importance of coalescence and breakup for selected minutes during two of the simulations is provided in Fig. 12. Both R1 cases consistently show that coalescence dominates breakup; additionally, as expected, coalescence results in a net increase of larger drops. At R2 in 2011 and 2012, the magnitude of breakup dominates over coalescence. Gains from coalescence are still seen in the larger/midsize drop region. Thus, coalescence is the dominant mechanism of DSD evolution in cases when there are interactions between the two distinct seeder and feeder drop populations. Breakup is dominant in cases with more convective characteristics and higher rainfall intensities throughout the column.

Fig. 12.

Contributions from coalescence and breakup recorded by the model for (right) case 1 at R2 and (left) case 2 at R1. (top) DSDs are represented at several different heights (500, 350, 250, 50, and 10 m AGL) in a semilog plot. (bottom) The left-hand side of the distribution is enlarged for (left) case 2, R1, at 500 m AGL and (right) case 2, R2, at 10 m AGL. In case 1, during minute 240, the rain rate was moderate and no low-level forcing was included. In case 2, during minute 61, low-level forcing was included, and the rain rate was very light (although nonzero).

Fig. 12.

Contributions from coalescence and breakup recorded by the model for (right) case 1 at R2 and (left) case 2 at R1. (top) DSDs are represented at several different heights (500, 350, 250, 50, and 10 m AGL) in a semilog plot. (bottom) The left-hand side of the distribution is enlarged for (left) case 2, R1, at 500 m AGL and (right) case 2, R2, at 10 m AGL. In case 1, during minute 240, the rain rate was moderate and no low-level forcing was included. In case 2, during minute 61, low-level forcing was included, and the rain rate was very light (although nonzero).

A survey of the underlying microphysics synthesized in terms of Dmσm relationships, based on model output at 1-min intervals and at several levels near the surface (ranging from 10 to 500 m AGL) is presented in Figs. 1315 for each simulation. The data show that, in general, model DSDs exhibit higher variability (width of distribution) for a given Dm than the disdrometer observations. However, when the simulated DSDs are binned as though the simulated rainfall was observed by the PARSIVEL disdrometer (i.e., with a much coarser resolution), there is better agreement between the observations and the model output. Note that the signature of the fog DSD in the model results has much smaller Dm values, although σm is about the same. Recall the findings reported by Tokay et al. (2013) regarding small drop bins for diameters below 0.76 mm with the PARSIVEL, which imply that with greater measurement accuracy, the disdrometer-based Dmσm observations would move to the left with respect to Dm, which is consistent with our results. Nevertheless, Dmσm parameters may not ultimately be the best way to describe the contribution of fog or cap clouds because of the very high number concentrations of very small drops. Detailed observations of fog microphysics in the prestorm environment are imperative to determine the best methods for analysis.

Fig. 13.

(top) Dmσm plots from the model, case 1, R1: (left) Dmσm calculations using all of the model output and (right) Dmσm calculations as though the PARSIVEL was observing the model output, with its discretization and observation thresholds. The red lines represent the fit for observed PARSIVEL disdrometer data from R1 (see map in Fig. 1) during spring and summer 2012 (see Table 3 for the fitted parameters). (bottom) Time series of (left) Dm and (right) σm, where the solid lines represent calculations using the model’s bins and resolution, and the dashed lines represent calculations with binning and resolution as though the PARSIVEL was observing the model output.

Fig. 13.

(top) Dmσm plots from the model, case 1, R1: (left) Dmσm calculations using all of the model output and (right) Dmσm calculations as though the PARSIVEL was observing the model output, with its discretization and observation thresholds. The red lines represent the fit for observed PARSIVEL disdrometer data from R1 (see map in Fig. 1) during spring and summer 2012 (see Table 3 for the fitted parameters). (bottom) Time series of (left) Dm and (right) σm, where the solid lines represent calculations using the model’s bins and resolution, and the dashed lines represent calculations with binning and resolution as though the PARSIVEL was observing the model output.

Fig. 14.

As in Fig. 13, but for case 2.

Fig. 14.

As in Fig. 13, but for case 2.

Fig. 15.

As in Fig. 13, but for case 3, V2. Red lines on the Dmσm plots represent the fit for observed PARSIVEL disdrometer data from V2 (see map in Fig. 1) during summer 2012 (see Table 3 for the fitted parameters).

Fig. 15.

As in Fig. 13, but for case 3, V2. Red lines on the Dmσm plots represent the fit for observed PARSIVEL disdrometer data from V2 (see map in Fig. 1) during summer 2012 (see Table 3 for the fitted parameters).

Recent research has pointed out the importance of relating different drop size distribution parameters to one another in order to help reduce degrees of freedom in retrieval algorithms (e.g., Zhang et al. 2003). Through analysis of DSD observations and the use of an explicit dynamical raindrop population model, the microphysical processes leading to the evolution of physically meaningful parameters describing the DSD, such as the mass spectrum mean diameter and standard deviation, as well as integral parameters related to the DSD such as rain rate and reflectivity, help us to understand the vertical structure of the atmosphere particularly within 2 km of the ground surface. These results corroborate previous work suggesting the dominance of coalescence processes when rain rates are less than 20 mm h−1 (Prat and Barros 2009). Our conclusions support using place-based regime-dependent DSDs (the mass spectrum parameters in particular, or some other parameters to be determined using fog observations) to inform retrievals instead of matching a functional form of the distribution depending on certain conditions.

A key contribution of this study is to illustrate and explain via modeling the importance of seeder–feeder interactions between stratiform rainfall systems and local low-level clouds and fog, and how enhanced coalescence can explain observed DSDs as well as the rainfall rates measured by collocated RGs. These processes are not unique to the Appalachians, and are expected to dominate in cloud forests, midmountains, and coastal regions around the world where there is persistent fog or low-level cloudiness locked to landform (e.g., Barros 2013).

It has been shown here through observations and modeling that processes occurring in the lowest 2 km in the atmosphere have profound impacts on the rain intensity and accumulation at the ground with enhancement factors ranging between 2 and 5 depending on the event duration and characteristics. These values are similar to observations in northern Europe and the British Isles reported in the literature [e.g., Bergeron 1960; see literature survey in Barros and Lettenmaier (1994)]. Yet, there is a great gap in our understanding of the full range of microphysical processes and interactions that govern precipitation processes from cloud condensation nuclei activation to raindrops. In particular, the role of turbulence and hydrodynamic interactions on the efficiency of collision–coalescence processes in cloud and fog evolution is relevant here. Pinsky et al. (1999, 2007, 2008) showed that turbulence and hydrodynamic interactions positively affect collision efficiency (i.e., turbulent conditions enhance seeder–feeder interactions), and Alfonso et al. (2013) investigated the impact of turbulence on the time scale of gelation and, thus, warm rainfall development. Furthermore, the impact of intermittency in the spatial field has also been shown to be highly significant (Pinsky et al. 2000; Kostinski and Shaw 2005). Nevertheless, detailed observations of the dynamics and microphysics of fog and low-level clouds that are essential to characterize the feeder DSDs, including turbulence, are largely unavailable. This research therefore highlights a pressing need for quantitative measurements and observations of fog and low-level clouds, their diurnal cycle, and their dynamic evolution in the presence of different storm regimes. Recording appropriate observations so that we are able to perform detailed analysis of all contributing physical processes is the focus of ongoing and future research.

Acknowledgments

The first author acknowledges support by NSF Graduate Research Program Grant 1106401 and the Pratt School of Engineering. The research was funded by NASA Grant NNX10AH66G with the second author. We are grateful to Greg Wilson and Pamela McCown for allowing us to set up the instrumentation at R2, to Asheville-Buncombe Technical Community College (V2), and to Paul Super and the National Park Service (R1). We thank the University of North Carolina at Asheville and Duke University students who assisted with maintenance and deployment of our RG network and the other instrumentation used in this research. Finally, we thank three anonymous reviewers and the editor Dr. Grabowski for their helpful comments and suggestions.

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