## Abstract

The current operational NCEP Global Forecast System (GFS) cumulus convection schemes are updated with a scale-aware parameterization where the cloud mass flux decreases with increasing grid resolution. The ratio of advective time to convective turnover time is also taken into account for the scale-aware parameterization. In addition, the present deep cumulus convection closure using the quasi-equilibrium assumption is no longer used for grid sizes smaller than a threshold value. For the shallow cumulus convection scheme, the cloud-base mass flux is modified to be given by a function of mean updraft velocity. A simple aerosol-aware parameterization where rain conversion in the convective updraft is modified by aerosol number concentration is also included in the update. Along with the scale- and aerosol-aware parameterizations, more changes are made to the schemes. The cloud-base mass-flux computation in the deep convection scheme is modified to use convective turnover time as the convective adjustment time scale. The rain conversion rate is modified to decrease with decreasing air temperature above the freezing level. Convective inhibition in the subcloud layer is used as an additional trigger condition. Convective cloudiness is enhanced by considering suspended cloud condensate in the updraft. The lateral entrainment in the deep convection scheme is also enhanced to more strongly suppress convection in a drier environment. The updated NCEP GFS cumulus convection schemes display significant improvements especially in the summertime continental U.S. precipitation forecasts.

## 1. Introduction

The cumulus convection schemes in the National Centers for Environmental Prediction’s (NCEP) Global Forecast System (GFS) model, the operational medium-range forecast model at NCEP, have been developed under the assumption that the fractional areas of convective updrafts over the grid box are negligibly small. The current version of the GFS deep convection scheme is based on Pan and Wu (1995), which uses Arakawa and Schubert’s (1974, hereafter AS) quasi-equilibrium assumption as a closure and is simplified by Grell (1993) with a saturated downdraft. Recent major changes to the scheme are found in Han and Pan (2011), where a new shallow convection scheme using a mass flux approach is also developed.

It has been recognized that a scale-aware parameterization is necessary for cumulus convection at the grid sizes (e.g., sizes of 500 m–10 km) where the convective updrafts are not negligibly small and are partially resolved (Hong and Dudhia 2012). Several scale-aware parameterizations for cumulus convection have been proposed (e.g., Arakawa and Wu 2013, hereafter AW; Grell and Freitas 2014, hereafter GF; Pan et al. 2014; Kwon and Hong 2017). In particular, the study by Kwon and Hong (2017) showed that the tests with a scale-aware convection scheme yielded a significant improvement in precipitation forecasts at 3- and 1-km resolutions compared to those without the scale-aware parameterization in the convection scheme and those without any convection scheme. With the GFS grid size approaching a range on the order of 10 km or less, in this study we modify the current operational GFS deep and shallow cumulus convection schemes to have scale awareness. On the other hand, GF investigated the interactions of aerosols with convective parameterizations and showed a positive impact from the aerosol-aware convective parameterization on the precipitation prediction. In this study, we employ a simple aerosol-aware parameterization that follows the studies by Lim (2011) and Han et al. (2016), where rain conversion and cloud condensate detrainment in the convective updraft are given by a function of the cloud condensation nuclei (CCN) number concentration.

Since the 2011 implementation, many issues in the GFS cumulus convection schemes have been raised. They include too much convective precipitation, too much light rain, and unrealistically noisy rainfall, especially over high terrain. To reduce these biases, in this study the convection schemes have been further modified in the convective adjustment time scale, rain conversion rate, convective trigger function, convective cloudiness, and lateral entrainment rate of the convective updraft.

Details of the parameterizations and modifications are described in sections 2 and 3. In section 4 we evaluate the impacts of the updated schemes on medium-range forecasts. Finally, in section 5 we summarize our study.

## 2. Scale- and aerosol-aware parameterizations

### a. Scale-aware parameterization

The present GFS cumulus convection schemes were developed under the assumption that the fractional areas of convective updrafts over the grid box are negligibly small. The assumption of a negligibly small updraft area may not be valid any longer as the model grid sizes become smaller and smaller (e.g., less than 10 km). For those small grid sizes where the convective updrafts are partially resolved, therefore, a scale-aware parameterization would be necessary.

The GFS deep and shallow cumulus convection schemes are modified to include AW’s scale-aware parameterization where the vertical convective eddy transport decreases with increasing fractional updraft area; that is,

where *w* is the vertical velocity, *ψ* is a variable such as moist static energy, *σ*_{u} is the fractional updraft area, an overbar indicates the grid average, a prime shows the perturbation from the grid average, a subscript *u* denotes an updraft, and a subscript *E* represents a value when *σ*_{u} ≪ 1. In this study, we slightly modify Eq. (1) as

where *M*_{BE} is the cloud-base mass flux when *σ*_{u} ≪ 1, and *M*_{B} is the cloud-base mass flux reduced with a finite *σ*_{u}. Equations (1) and (2) are conceptually identical, and we find that Eq. (2) is more easily applicable for the scale-aware parameterization.

As AW points out, the key parameter for the scale-aware parameterization is *σ*_{u}. In AW, *σ*_{u} is determined as

Since Eq. (1) is derived based on Eq. (3), it would be desirable to determine *σ*_{u} using Eq. (3) for consistency. In tests with a very high-resolution (2 km) run of the Hurricane Weather Research and Forecasting (HWRF) Model, however, we find that Eq. (3) tends to produce unrealistically small *σ*_{u} even for 2-km resolution, having *σ*_{u} < 0.1 in most convective regions and consequently resulting in precipitation mostly from subgrid cumulus convection, which is unrealistic. It appears that this small *σ*_{u} is mainly due to the inability to produce sufficiently large grid-scale vertical velocity in the high-resolution model run. To obtain more realistic *σ*_{u} results from Eq. (3), on the other hand, one should also have a good estimation of *w*_{u}, *ψ*_{u}, and grid-scale *ψ*, as well as more realistic grid-scale *w* from the model forecast, which may not be easily achievable considering the uncertainties in the current existing cumulus convection parameterizations. While it is worthwhile to seek an improved method for obtaining more realistic *σ*_{u} from Eq. (3), for a practical use in an operational numerical weather prediction model such as GFS we adopt the GF method to determine *σ*_{u}. The GF method is simple, yet it has a smooth transition on increasing grid resolution. Note that GF also uses Eq. (1) for their scale-aware parameterization.

Following GF, *σ*_{u} is given as

where *A*_{grid} is the grid-box area, *R*_{c} the radius of the convective updraft, and *ε*_{0} the turbulent lateral entrainment rate of the updraft. We consider the entrainment rate at the cloud base as representative for a cumulus cloud, which is given as (Han and Pan 2011)

where *z*_{B} is the cloud-base height and the constant *c*_{0} is 0.1 for deep convection and 0.3 for shallow convection. Although it is quite uncertain that Eqs. (2), (4), and (5) may yield the grid-size-dependent convective eddy fluxes consistent with AW’s study (which warrants further investigation), they provide an essential feature of scale awareness that the convective eddy fluxes decrease with increasing grid resolution for the grid sizes where the convective updrafts are partially resolved.

Figure 1 shows the *σ*_{u} distributions using Eqs. (4) and (5) from high-resolution (2 and 6 km) runs of the HWRF Model for hurricanes in the Atlantic Ocean. The use of the HWRF Model for the tests of scale-aware parameterization is motivated by a practical difficulty in the GFS model runs with such fine grid sizes of 2 and 6 km. While for the grid size of 2 km (Fig. 1a) the fractional areas from most of the cumulus clouds are larger than 0.3 and some of them are even larger than 0.9, for the grid size of 6 km (Fig. 1b) they are rarely larger than 0.3. Figures 2 and 3 display 6-h accumulated convective and total precipitation with the scale-aware parameterization [i.e., Eqs. (2), (4), and (5)] compared to the precipitation without the scale-aware parameterization. As is consistent with the fractional areas, the convective precipitation for the grid size of 2 km is much smaller with the scale-aware parameterization than without it (Figs. 2a,b) while there is not much difference in the total precipitation (Figs. 2c,d). This implies that using the scale-aware parameterization with reduced grid size, which reduces convective precipitation, leads to increased grid-scale precipitation resolved by cloud microphysics, as may be expected to happen at high grid resolution. For the grid size of 6 km (Fig. 3), on the other hand, the convective precipitation is just a little smaller with the scale-aware parameterization than without it (Figs. 3a,b), as expected from much smaller cumulus fractional areas in Fig. 1b, indicating much weaker impacts of the scale-aware parameterization in this grid size.

As another consideration for scale awareness, the ratio of the advective time (ADT) to the convective turnover time (CTT) of a cumulus cloud is taken into account. When the CTT is larger than the ADT, the convective mixing is not fully conducted before the cumulus cloud is advected out of the grid cell. In this case, therefore, the cloud-base mass flux is further reduced in proportion to the ratio of ADT to CTT.

On the other hand, as the grid size becomes smaller and smaller, the quasi-equilibrium closure assumption in AS may not be valid any more. For grid sizes smaller than a threshold value (currently set to 8 km), therefore, we propose that the cloud-base mass flux in the deep convection scheme is given by a function of mean updraft velocity rather than by AS’s quasi-equilibrium; that is,

for grid sizes less than 8 km, where 〈*w*_{u}〉 is the cumulus updraft velocity averaged over the whole cloud depth. The updraft velocity *w*_{u} is computed using (Simpson and Wiggert 1969)

with the buoyancy as a source term (where *θ*_{υ} is the virtual potential temperature and *g* is the gravity). In Eq. (7), *ε* is the lateral entrainment rate described in Eqs. (4), (5), and (14), and the optimal values of the coefficients, *c*_{1} and *c*_{2}, are given as 4.0 and 0.8, respectively. Although not shown, the convective precipitation from Eq. (6) is found to be not much different from AS’s quasi-equilibrium for the current operational GFS grid resolution of 13 km.

In the current operational shallow convection scheme, the value of *M*_{BE} is obtained as a function of the unstable boundary layer velocity scale (Grant 2001; Han and Pan 2011), and thus the shallow convection is never triggered in the stable boundary layer. In the update, Eq. (6) is used to compute *M*_{BE} for all grid sizes, allowing shallow convection even for the stable boundary layer. On the other hand, the threshold of cloud thickness that distinguishes shallow from deep convection is increased from 150 to 200 hPa.

### b. Aerosol-aware parameterization

In the current operational cumulus convection schemes, the rain conversion rate of the parcel in updrafts *d*_{0} is set to be a constant with a value of 0.002 m^{−1}. From analysis of a three-dimensional (3D) cloud-resolving simulation dataset of a convective storm, Lim (2011) and Han et al. (2016) have shown that the conversion rate *d*_{0} from cloud liquid water and ice to rain, snow, and graupel not only decreases exponentially above the freezing level but also increases with decreasing aerosol number concentration (e.g., see Fig. 1 in Han et al. 2016). Furthermore, Han et al. 2016) found overall improvement in forecast skill in precipitation and large-scale fields with the exponential decrease of the rate above the freezing level. Following Han et al. (2016), in this study the exponential decrease of the rate is given by

where *a* (=0.002 m^{−1}) and *b* (=0.07) are constants and *T*_{0} (=0°C) is the freezing temperature. In the update, *b* is set to the much smaller value of 0.01 to avoid too much of an increase in cloud condensate in the upper troposphere. Note that the reduction of *d*_{0} in the upper troposphere increases the detrainment of cloud condensate in the updrafts into grid-scale condensate, enhancing cloud ice there, as shown in Fig. 4. Since the GFS tends to underestimate high clouds (C. R. Jones et al. 2017, unpublished manuscript), the increased cloud condensate in the upper troposphere would be desirable.

Based on the 3D cloud-resolving simulation of a convective storm conducted by Lim (2011), we formulate *a* as a function of the aerosol number concentration *N*_{ccn} (cm^{−3}), decreasing with increasing *N*_{ccn}:

where *a*_{1} and *a*_{2} are tunable parameters, given as −0.7 and 24 in this study, respectively.

Because *N*_{ccn} varying with time and space is not currently available, in this study we use typical values of *N*_{ccn} = 100 and *N*_{ccn} = 7000 for sea and land, respectively, which gives rise to *a* = 0.002 for sea and *a* = 0.001 78 for land. In the future, use of double-moment cloud microphysics scheme would produce *N*_{ccn} varying with time and space, providing more realistic *a* values.

## 3. Other updates

### a. Quasi-equilibrium closure

The updraft mass flux at the cloud base in the deep cumulus convection scheme is obtained using AS’s quasi-equilibrium assumption:

where *A* is the cloud work function, a measure of the buoyancy of cloud; *A*_{0} is a reference cloud work function derived from observations by Lord (1978); *α*(*w*) is a function of vertical velocity *w*, modifying *A*_{0}; *τ* is a convective adjustment time scale with a range of 20–60 min inversely proportional to *w*; is the cloud work function after the modification of the thermodynamic fields by an arbitrary amount of mass flux ; *T* is the environmental temperature; *c*_{p} is the specific heat at constant pressure; *L* is the latent heat of the vaporization of water; *h* is the moist static energy of a parcel; *q* is the moisture; the subscripts *u* and *s* stand for updraft and saturation, respectively; the overbar represents the environmental mean value; *z* is the height; and *z*_{T} and *z*_{B} are the heights at the cloud top and base, respectively.

Following the study by Bechtold et al. (2008), in the update the convective adjustment time scale *τ* is set proportional to the convective turnover time, which is computed using the mean updraft velocity averaged over the entire cloud layers from Eq. (7). The value of *A*_{0} is set to zero, implying that the instability is completely eliminated after the convective adjustment time.

### b. Convection trigger function

Unrealistically spotty rainfall, especially over high terrain during summertime, has been often reported in the GFS forecasts. Figure 5 displays one example of such a case. [For example, see the control precipitation forecast (Fig. 5b) compared to the observations (Fig. 5a) over the Rocky Mountains.] This unrealistically spotty rainfall is found to be mainly from convective rain, indicating that convection triggering may be too easy over mountainous regions. The triggering condition in the GFS convection schemes is that a parcel lifted from the convection starting level (CSL) without entrainment must reach the level of free convection within the range of 120–180 hPa, in proportion to the large-scale vertical velocity. One problem with this condition is that it does not have information for the environmental profile feature below cloud base. To account for environmental profile information, in the update we introduce the convective inhibition (CIN) as

where *Z*_{S} is the height of the CSL. In the update, the convection is not triggered if CIN is less than a critical value, which is currently set to a range from −120 to −80 m^{2} s^{−2} and inversely proportional to the large-scale vertical velocity. With the additional trigger condition of the CIN, the unrealistically spotty precipitation has been largely suppressed (Fig. 5c).

### c. Convective cloudiness enhancement

The cloud fraction in the GFS used in the radiation computation is determined following Xu and Randall (1996), which is proportional to the grid-scale cloud condensate and relative humidity. On the other hand, the convective cloudiness in the GFS is taken into account by detraining cloud water from upper cumulus layers into the grid-scale cloud condensate, which helps to increase high cirrus clouds. To take into account the convective cloudiness contribution from all the cumulus layers, in this study we add the suspended cloud water in every cumulus layer into the grid-scale cloud condensate only for the cloud fraction and radiation computations. Figure 6 shows that cloudiness enhancement by the suspended cloud water in the convective updraft is evident in the low and middle clouds over the tropical convective regions. In particular, the larger increase in the low clouds (Fig. 6a) appears to be due to contributions from shallow convection in addition to deep convection. For the high clouds (Fig. 6c), the impact of the suspended cloud water on convective cloudiness enhancement appears to be rather small, indicating that the convective cloudiness for high clouds is mainly from the detrained cloud condensate from upper cumulus layers.

### d. Entrainment enhancement in dry environments

The entrainment rate in the current GFS deep cumulus convection scheme is given by (Bechtold et al. 2008; Han and Pan 2011)

where *ε*_{0} is the turbulent entrainment rate at the cloud base given in Eq. (5); RH is the environmental relative humidity; *d*_{1} is a tunable parameter with a value of 1.0 × 10^{−4} m^{−1}; *q*_{s} and *q*_{sb} are the saturation specific humidity at the parcel level and the cloud base, respectively; and *F*_{0} and *F*_{1} are dimensionless vertical scaling functions that decrease strongly with height. In the update, we increase *d*_{1} 10 times (i.e., *d*_{1} = 1.0 × 10^{−3} m^{−1}) to more strongly suppress convection in a drier environment. Bechtold et al. (2014) report that *d*_{1} is also greatly increased in the recent update of the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System (IFS). Although *d*_{1} is an order of magnitude larger, the impact in a moist environment (i.e., where RH is close to 1.0) is small.

## 4. Medium-range forecast results

To assess the impacts of the updated schemes on forecast skill, 6-day forecasts for the period of 1 July–31 October 2015 were conducted. The initial forecast time was at 0000 UTC for each day. The GFS used in this test has 64 vertical sigma-pressure hybrid layers and semi-Lagrangian T1534 (about 13 km) horizontal resolution. Since the forecasts were performed with no data assimilation, the analysis data from the operational GFS were used as the initial conditions. Although tests with data assimilation would be desirable, they not only cost too much in computing time but Park and Hong’s (2013) study indicates that initial conditions are not always essential when evaluating the impact of model physics changes.

Figure 7 shows that the updated schemes have a neutral impact on anomaly correlation for the 500-hPa height that illustrates how well synoptic-scale systems are represented over the globe. Although there is slight improvement in the Southern Hemisphere in later forecast hours, the improvement is not statistically significant. Comparisons of the mean equitable threat and bias scores (Gandin and Murphy 1992) for the 60–84-h precipitation forecasts over the continental United States are displayed in Fig. 8. The forecast scores for the 12–36- and 36–60-h precipitation forecasts (not shown) were similar to those for the 60–84-h precipitation forecasts. Compared to the control forecasts, the updated schemes show a significant improvement in the continental U.S. precipitation forecasts. The equitable threat score (Figs. 8a,c) is better with the updated schemes for all rain threshold ranges, although it is statistically not significant for the heavy rain ranges larger than 25 mm day^{−1}. For the bias (Figs. 8b,d), the updated schemes reduce both the wet bias for light rain (e.g., rain less than the threshold of 5 mm day^{−1}) and the dry bias for moderate rain (e.g., rain within the threshold of 10–35 mm day^{−1}). Although they tend to produce more heavy rain (e.g., increased wet bias for rain over 35 mm day^{−1}), it is statistically not significant. The mean precipitation for the forecasts with the updated schemes and its difference with respect to that for the control forecasts are shown in Fig. 9. Although overall the differences are small, there are some areas with a significant precipitation difference over the tropics. A detailed investigation for these differences is outside of the scope of this study and remains as a topic for future study.

The performance of the updated schemes for hurricane forecasts is shown in Figs. 10 and 11 in terms of hurricane track and intensity errors, respectively, and shows a mixed signal. The numbers of hurricanes used in the statistics are 8, 12, and 20 for Atlantic, east Pacific, and west Pacific hurricanes, respectively. Compared to the control track forecasts, the updated schemes show smaller errors after 72 forecast hours for 2015 Atlantic hurricanes (Fig. 10a), while they display larger errors after 48 forecast hours for 2015 east Pacific hurricanes (Fig. 10b). For the intensity forecasts, the updated schemes generally show larger errors than the control for 2015 Atlantic (Fig. 11a) and west Pacific (Fig. 11c) hurricanes except for during the early forecast hours, while for 2015 east Pacific hurricanes (Fig. 11b), they display smaller errors than the control during most forecast hours.

## 5. Summary and discussion

The current operational GFS deep and shallow cumulus convection schemes have been updated with a scale-aware parameterization based on AW where the cloud mass flux decreases with increasing fractional convective updraft area *σ*_{u}. The parameterization of *σ*_{u} (which is a key parameter in AW) is given as an inverse function of the entrainment rate following GF. For the cases when the convective turnover time is larger than the large-scale advective time scale of cumulus clouds (which may occur more often for smaller grid sizes), the cloud mass flux is further reduced in proportion to the ratio of the advective time to the convective turnover time. In addition, for grid sizes smaller than a threshold value (currently set to 8 km), the cloud-base mass flux in the deep convection scheme is given by a function of mean updraft velocity rather than by the current closure using AS’s quasi-equilibrium assumption. The cloud-base mass flux in the shallow convection scheme is also modified, given by a function of mean updraft velocity rather than by the current function of the positive boundary layer convective velocity scale. A simple aerosol-aware parameterization based on the studies by Lim (2011) and Han et al. (2016), where rain conversion in the convective updraft is given by a function of aerosol number concentration, is also included in the update.

Along with the scale- and aerosol-aware parameterizations, the cumulus convection schemes have been further modified to improve the forecast skill. The cloud-base mass-flux computation in the deep convection scheme is modified to use convective turnover time as the convective adjustment time scale. The rain conversion rate is modified to decrease with decreasing air temperature above the freezing level, which gives rise to more detrainment of cloud condensate in the upper updraft layers and consequently more high clouds. A convective inhibition in the subcloud layer is used as an additional trigger condition. Convective cloudiness is enhanced by considering suspended cloud condensate in the updraft. The lateral entrainment in the deep convection scheme is also enhanced to more strongly suppress convection in drier environments.

The updated GFS cumulus convection schemes yield a significant improvement in the U.S. continental precipitation forecasts, especially during the summertime, while they show neutral impact on the 500-hPa height forecast skill. For the hurricane track forecasts, the performance of the updated schemes shows a mixed signal: with a limited sample size during 2015, the track forecasts are improved for the Atlantic hurricanes while they worsen for the east Pacific hurricanes.

The updated GFS cumulus convection schemes have been operationally implemented as part of the July 2017 NCEP GFS upgrade. Preoperational tests with the bundle of GFS changes including the updated schemes in this study have shown a positive impact on the medium-range forecasts. While how a particular change plays its role in the forecast improvement warrants further investigation, our study is limited to a statistical evaluation of medium-range forecasts over a longer term for combined effects from all the changes outlined in sections 2 and 3 of this study. With the current GFS resolution of 13 km, on the other hand, the impact of the scale-aware parameterization on forecasts is rather small. The GFS resolution will be below 10 km over the next few years, when the impact of the scale-aware parameterization will be more significant.

## Acknowledgments

This work was conducted collaboratively between NCEP and Korea Institute of Atmospheric Prediction Systems (KIAPS). Internal reviews from Shrinivas Moorthi, Helin Wei, and Mary Hart at NCEP/EMC are highly appreciated. We also thank the anonymous reviewers for valuable comments that helped to improve the manuscript.

## REFERENCES

## Footnotes

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