1. Introduction
The planetary boundary layer (PBL) scheme in the National Centers for Environmental Prediction’s (NCEP) Global Forecast System (GFS) adopts an eddy-diffusivity countergradient (EDCG) mixing approach (Deardorff 1966; Troen and Mahrt 1986; Hong and Pan 1996; Han and Pan 2011) to take into account nonlocal transport by strong updrafts in the daytime convective boundary layer (CBL). Although the GFS EDCG PBL scheme provides a realistic CBL development projection despite its simplicity, it has been found to underpredict the daytime CBL growth (Noh et al. 2003; Siebesma et al. 2007). Siebesma et al. (2007) show that the underestimation of the CBL growth by the EDCG approach is due to too weak entrainment flux at the CBL top, which is caused by a positive countergradient (CG) term over the entire CBL.
To have better PBL growth in the CBL, an eddy-diffusivity mass-flux (EDMF) PBL scheme is developed based on Siebesma and Teixeira (2000), Soares et al. (2004), and Siebesma et al. (2007). In the EDMF approach, the nonlocal subgrid transport due to the strong updrafts is taken into account by a mass-flux (MF) scheme and the remaining transport by smaller eddies is handled by an eddy-diffusivity (ED) scheme. The nonlocal momentum transport is also developed with the MF approach, which is missing in the original Siebesma et al. (2007) EDMF scheme.
PBL models based on the EDMF concept are being more widely used in weather and climate prediction models. For example, Köhler et al. (2011) implemented an EDMF scheme in the European Centre for Medium-Range Weather Forecasts (ECMWF) model. Hourdin et al. (2013) presented an implementation of the thermal plume model (Rio and Hourdin 2008), an EDMF-type scheme, in the Laboratoire de Météorologie Dynamique model (LMDZ; with Z standing for the model zoom capacity). Recently, Sušelj et al. (2014) developed a unified boundary layer and shallow convection parameterization based on a stochastic EDMF approach and implemented the scheme in the Navy Global Environmental Model (NAVGEM).
Initial tests indicate that while the EDMF scheme predicts the CBL growth well, it tends to have too much vertical mixing in wind and moisture over the tropics, giving rise to unrealistically large low cloud amounts as well as largely degraded wind vector forecasts in the tropics. This led to the development of a hybrid EDMF PBL scheme, which will be discussed in detail later. On the other hand, heating by the dissipation of turbulent kinetic energy (TKE) is usually neglected in numerical weather and climate prediction models because of its small magnitude. Bister and Emanuel (1998) first recognized that the viscous dissipation of TKE can be a significant source of heat in hurricanes. Afterward, many studies (e.g., Zhang and Altshuler 1999; Jin et al. 2007) have shown that the inclusion of TKE dissipative heating increases the surface maximum wind by about 10%–30% in hurricane forecasts. In this study, the TKE dissipative heating is parameterized and included in the hybrid EDMF PBL scheme. In addition, the current local scheme in the stable boundary layer (SBL) is modified to improve vertical turbulent mixing for weakly and moderately stable conditions.
The implementation procedure associated with model physics changes in the NCEP operational numerical weather prediction (NWP) system is as follows: 1) identify model forecast biases or problems (e.g., the present study was motivated by the problem of underestimated PBL growth with the current EDCG PBL scheme in strongly unstable conditions), 2) reduce the bias by updating the model physics with more realistic parameterizations (e.g., develop the EDMF PBL scheme to have better PBL growth in strongly unstable conditions), 3) test the updated scheme in the operational mode, and, finally, 4) implement the scheme into the operational model if the forecast skill is improved or at least neutral, otherwise, reject the scheme even though it may be more realistic in many respects. A better scheme is not always guaranteed to produce a statistically improved forecast skill due to adverse interactions among model physics parameterizations and with the dynamics. While more specific case studies are helpful for a detailed investigation on how and why a particular scheme improves the forecast, our study focuses on statistical forecast skill improvement over the longer term, which is the main concern in operational NWP centers.
Details of the parameterizations and modifications are described in section 2. In section 3, we evaluate the impacts of the new scheme on medium-range forecasts. Finally, in section 4 we summarize our study.
2. Development of hybrid EDMF PBL parameterization with dissipative heating and modified SBL mixing
The NCEP GFS is a global spectral model for weather and climate prediction. The radiation parameterization uses the Rapid Radiative Transfer Model (RRTM) adapted from Atmospheric and Environmental Research (AER), Inc. (e.g., Mlawer et al. 1997; Iacono et al. 2000; Clough et al. 2005). The orographic gravity wave drag parameterization is adopted from Kim and Arakawa (1995). The cloud condensate is a prognostic quantity with a simple cloud microphysics parameterization (Zhao and Carr 1997; Sundqvist et al. 1989; Moorthi et al. 2001). The fractional cloud cover used for radiation purposes is diagnostically determined by the predicted cloud condensate, following the study by Xu and Randall (1996). The Noah land surface model (Ek et al. 2003) is used and the surface layer similarity formulations are based on Long (1986, 1989).
Recently, parameterizations for deep and shallow cumulus convection and vertical turbulent mixing have been greatly revised (Han and Pan 2011). The shallow cumulus convection scheme now employs a mass-flux parameterization replacing the old turbulent diffusion–based approach. For deep cumulus convection, the scheme was revised to make convection stronger and deeper to suppress excessive grid-scale precipitation. The PBL model was revised to enhance turbulent diffusion in stratocumulus regions. In this study, we further update the PBL model using the EDMF approach.
a. Hybrid EDMF parameterization for the CBL
Figure 1 compares the vertical profiles of potential temperature and turbulent heat flux simulated after 8 h by a large-eddy simulation (LES) with the GFS single-column model (SCM) result using the EDCG scheme. For the simulations, an initial potential temperature profile θ = 288 K + (3 K km−1)z is used and a constant surface buoyancy flux (8 × 10−3 m2 s−3) is specified. The LES model used here is version 6 of the System for Atmospheric Modeling (SAM; Khairoutdinov and Randall 2003). The vertical grid size for the SCM is 50 m, which is the same as that in the LES. Figure 1b clearly shows that the underprediction of the CBL growth seen in Fig. 1a is caused by a too weak entrainment flux at the CBL top due to the positive CG term over the entire CBL, as also shown by the Siebesma et al. (2007) study. Note that the CBL growth is influenced by both the heat flux at the surface and the entrainment flux at the top.
SCM results with the EDCG scheme compared with LES results after an 8-h simulation. Vertical profiles of (a) potential temperature and (b) total turbulent heat fluxes normalized by surface heat flux with the breakdown of heat flux of the EDCG scheme into ED and CG contributions.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
In the EDMF scheme, two different PBL heights are used. The h in Eq. (3) is determined from Eq. (11) but without the thermal excess in Eq. (12). For the MF part in Eq. (4), h is defined as the height of wu = 0 from Eq. (6), so that the updraft overshoots into the upper stable layer and produces negative entrainment fluxes in the upper PBL. For diagnostic purposes, h is defined as Eq. (11) but without the thermal excess in Eq. (12).
The tunable coefficients of this implementation, au, b1, b2, and cε, are somewhat sensitive to the vertical grid resolution. The optimal coefficient values used for the SCM, for which the vertical grid size is 50 m, are au = 0.1, b1 = 1.8, b2 = 4.0, and cε = 0.37. These values are comparable to those used in Soares et al. (2004) and Siebesma et al. (2007). In Soares et al. (2004), for example, au = 0.1, b1 = 2.0, b2 = 4.0, and cε = 0.5, whereas in Siebesma et al. (2007), au = 0.12, b1 = 1.43, b2 = 2.86, and cε = 0.4. Figure 2a displays the GFS-SCM result using the EDMF scheme and shows that the EDMF scheme predicts the CBL growth well compared to the LES. As shown in Fig. 2b, the better CBL growth in the EDMF scheme compared to the EDCG scheme (Fig. 1b) is mainly due to a larger entrainment flux induced by the mass-flux term. However, when we use the operational GFS vertical grid spacing, which is ~200 m at about 1-km level above the surface, the CBL growth with the EDMF scheme tends to be overpredicted, as seen in Fig. 3. Therefore, the coefficients were optimized again for better CBL growth at the operational GFS resolution as au = 0.08, b1 = 1.8, b2 = 3.5, and cε = 0.38 (Fig. 3). Although not shown, one of the reasons for the sensitivity to the vertical resolution is the dependency of the entrainment rate [Eq. (8)] on the vertical grid size (Δz). Thus, entrainment parameterizations that are not directly related to the vertical grid size (e.g., Köhler et al. 2011; Witek et al. 2011) will be investigated in future studies.
As in Fig. 1, but for SCM results with the EDMF scheme.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
Vertical profiles of potential temperature from the SCM results with the current operational GFS vertical resolution compared with LES results after an 8-h simulation. The green line (EDMF1) represents the EDMF scheme result with the same tunable parameters as those used in high-resolution SCM tests (i.e., a1 = 0.1, b1 = 1.8, b2 = 4.0, and ce = 0.37), the blue line (EDMF2) shows the EDMF scheme result with parameters optimized for the operational GFS vertical resolution (i.e., a1 = 0.08, b1 = 1.8, b2 = 3.5, and ce = 0.38), and the red line shows the EDCG scheme result.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
To investigate the impact of the new EDMF PBL scheme, medium-range forecasts have been conducted with initial conditions from the operational GFS analysis data. Initial tests indicated that the EDMF scheme improved the forecast skill in the 500-mb height anomaly correlation over the midlatitudes and in precipitation (especially light rain) over the continental United States. However, it degraded the tropical wind vector forecast and also increased low cloud amounts too much over the tropics, as shown in Fig. 4. In order not to degrade the forecast skill over the tropics, a hybrid EDMF scheme is developed, where the EDMF scheme is applied only for the strongly unstable PBL (i.e., CBL) while the EDCG scheme is retained for the weakly unstable PBL. Note that unlike for strongly unstable conditions, the EDCG scheme does not underestimate the PBL growth for weakly unstable conditions (Noh et al. 2003). A detailed investigation on why using this version of EDMF for weakly unstable conditions may slightly degrade forecast skill in the GFS will be performed in a future study. The z/L [L is the Monin–Obukhov length, defined as 
Mean differences between EDMF scheme and control forecasts of (a) wind vector RMSE over the tropics (30°S–30°N) and (b) low cloud fraction. The forecast period for the mean difference calculation is from 7 Jul to 31 Oct 2012, and the low cloud fraction is the average of 102, 108, 114, and 120 forecast hours.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
As in Fig. 4, but for the hybrid EDMF scheme.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
b. Dissipative heating parameterization
Viscous dissipation of TKE can be a significant source of heat especially in strong wind conditions such as hurricanes, but its effect is not taken into account in the current operational GFS. On the other hand, excluding the dissipative heating can cause a significant energy imbalance in an atmospheric model (Fiedler 2000).
Figure 6 displays the impact of TKE dissipative heating on a hurricane forecast. It shows that the hurricane intensity (e.g., maximum 10-m wind speed) can be significantly increased in the GFS forecast with TKE dissipative heating, consistent with the studies such as those of Zhang and Altshuler (1999) and Jin et al. (2007). Note that the hurricane intensity in the GFS is generally much weaker than the observations, which is mainly due to coarse resolution and missing physics such as the TKE dissipative heating in the present study. On the other hand, the energy balance test results with TKE dissipative heating (Table 1) show that a large atmospheric energy loss (about 4–5 W m−2) found in the GFS atmosphere–ocean coupled run can be largely reduced with the inclusion of TKE dissipative heating, leading to an improved global energy balance.
Time evolution of (a) min central mean sea level pressure and (b) max 10-m wind speed with (dashed line) and without (solid line) TKE dissipative heating for Hurricane Ike (2008). The observed intensities from the best-track data are displayed as light solid lines.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
Energy balance test results from the GFS atmosphere–ocean coupled (1 yr) run with (EXP) and without (CTL) TKE dissipative heating. TOA represents the mean energy budget at the top of the atmosphere, SFC is the mean energy budget at the surface, and difference is the mean energy budget at the TOA minus the SFC [indicating mean energy loss (positive) or gain (negative) in the model atmosphere].
c. Modification in the SBL vertical mixing scheme
3. Medium-range forecast results
To assess the impacts of the new scheme (i.e., the hybrid EDMF PBL scheme with TKE dissipative heating and modification in the SBL) on forecast skill, 7-day forecasts for the period from 1 July to 31 October 2012 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 T574 (about 35 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 the study of Park and Hong (2013) seems to indicate that initial conditions are not always essential when evaluating the impact of model physics changes.
A comparison of anomaly correlations for the 500-hPa height, which illustrates how well synoptic-scale systems are represented around the globe, is shown in Fig. 7 as a function of forecast length. In both the Northern (20°–80°N) and Southern (20°–80°S) Hemispheres, the new scheme displays better anomaly correlations than does the control forecast. Although the improvement is somewhat small, it is still significant.
Mean difference in anomaly correlations of 500-hPa height for the forecasts with the new scheme with respect to the control forecasts in the (a) Northern Hemisphere (20°–80°N) and (b) Southern Hemisphere (20°–80°S) from 7 Jul to 31 Oct 2012. The differences outside the rectangle bars are statistically significant at the 95% confidence level.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
Comparisons of equitable threat and bias scores (Gandin and Murphy 1992) for the 12–36-h precipitation forecasts over the continental United States are shown in Fig. 8. Compared to the control forecasts, the equitable threat score (Fig. 8a) is slightly better for light rain but slightly worse for heavy rain. For the bias (Fig. 8b), the new scheme is wetter than the control for heavy rain. However, the forecast skill differences for both equitable threat and bias scores are not significant. Although not shown, the forecast skill differences for the 36–60- and 60–84-h precipitation forecasts were very similar to those in the 12–36-h precipitation forecasts.
Mean (a) equitable threat score and (b) bias score for 12–36-h precipitation forecasts over the continental United States for the control forecasts (solid line) and forecasts with the new scheme (dashed line) from 7 Jul to 31 Oct 2012.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
The performance of the new scheme for hurricane forecasts is shown in Fig. 9 in terms of hurricane track and intensity errors, and shows a mixed signal. Compared to the control track forecast (Figs. 9a,b), the new scheme shows a smaller error at 120 forecast hours for 2012 Atlantic hurricanes, but it displays a larger error at 72 and 96 forecast hours for 2012 eastern Pacific hurricanes. For the intensity forecast (Figs. 9c,d), the new scheme shows a smaller error than the control for 2012 eastern Pacific hurricanes for most forecast times, but for 2012 Atlantic hurricanes it displays a slightly larger error than the control, especially at forecast times of less than 24 h. In general, the error differences between the new scheme and the control are quite small for both hurricane track and intensity forecasts.
As in Fig. 8, but for mean hurricane (a),(b) track and (c),(d) intensity errors for the (left) Atlantic and (right) eastern Pacific Ocean regions.
Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0053.1
4. Summary and conclusions
The current operational eddy-diffusivity countergradient (EDCG) planetary boundary layer (PBL) scheme in the NCEP GFS underestimates PBL growth in the convective boundary layer (CBL) because of a positive countergradient (CG) mixing term in the entrainment zone. To improve the PBL growth in the CBL, an eddy-diffusivity mass-flux (EDMF) PBL scheme is developed for the GFS, where the constant CG mixing term is replaced by a mass-flux (MF) scheme. For the vertical momentum mixing, the MF scheme is further modified to include the effect of the updraft-induced pressure gradient force, which weakens the vertical momentum exchange.
The EDMF scheme predicts the CBL growth well compared to large-eddy simulation results. Over the tropics, however, the EDMF scheme tends to overproduce the amounts of low clouds and degrades the wind vector forecasts. Note that the tropics are mostly composed of oceans where a strongly unstable PBL is rarely found. In order not to degrade the forecast skill in the tropics a hybrid scheme is developed, where the EDMF scheme is applied only for the strongly unstable PBL (i.e., CBL), while the EDCG scheme is used for the weakly unstable PBL. As a result, the EDCG scheme is mostly called over the tropics in the hybrid scheme.
To improve a significant energy imbalance in the coupled GFS, the heating by turbulent kinetic energy (TKE) dissipation, which is a missing physics term in the GFS model, is parameterized and included in the hybrid EDMF PBL scheme. With the inclusion of TKE dissipative heating, a large atmospheric energy loss in the GFS atmosphere–ocean coupled run is largely reduced and the hurricane intensity is significantly increased in the GFS forecast. To enhance the vertical turbulent mixing that is too weak in the case of a stable boundary layer (SBL), the current local scheme is modified to use an ED profile method for weakly and moderately stable conditions with varying critical bulk Richardson numbers for determining the PBL height.
Medium-range forecasts have been conducted to assess the impacts on forecast skill of the new hybrid EDMF PBL scheme with the TKE dissipative heating and the modifications in the SBL mixing scheme. The new scheme presents a significant improvement for the 500-hPa height anomaly correlations, which is one of the most important forecast skill measurements since it illustrates how well synoptic-scale systems are represented around the globe. For precipitation forecasts over the continental United States and hurricane forecasts, the new scheme shows a neutral skill. The hybrid EDMF PBL scheme with TKE dissipative heating and modification in the SBL was operationally implemented in the January 2015 NCEP GFS upgrade as tests with this scheme showed a positive impact on the GFS medium-range forecasts.
This study is a first step toward including an EDMF parameterization of the PBL in the GFS. For further improvement of the EDMF parameterization, we are currently developing an EDMF scheme based on a TKE closure model that requires a prognostic equation for TKE. With the memory of turbulence retained, the TKE-based EDMF scheme may better handle vertical turbulent mixing, especially for a PBL with weak turbulence, which in turn, may help to improve model performance for weakly unstable conditions over the tropical ocean. A unified EDMF model of PBL and shallow convection is also under development.
This work was supported by the NOAA MAPP/CPO program as part of the Sc-to-Cu Transition Climate Process Team. Internal reviews from Helin Wei, Qingfu Liu, 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
Bister, M., , and Emanuel K. A. , 1998: Dissipative heating and hurricane intensity. Meteor. Atmos. Phys., 65, 233–240, doi:10.1007/BF01030791.
Brown, A. R., , and Grant A. L. M. , 1997: Non-local mixing of momentum in the convective boundary layer. Bound.-Layer Meteor., 84, 1–12, doi:10.1023/A:1000388830859.
Clough, S. A., , Shephard M. W. , , Mlawer E. J. , , Delamere J. S. , , Iacono M. J. , , Cady-Pereira K. , , Boukabara S. , , and Brown P. D. , 2005: Atmospheric radiative transfer modeling: A summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244, doi:10.1016/j.jqsrt.2004.05.058.
Cuxart, J., and Coauthors, 2006: Single-column model intercomparison for a stably stratified atmospheric boundary layer. Bound.-Layer Meteor., 118, 273–303, doi:10.1007/s10546-005-3780-1.
Deardorff, J. W., 1966: The counter-gradient heat flux in the lower atmosphere and in the laboratory. J. Atmos. Sci., 23, 503–506, doi:10.1175/1520-0469(1966)023<0503:TCGHFI>2.0.CO;2.
de Roode, S. R., , Siebesma A. P. , , Jonker H. J. J. , , and de Voogd Y. , 2012: Parameterization of the vertical velocity equation for shallow cumulus clouds. Mon. Wea. Rev., 140, 2424–2436, doi:10.1175/MWR-D-11-00277.1.
Ek, M. B., , Mitchell K. E. , , Lin Y. , , Rogers E. , , Grummann P. , , Koren V. , , Gayno G. , , and Tarplay J. D. , 2003: Implementation of the Noah land-use model advances in the NCEP operational mesoscale Eta Model. J. Geophys. Res., 108, 8851, doi:10.1029/2002JD003296.
Fiedler, B. H., 2000: Dissipative heating in climate models. Quart. J. Roy. Meteor. Soc., 126, 925–939, doi:10.1002/qj.49712656408.
Frech, M., , and Mahrt L. , 1995: A two-scale mixing formulation for the atmospheric boundary layer. Bound.-Layer Meteor., 73, 91–104, doi:10.1007/BF00708931.
Gandin, L. S., , and Murphy A. H. , 1992: Equitable skill scores for categorical forecasts. Mon. Wea. Rev., 120, 361–370, doi:10.1175/1520-0493(1992)120<0361:ESSFCF>2.0.CO;2.
Han, J., , and Pan H.-L. , 2006: Sensitivity of hurricane intensity forecast to convective momentum transport parameterization. Mon. Wea. Rev., 134, 664–674, doi:10.1175/MWR3090.1.
Han, J., , and Pan H.-L. , 2011: Revision of convection and vertical diffusion schemes in the NCEP Global Forecast System. Wea. Forecasting, 26, 520–533, doi:10.1175/WAF-D-10-05038.1.
Hong, S.-Y., 2010: A new stable boundary-layer mixing scheme and its impact on the simulated East Asian summer monsoon. Quart. J. Roy. Meteor. Soc., 136, 1481–1496, doi:10.1002/qj.665.
Hong, S.-Y., , and Pan H.-L. , 1996: Nonlocal boundary layer vertical diffusion in a medium-range forecast model. Mon. Wea. Rev., 124, 2322–2339, doi:10.1175/1520-0493(1996)124<2322:NBLVDI>2.0.CO;2.
Hourdin, F., and Coauthors, 2013: LMDZ5B: The atmospheric component of the IPSL climate model with revisited parameterizations for clouds and convection. Climate Dyn., 40, 2193–2222, doi:10.1007/s00382-012-1343-y.
Iacono, M. J., , Mlawer E. J. , , Clough S. A. , , and Morcrette J.-J. , 2000: Impact of an improved longwave radiation model, RRTM, on the energy budget and thermodynamic properties of the NCAR Community Climate Model, CCM3. J. Geophys. Res., 105, 14 873–14 890, doi:10.1029/2000JD900091.
Jin, Y., , Thompson W. T. , , Wang S. , , and Liou C.-S. , 2007: A numerical study of the effect of dissipative heating on tropical cyclone intensity. Wea. Forecasting, 22, 950–966, doi:10.1175/WAF1028.1.
Khairoutdinov, M. F., , and Randall D. A. , 2003: Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607–625, doi:10.1175/1520-0469(2003)060<0607:CRMOTA>2.0.CO;2.
Kim, Y.-J., , and Arakawa A. , 1995: Improvement of orographic gravity wave parameterization using a mesoscale gravity wave model. J. Atmos. Sci., 52, 1875–1902, doi:10.1175/1520-0469(1995)052<1875:IOOGWP>2.0.CO;2.
Köhler, M., , Ahlgrimm M. , , and Beljaars A. , 2011: Unified treatment of dry convective and stratocumulus-topped boundary layers in the ECMWF model. Quart. J. Roy. Meteor. Soc., 137, 43–57, doi:10.1002/qj.713.
Long, P. E., 1986: An economical and compatible scheme for parameterizing the stable surface layer in the medium range forecast model. NCEP Office Note 321, 24 pp. [Available online at http://www.lib.ncep.noaa.gov/ncepofficenotes/files/01408602.pdf.]
Long, P. E., , 1989: Derivation and suggested method of the application of simplified relations for surface fluxes in the medium-range forecast model: Unstable case. NCEP Office Note 356, 53 pp. [Available online at http://www.lib.ncep.noaa.gov/ncepofficenotes/files/0140893E.pdf.]
Mlawer, E. J., , Taubman S. J. , , Brown P. D. , , Iacono M. J. , , and Clough S. A. , 1997: RRTM, a validated correlated-k model for the longwave. J. Geophys. Res., 102, 16 663–16 682, doi:10.1029/97JD00237.
Moorthi, S., , Pan H.-L. , , and Caplan P. , 2001: Changes to the 2001 NCEP operational MRF/AVN global analysis/forecast system. NWS Tech. Procedures Bull. 484, 14 pp. [Available online at http://www.nws.noaa.gov/om/tpb/484.pdf.]
Neggers, R. A. J., , Siebesma A. P. , , and Jonker H. J. J. , 2002: A multiparcel model for shallow cumulus convection. J. Atmos. Sci., 59, 1655–1668, doi:10.1175/1520-0469(2002)059<1655:AMMFSC>2.0.CO;2.
Neggers, R. A. J., , Köhler M. , , and Beljaars A. C. M. , 2009: A dual mass flux framework for boundary layer convection. Part I: Transport. J. Atmos. Sci., 66, 1465–1487, doi:10.1175/2008JAS2635.1.
Noh, Y., , Cheon W. G. , , Hong S.-Y. , , and Raasch S. , 2003: Improvement of the profile model for the planetary boundary layer based on large eddy simulation data. Bound.-Layer Meteor., 107, 401–427, doi:10.1023/A:1022146015946.
Park, B.-K., , and Hong S.-Y. , 2013: Effects of physics packages on medium-range forecasts in a global forecasting system. J. Atmos. Sol.-Terr. Phys., 100–101, 50–58, doi:10.1016/j.jastp.2013.03.027.
Rio, C., , and Hourdin F. , 2008: A thermal plume model for the convective boundary layer: Representation of cumulus clouds. J. Atmos. Sci., 65, 407–425, doi:10.1175/2007JAS2256.1.
Siebesma, A. P., , and Teixeira J. , 2000: An advection–diffusion scheme for the convective boundary layer: Description and 1d-results. Preprints, 14th Symp. on Boundary Layers and Turbulence, Aspen, CO, Amer. Meteor. Soc., 133–136.
Siebesma, A. P., , Soares P. M. M. , , and Teixeira J. , 2007: A combined eddy-diffusivity mass-flux approach for the convective boundary layer. J. Atmos. Sci., 64, 1230–1248, doi:10.1175/JAS3888.1.
Soares, P. M. M., , Miranda P. M. A. , , Siebesma A. P. , , and Teixeira J. , 2004: An eddy-diffusivity/mass-flux parameterization for dry and shallow cumulus convection. Quart. J. Roy. Meteor. Soc., 130, 3365–3384, doi:10.1256/qj.03.223.
Sorbjan, Z., 1989: Structure of the Atmospheric Boundary Layer. Prentice Hall, 317 pp.
Sundqvist, H., , Berge E. , , and Kristjansson J. E. , 1989: Condensation and cloud studies with mesoscale numerical weather prediction model. Mon. Wea. Rev., 117, 1641–1657, doi:10.1175/1520-0493(1989)117<1641:CACPSW>2.0.CO;2.
Sušelj, K., , Hogan T. F. , , and Teixeira J. , 2014: Implementation of a stochastic eddy-diffusivity/mass-flux parameterization into the Navy Global Environmental Model. Wea. Forecasting, 29, 1374–1390, doi:10.1175/WAF-D-14-00043.1.
Troen, I. B., , and Mahrt L. , 1986: A simple model of the atmospheric boundary layer sensitivity to surface evaporation. Bound.-Layer Meteor., 37, 129–148, doi:10.1007/BF00122760.
Vickers, D., , and Mahrt L. , 2004: Evaluating formulations of stable boundary layer height. J. Appl. Meteor., 43, 1736–1749, doi:10.1175/JAM2160.1.
Witek, M. L., , Teixeira J. , , and Matheou G. , 2011: An integrated TKE-based eddy-diffusivity/mass-flux boundary layer closure for the dry convective boundary layer. J. Atmos. Sci., 68, 1526–1540, doi:10.1175/2011JAS3548.1.
Xu, K.-M., , and Randall D. A. , 1996: A semiempirical cloudiness parameterization for use in climate models. J. Atmos. Sci., 53, 3084–3102, doi:10.1175/1520-0469(1996)053<3084:ASCPFU>2.0.CO;2.
Zhang, D.-L., , and Altshuler E. , 1999: The effects of dissipative heating on hurricane intensity. Mon. Wea. Rev., 127, 3032–3038, doi:10.1175/1520-0493(1999)127<3032:TEODHO>2.0.CO;2.
Zhang, G. J., , and Wu X. , 2003: Convective momentum transport and perturbation pressure field from a cloud-resolving model simulation. J. Atmos. Sci., 60, 1120–1139, doi:10.1175/1520-0469(2003)060<1120:CMTAPP>2.0.CO;2.
Zhao, Q., , and Carr F. H. , 1997: A prognostic cloud scheme for operational NWP models. Mon. Wea. Rev., 125, 1931–1953, doi:10.1175/1520-0493(1997)125<1931:APCSFO>2.0.CO;2.
