1. Introduction
Limitations in horizontal resolution, resources, and other complexities force numerical weather prediction models to parameterize some physical and dynamical processes. Examples of parameterizations used in mesoscale models include turbulence and convective precipitation physics. Furthermore, today’s cumulus parameterization schemes are sophisticated and include physical processes like moist downdrafts (e.g., Johnson 1976; Molinari and Corsetti 1985; Grell 1993). These convective downdrafts are driven by evaporation and precipitation drag; thus, they provide a heat sink (Johnson 1976; Gregory and Miller 1989; Gregory 1995). In nonhydrostatic models with comprehensive physics, the occurrence of downdrafts helps to limit the precipitation production (Srivastava 1967; Schlesinger 1973)1 and transport downward cooler air from the midtroposphere. The parameterization of downdrafts in mesoscale numerical weather prediction models tends to neutralize the profile and this has significant impact on the grid-scale precipitation production. A frequently used method for modeling downdrafts uses the entraining inverted or downward plume (Johnson 1976). Modifications predicted by the parameterized downdraft are added to those ascribed to the updraft.
An alternative technique for including the effects of precipitation drag and diabatically related subgrid-scale mixing into a hydrostatic forecast model is introduced in this study. Our parameterization of subgrid-scale and grid-scale precipitation drag and diabatic related mixing introduces a damping nonhydrostatic vertical acceleration that accomplishes some of the same features credited to downdrafts in cumulus parameterization. Our decision to modify the vertical motion follows from the fact that in nonhydrostatic models that resolve both convective and nonconvective rainfall, it is the vertical motion that is directly impacted by precipitation drag. In the model equations, this modification is linearly dependent upon the magnitude of the atmospheric liquid water.
Srivastava (1967) has shown that sufficient precipitation drag will induce downdrafts, while Schlesinger (1973) found in his 3D nonhydrostatic model that the liquid water drag “serves much more to curb the updrafts than to initiate the downdrafts.” Clearly, convective precipitation drag plus other subgrid-scale and cloud-scale mixing processes discourage excessive vertical transport which helps moderate the total amount of condensation. However, in nonhydrostatic models, convective precipitation accounts for only part of the atmospheric liquid water. The mesoscale nonconvective grid-scale precipitation is also important. In this study it is shown that mesoscale models need to include grid-scale precipitation drag. This is especially important when highly selective horizontal smoothing, filtering, or diffusion processes are used.
Cumulus parameterization must be concerned with the moistening and warming of the upper troposphere, the redistribution of moisture that moistens the upper troposphere and dries the lower troposphere, the production of convective precipitation, and the removal of instabilities to help reduce the grid-scale precipitation production. It also keeps grid-scale vertical motions within reasonable bounds. An important component in this process results from precipitation drag. This nonhydrostatic process directly impacts the vertical acceleration. Evaporation and the presence of turbulence or convective mixing further enhances the stabilizing process by fueling downdrafts.
The parameterization of subgrid-scale effects on the grid scale requires enough rigor to capture the general consequences without requiring exactness (which is usually not mathematically or computationally feasible). Molinari and Dudek (1992) point out several areas in cumulus parameterization where mathematical/physical difficulties require simplifying assumptions. To make computation possible, downdraft calculations also are simplified. Observational measurements for verification of the downdraft, independent of the total convective process, are lacking, but their presence has been identified (Paluch 1979) while their addition to cumulus parameterization tends to improve numerical forecasts (Gregory 1995). Emanuel (1991) cautions that the validation of convective schemes by using precipitation and the vertical profile of the heating rate is difficult, since in the former (precipitation) vertical integrals only are compared, while the latter (heating rate) is largely determined by large-scale atmospheric adiabatic and radiative cooling processes. Nevertheless, Emanuel argues that convective schemes should include “physical and microphysical cloud processes as deduced from observations, numerical cloud models, and theory.”
As noted by many authors, including Zhang et al. (1994), cumulus parameterization can help eliminate the overproduction of excessive grid-scale rainfall by stabilizing the unstable column.2 Including more physics in the cumulus parameterization should improve its performance (Emanuel 1991). However, the “goodness” of a cumulus parameterization must be interpreted relative to other parameterizations used in the forecast model. For example, the boundary layer turbulence parameterization is quite important. Raymond and Stull (1990) found differences in forecasts of precipitation and cloud pattern between simulations using nonconservative K theory and those based on the conservative transilient turbulence theory. In the same way, the nature of the horizontal smoothing (or the horizontal diffusion) is very important, especially to the water vapor field that characteristically contains sharp gradients. In numerical modeling, smoothing or diffusion is used (required) to improve (maintain) the numerical stability, aid grid coupling (Mesinger 1973), and assist with the parameterization of turbulence. Unfortunately, second- and fourth-order horizontal diffusion can substantially smooth the water vapor field, causing significant reductions in the precipitation. For precipitation events, moisture losses are more important than losses in the temperature and momentum fields since diabatic contributions can be significant. Consequently, modelers may falsely think that the model precipitation physics, based on precipitation and heating rates, are realistic when in fact they have been formulated to work within a specific model’s capabilities, which includes a number of parameterizations that interact in complicated ways.
Accurate forecasting of moisture patterns requires modeling techniques that conserve moisture as much as possible. The use of a selective filter in place of a diffusion operator is one option that can help preserve more of the integrity of the moisture field, since only high-wavenumber features are extensively dampened (see the appendix). This may, however, dictate some adjustment in the convective and liquid water physics. This was required in our case when a highly selective sixth-order tangential filter (Raymond 1988) replaced the second-order horizontal diffusion in the CIMSS’s (Cooperative Institute for Meteorological Satellite Studies) Regional Assimilation System (CRAS) model (see section 3). The artificial damping of meteorologically important features was reduced by concentrating the heaviest smoothing on the unresolvable high-frequency noise, which resulted in an increase in precipitation production. Aside from this consequence on the precipitation, selective smoothing is still preferred because there is evidence that formulations of horizontal smoothing that act exclusively on the smaller scales provide a more realistic solution, with smaller validated errors (Laursen and Eliasen 1989).
Enhancing the convective physics by including downdrafts has been shown to help control excessive grid-scale precipitation (Johnson 1976; Molinari and Corsetti 1985; Grell 1993). In this study an alternative approach is introduced that parameterizes both convective and grid-scale precipitation drag. These processes have a negating or counterbalancing effect upon the (grid-scale) vertical motion. However, in a hydrostatic atmosphere the exact mathematical formulation for the precipitation drag is unclear. This distinction is also evident in the complex differences between hydrostatic and nonhydrostatic buoyancy. Consequently, the known expression for the nonhydrostatic precipitation drag is not directly appropriate in hydrostatic modeling. Even a rescaling of the nonhydrostatic drag is found to be problematic, making it undesirable. In contrast, it is shown that an empirical parameterization of the drag in the form of a Rayleigh drag, with a coefficient that is linearly dependent upon the atmospheric liquid water, can be used. Even though the Rayleigh drag formulation is not obtained from first principles, forecasts made using this drag formulation contain far less objectionable features than those produced using the rescaled nonhydrostatic precipitation drag, or from the use of artificial horizontal smoothing by low-order diffusion. Our forecasts with the Rayleigh drag parameterization show improved rainfall statistics (and most often improvements in other areas also) when compared with observations. As with other parameterizations, our formulation is based on the resolvable model variables, but the parameterization scheme requires the use of the quasi-hydrostatic approximation (Orlanski 1981), so its application mechanics differ from techniques employed in implementing downdrafts within cumulus parameterization schemes.
2. Discussion
For mesoscale flow, Tarbell et al. (1981) have shown that the hydrostatic omega equation is dominated by the diabatic term when diabatic heating is significant. This is the reason that diabatic heating rates (used to simulate latent heating in diabatic initialization and diabatic forcing) substantially increase the vertical motion (Raymond et al. 1995). This direct coupling between the vertical motion and the heating rate also provides a method to control excessive precipitation. That is, physical and dynamical processes that dampen (excessive) vertical motion will also tend to limit the diabatic heating and the resulting precipitation. The removal of excessive precipitation and vertical motion also allows, in a relative sense, evaporation and other processes to influence the dynamics at realistic levels.
Simplified models, derivable from first principles with some assumptions, like the quasi-one-dimensional plume models (e.g., Kuo and Raymond 1980) relate entrainment directly to a Rayleigh drag term involving vertical motion (e.g., Simpson and Wiggert 1969; Wallace and Hobbs 1977, 121). The influence of the moist downdraft can also be parameterized in a similar manner as presented in Kuo and Raymond (1980). It is shown below that the Rayleigh drag approach is quite well behaved and easy to use but somewhat more removed from first principles than (4). Nevertheless, in hydrostatic models it is a useful alternative to (4).
Diabatically induced mixing or cloud-scale turbulence can also be parameterized. If the mixing is assumed to be related to the heating rate, which in turn is related to vertical moisture flux (moisture convergence), as assumed in Kuo-type cumulus parameterizations, then the consequences of the turbulence on the nonhydrostatic vertical motion can be expressed in terms of a Rayleigh drag similar in form to (5) but with a coefficient that is somewhat different. In this study, a satisfactory choice for a in (6) is assumed to parameterize some of the effects of diabatic mixing on the nonhydrostatic vertical motion.
The precipitation damping is subject to the restriction that the change computed at any level using (5) will not alter the sign of the vertical motion at that level. This condition is automatically satisfied when the ratio qliq/qcrit ⩽ 1 and it is consistent with our overall goal of controlling excessive precipitation and with Schlesinger’s (1973) finding that precipitation drag diminishes the updraft rather than initiating the downdraft. Also, in the presence of heavy precipitation our approach avoids stimulating large unrealistic downward motion with scales that are much larger than Fujita’s (1985) estimates for moist downbursts. So, at the scales used in this study, it is reasonable to assume that precipitation drag and subgrid-scale diabatic mixing reduce both downdrafts and updrafts, which generates a reduction in the mesoscale circulation. For our calculations with either (4) or (5), ωnon = 0 except when qliq ≠ 0.
Orlanski (1981) developed a rigorous approach to include nonhydrostatic contributions in existing hydrostatic models. His technique resolves nonhydrostatic contributions in the quasi-hydrostatic model for horizontal grid resolutions with appropriate non-hydrostatic scales. However, in his study, the nonhydrostatic contributions remained small until the horizontal grid size approaches 10 km. The order in which Orlanski performed his calculations is, first, find the perturbation pressure (and temperature) at each level from a vertical integration of dω/dt, then compute the modification to the horizontal wind components from the perturbation pressure, then find the vertical velocity correction from the modified horizontal wind. All quasi-hydrostatic calculations are completed over just one time step, that is, at time t = τ + 1. It is important to understand that the pressure and temperature are modified by the vertical acceleration while the wind components are indirectly influenced because of these changes in the pressure and temperature fields. Consequently, the order in which the calculations are performed is very important.
The Orlanski quasi-hydrostatic procedure is a good framework within which to implement our proposed parameterization. However, in the CRAS forecast model this implementation is simplified because of the combined semi-implicit leapfrog time-stepping formulation. This allows us to modify the quasi-hydrostatic methodology. Because of the simplicity of (5), the nonhydrostatic correction to the hydrostatic vertical velocity is known at each grid point centered at time t = τ. These centered in time estimates of the nonhydrostatic vertical accelerations can be used directly in the leapfrog calculation of the new pressure and temperature variables valid at time t = τ + 1. The effect on the horizontal wind components is however indirect but occurs at the same time step because changes in the pressure field impact the τ + 1 semi-implicit calculation of the horizontal velocity. This indirect feedback through the mass field calculations avoids the generation of excessive gravity wave noise in the horizontal wind components that might otherwise occur if the horizontal winds were directly modified.
It is our contention that hydrostatic calculations can produce inflated vertical motions (especially when the numerical model uses selective filtering) because precipitation drag and diabatically induced subgrid-scale mixing are missing. If these processes are included either by cumulus parameterization containing downdrafts or as proposed here, then the production of excessive precipitation (excessive vertical motion) will be greatly reduced or eliminated. This physically based approach to control excessive precipitation is viewed as being more realistic than the application of nonselective or low-order horizontal smoothing. Controlling the excessive precipitation usually helps improve the forecast statistics. This is shown in the results presented below.
3. Model description
CRAS is an improved version of the original Bureau of Meteorology Research Centre’s operational analysis (Mills and Seaman 1990), vertical mode initialization (Bourke and McGregor 1983), and semi-implicit forecast model (Leslie et al. 1985). This assimilation system is described further in Diak et al. (1992) and Raymond et al. (1995). In this study, our calculations use horizontal grid spacing of 80 and 150 km on a 85 × 67 and 73 × 53 lattice, while the vertical sigma coordinate is subdivided into 24 and 20 levels, respectively. Climatological values are used for surface roughness, ice cover, deep soil temperature, and albedo. Surface topography is interpolated from the high-resolution navy 10′ dataset. Lateral boundary conditions use the National Centers for Environmental Prediction’s aviation initialized dataset.
Information pertinent to this study includes the fact that unresolvable features or noise, which are created because of the presence of the lateral boundaries and finite numerics, is eliminated or controlled by the application of a sixth-order implicit low-pass tangent filter (Raymond 1988). The filtering is applied at every time step, but because it is very selective it provides only a small amount of horizontal mixing to the horizontal wind components, temperature, and mixing ratio fields. Cloud water and rainwater are horizontally discontinuous quantities and are not filtered, nor is horizontal diffusion applied. In our forecasts, cloud liquid water quantities appear to be of the proper magnitude (Raymond et al. 1995) and the lack of horizontal mixing has so far not presented any numerical stability problems. Horizontal diffusion is not used when the sixth-order tangential filter is applied.
Some discussion about the cloud physics parameterization is also relevant to this study. The forecast model contains explicit conservation equations for cloud, ice, and rainwater (Diak et al. 1992; Raymond et al. 1995). The explicit cloud physics parameterization uses a modified Kessler-type scheme (Kessler 1974) while the ice physics follows the developments in Dudhia (1989), and the condensation and fall velocity calculations are similar to those described in Asai (1965), Liu and Orville (1969), and Dudhia (1989). The mixing ratios for the liquid and ice clouds and rainwater quantities are used in all calculations involving the virtual temperature, that is, water loading is included. Water loading is one feedback mechanism that helps limit excessive rainfall production, so its inclusion in all virtual temperature calculations, including the turbulence parameterization, is relevant to our discussion.
Since the publications of Diak et al. (1992) and Raymond et al. (1995), some improvements in the turbulence parameterization and explicit cloud physics have been added to the forecast model. Now, several different options are available to parameterize turbulent mixing in the vertical. Two are K-theory schemes; one depends upon a Richardson number stability criteria while the other depends upon a modified turbulent kinetic energy expression evaluated using nonlocal techniques. Another choice is a nonlocal turbulence parameterization, denoted by the acronym NTAC, which stands for the nonlocal parameterization of turbulence using convective adjustment concepts. In our tests, the NTAC scheme was as good as or better than other methods tested. Details about the latter two new schemes will be presented elsewhere.
Another new feature is that two parameterizations are now available to describe deep cumulus convection. One choice is the modified Kuo scheme (Sundqvist et al. 1989) while the other choice is Emanuel’s (1991) mixing/buoyancy approach. Comparisons between each of these schemes show reasonable agreement. However, the overproduction of mesoscale precipitation occurs with all cumulus convective schemes whenever the filtering is highly selective, for example, with application of the sixth-order horizontal filter, unless the drag parameterization is utilized.
A recent addition that improves our precipitation modeling detrains the convective cloud water and rainwater to the grid-scale cloud variables as suggested in the hybrid scheme described in Molinari and Dudek (1992). By not instantaneously converting the convective rainwater directly into surface precipitation, the opportunity for (additional) evaporation is increased in the hybrid approach since the explicit cloud calculations contain both time-dependent liquid water advection and rainwater fallout calculations.
The proposed damping parameterization, described in the previous section, is tested in detail in a number of forecasts. These include forecasts for the 72-h period starting 0000 UTC 25 April 1994 (case 1) and the 48-h period during the great blizzard of 1993 (Gilhousen 1994; Uccellini et al. 1995), beginning at 1200 UTC 12 March 1993 (case 2). In both forecasts, a 12-h preforecast is used as the guess field in the analysis. For the simulation of the blizzard of 1993, Newtonian nudging (Hoke and Anthes 1976) is used during the preforecast to steer the model calculations toward the initial 1200 UTC analysis fields. Statistics or profiles are also presented from a number of other forecasts.
4. Numerical results
a. Comparing drag formulations
Vertical profiles of the accelerations described by (4) and (5) on sigma surfaces are presented in Fig. 1b. The precipitation (prec) drag, as defined by the right-hand side of (4), does not depend upon either ω or W but its vertical profile in Fig. 1b parallels that of the liquid water shown in Fig. 1a. As expected, the vertical profile of the Rayleigh drag in Fig. 1b is similar in shape to the negative of the vertical motion (Fig. 1a). However, as seen in Fig. 1b, the acceleration produced by the nonhydrostatic precipitation drag (4) is more than 100–200 times larger in magnitude than that predicted using the Rayleigh damping scheme described in (5).
In tests using the precipitation drag as expressed in (4), it is very common for heavy precipitation to induce large unrealistic descending motion over a significant portion of the vertical column. This descending motion is so strong that numerical stability criteria are quickly violated and the model calculations become unstable. A rescaling of (4), as discussed earlier, allows numerical stability to be retained, but our forecasts (not explicitly shown) reveal that this approach lacks sensitivity which allows either too little or too much damping unless the ceiling is uniquely adjusted for every profile. Consequently, in our calculations a simple rescaling does not guarantee that the precipitation forecasts will not be excessive. Because of this undesirable characteristic, the Rayleigh drag given in (5) is much preferred and is used in the results presented below.
b. Case 1 (beginning 0000 UTC 25 April 1994)
Figure 2 shows the verifying precipitation (mm), analyzed to our grid, reported by the rain gauge network during the 24-h period ending at 1200 UTC 26 April 1994. These precipitation values are in agreement with the rainfall presented in the Daily Weather Maps, weekly series 25 April–1 May 1994 (NOAA 1994). From Fig. 2 it is clear that significant precipitation has occurred in the northern Rocky Mountains and across the northern Great Plains, and the Great Lakes, with lesser amounts in parts of Texas, Oklahoma, Florida, and New England states. Light amounts of precipitation are also found across much of Canada and the Mississippi Valley. This rainfall event is associated with a major trough situated over the eastern extent of the Rocky Mountains, with a surface cold front aligned just south and parallel to the United States–Canada border from Montana to near the United States east coast.
Figure 3a shows the final 24-h accumulated precipitation (mm) predicted in the 36-h control forecast that uses a modified Kuo cumulus parameterization. The new (precipitation) damping scheme is not employed in the calculations used to produce Fig. 3a. The shading in Fig. 3a shows a precipitation center southwest of Lake Winnipeg in the northern Great Plains. Excessive precipitation occurs in this center according to the verifying precipitation field in Fig. 2. A second center with excessive precipitation is located near Lake Superior, with smaller amounts in California and over the mountains. A little precipitation is predicted for Texas and Illinois but much of the light precipitation evident in Fig. 2 is not forecasted in our coarse mesh simulation. Two small centers of excessive rainfall are indicated at the northern tip of South America. Some preliminary testing has found these two precipitation centers to be somewhat diminished when the mixing ratio tendency is calculated with sufficiently greater accuracy and with a technique that prevents numerical (overshooting) inaccuracies caused by aliasing and spurious Gibbs phenomena (Rasch and Williamson 1990). (These numerical modeling results with be presented elsewhere.) Even then, the precipitation drag is beneficial in reducing the over-production of excessive rainfall.
An additional forecast is now presented to show that the excessive rainfall illustrated in Fig. 3a is not inordinately related to obvious shortcomings in the Kuo convective scheme. Figure 3b illustrates the 24-h precipitation forecast using Emanuel’s cumulus parameterization (Emanuel 1991).4 Emanuel’s scheme shows some improvement in predicting the areal coverage of light precipitation, especially in Canada; however, the excessive rainfall illustrated above in Fig. 3a remains.
Use of the precipitation drag parameterization in (5) partially reduces the centers of heavy precipitation without impacting the light precipitation. This is illustrated in Fig. 4, which used qcrit = 0.001 g g−1. The center of heavy rainfall concentrated near Lake Superior is displaced slightly too far north when compared with Fig. 2, and all major precipitation centers are still heavier than observed but much improved over that displayed in Fig. 3a. Clearly, no one single improvement in the model physics can cure all modeling deficiencies. Grid resolution, finite differencing errors, inaccurate initial conditions, overspecification of boundary conditions, and approximations in model physics all contribute in complicated ways to errors in the forecast. The differences between Figs. 3 and 4 demonstrate that precipitation drag is important and that it should be included in grid-scale precipitation calculations.
The results in Fig. 4 can be contrasted with those in Fig. 5, which are computed using second-order diffusion with an eddy diffusivity of 6.25 × 105 m2 s−1. This amount of eddy diffusivity is needed to maintain numerical stability during the 72-h forecast. Even with this minimum amount of diffusion, the mesoscale production of light precipitation is significantly reduced in the Midwest and in Canada. But there is some improvement in areal coverage in the New England states and in Wisconsin, even though for the latter the predicted amounts are too large. It is quite clear in Fig. 5 that the two small but strong centers in South America and the developments along the lateral boundary are not damped enough. These findings are entirely consistent with the characteristics of the low-order diffusion amplitude response functions shown in Raymond and Garder (1991).
Comparing the forecasted precipitation using 150-km horizontal resolution against rain gauge reports shows (Fig. 6) that the model is correct less than 50% of the time. A large number of light rain episodes are completely missed. The parameterized drag improved this percentage very slightly. This improved result can be contrasted with the diffusion case, which shows lower verification percentages even though the diffusion coefficient is not excessively large in magnitude. Diffusion especially hinders the model in forecasting light precipitation events. In contrast, reducing the magnitude of the low-pass sixth-order filter parameter actually worsens the results slightly. However, our experience shows us that values of the filter parameter larger than approximately ε = 1.00 introduce excessive mixing over mountainous topography in the sigma vertical coordinate system used in the CRAS. Results for this case and many others have convinced us that a value in the range 0.10–0.25 for the filter parameter epsilon (abbreviated as “ep” in the legend in Fig. 6) is satisfactory for our model. However, other models may have dissimilar optimal selections because of different physical and numerical characteristics, different requirements for the number of filter applications, or because different criteria are used to gauge or measure the forecasts.
In Fig. 7 the maximum forecasted rainfall (mm), accumulated over 24-h periods, is compared with raingauge reports. As expected, the larger the drag coefficient (a−1qliq), the smaller the maximum forecasted precipitation. Remember that because qcrit occurs in the denominator, see (4b) and (5), more damping is implied as qcrit is decreased in magnitude. If qcrit is too small, the disturbances will then be damped too much and prevented from interacting with the environment at the proper vertical levels, causing improper steering of the disturbance. The value of qcrit = 0.001 g g−1 is satisfactory for the 150-km grid resolution used in this case. Also, in Fig. 7 it is clear that diffusion is unable to effectively control the runaway behavior of the intense diabatic activity. The sixth-order tangent filter is effective the first 48 h of the forecast, but is less effective late in the forecast unless the drag parameterization is included. The placement of the most intense diabatic activity is seen in Fig. 5 in the two small but strong centers found in South America.
When both forecast and observation indicate the occurrence of surface rain, rms error statistics are compiled. The rms errors, illustrated in Fig. 8, show that the parameterized drag decreases the precipitation rms error slightly, but diffusion is also effective during the last half of the forecast. The rms statistics presented in Fig. 9 show that the 500-mb temperature field improved only minutely with the new parameterized drag (qcrit = 0.001 g g−1). Calculations with no drag and drag with qcrit = 0.003 g g−1 yield nearly identical results and are indistinguishable in Fig. 9.
c. Case 2 (beginning 1200 UTC 12 March 1993)
The 48-h forecasted mean sea level pressure is presented, as a function of qcrit, in Figs. 10a–d for the blizzard of 1993. The central pressure of 968 mb in Fig. 10a is approximately correct in the control forecast (no parameterized damping) and its location just east of New England is correct (deAngelis 1993). The pressure field remains unchanged using a drag with qcrit = 0.005 g g−1 (Fig. 10b), a satisfactory value for qcrit for this simulation that uses an 80-km horizontal resolution. When the drag is set too heavy (qcrit = 0.001 g g−1, Fig. 10c), the location of the center of the disturbance is shifted inland and to the south but the central pressure remains nearly unaffected. In contrast, the simulation with second-order diffusion shows no shift in position but the diffusion fills in or reduces the central pressure from 968 to 980 mb, as seen in Fig. 10d.
Significant differences in predicted 48-h accumulated precipitation exist between the control (Fig. 11a) and parameterized drag (qcrit = 0.005 g g−1, Fig. 11b) forecasts. When comparing these figures, note that the drag parameterization does not influence the light rainfall but does substantially reduce the heavier precipitation in the Gulf of Mexico and along the east coast of the United States. The predicted precipitation shown in Fig. 11c is too light when qcrit = 0.001 g g−1. The forecast using qcrit = 0.005 g g−1 (Fig. 11b) produced the best precipitation prediction based on the 48-h accumulated precipitation analysis by Uccellini et al. (1995, see their Fig. 2).
Twenty-four-hour accumulations from the control forecast, ending at 1200 UTC 14 March 1994, are illustrated in the scatter diagram presented in Fig. 12a, where the predicted precipitation (ordinate) is plotted against the observations (abscissa). The large amount of scatter present in Fig. 12a is significantly reduced when the drag parameterization (qcrit = 0.005 g g−1) is used (see Fig. 12b).
Statistics for accumulated precipitation are given in Table 1 for two 24-h time periods. Bias (model minus observed) and rms error statistics (Anthes 1983) are shown as a function of the type of horizontal smoothing and the magnitude of qcrit. During the first 24-h time period analyzed, the drag parameterization (qcrit = 0.005 g g−1) reduces the rms error from 46.1 to 29.1 (mm) while the bias is reduced from 23.2 to 14.19. Additional improvement occurs as qcrit is reduced in magnitude, but as already pointed out, too much damping causes the storm track to be altered. During the second 24-h time period analyzed, second-order diffusion also improves the statistics and yields the best precipitation forecast. But this reduction in excessive precipitation comes at the expense of reducing the central pressure of the storm.
Temperature errors at the 500-mb pressure level associated with the 24- and 48-h forecasts are given in Table 2. Note that the proposed precipitation damping scheme actually improves the rms scores 48 h into the forecast and that second-order diffusion always gives a larger rms error. The excessive error in the 48-h forecast associated with qcrit = 0.001 g g−1 reflects the fact that the storm track is incorrect.
d. Statistics from other cases
The parameterized drag term discussed above has been used in the “operational” CRAS since March 1994. The “operational” CRAS is composed of two independent forecast systems that have been running automatically on UNIX workstations at CIMSS since February 1992. A 72-h forecast is produced twice daily on a coarse-mesh grid of 150-km spacing that covers all of North America, the eastern Pacific to l70°W, and the western Atlantic to 50°W. This forecast uses a drag coefficient with qcrit = 0.001 g g−1. A 48-h forecast that uses qcrit = 0.005 g g−1 is produced twice daily on a fine-mesh grid of 80-km spacing. This grid covers the contiguous United States, southern Canada, and Mexico.
To illustrate how the drag parameterization affects the daily forecast runs, a sequence of six forecasts were examined from the coarse-mesh system. Parallel forecasts were run with the drag parameterization turned off for the 0000 UTC runs from 11 January to 16 January 1995. During this period a low pressure system formed along the Gulf Coast states and moved northeast into New England producing substantial precipitation. The impact of the drag parameterization on the storm intensity is shown in Fig. 13. The maximum vertical motion in microbars per second produced by the storm is shown in Fig. 13a. The values designated by triangles were generated with no drag present. The values produced with the drag term present are more reasonable for a forecast of this horizontal scale. To illustrate the improvement in the precipitation forecasts for this case, the forecasted 24-h accumulated precipitation was verified against surface rain gauge reports. Figure 13b shows the rms error for the six forecasts. A few reports above 75 mm were omitted from the verification because they represent subgrid-scale precipitation processes that cannot be accurately predicted with this forecast system. Reports below 6 mm were also omitted because they generally show little change between the two forecasts. The rms errors for the forecasts with the drag parameterization term (line containing squares) were significantly less than the forecasts without the drag parameterization (line containing triangles). The maximum 24-h accumulation for each forecast is plotted in Fig. 13c. The line containing triangles denotes the forecasts without drag, the line with squares shows the drag forecasts, and the line with dots shows the maximum observed from the surface observation network. Clearly the forecasts that include the drag term are superior.
5. Summary
This study examines the stabilizing nonhydrostatic feedback effects from precipitation drag and diabatic mixing. To reduce the tendency of excessive precipitation in hydrostatic forecast models, precipitation drag and diabatic mixing are parameterized in terms of a Rayleigh drag and used to predict the calming contribution from the nonhydrostatic vertical acceleration toward the vertical motion field. When used in the CIMSS Regional Assimilation System (CRAS) hydrostatic forecast model, this parameterization reduces excessive mesoscale precipitation. This parameterization of the nonhydrostatic vertical acceleration uses the total convective and grid-scale precipitating ice and liquid water as a coefficient and is applied in a manner similar to the quasi-hydrostatic approximation (Orlanski 1981). The parameterized drag has a direct effect upon the pressure, temperature, and moisture fields, which then indirectly impact the dynamics. Thus the wind field is influenced obliquely so that artificial noise and gravity waves are not a problem.
Horizontal smoothing can also modify the precipitation production. However, second-order diffusion is not selective enough to control excessive precipitation production without greatly impacting the meteorological fields. Higher-order schemes, like the sixth-order tangent filter regularly used in the CRAS model, perform better. But, even they cannot control excessive mesoscale precipitation adequately because the filtering is selective and must be minimized on sigma surfaces. Otherwise, in regions with mountainous topography unrealistic vertical motions will be generated. The drag parameterization proposed in this study avoids these problems. It allows excessive precipitation to be significantly reduced, yet will not impact the light precipitation. Clearly, parameterizing the nonhydrostatic feedbacks from precipitation drag and small-scale diabatic entrainment/mixing is physically correct and more appealing than using an enhancing horizontal smoother to artificially control excessive precipitation.
Acknowledgments
This work was supported in part by NOAA Grants NA17EC0427-0 and NA47EC0420. One author (WHR) also benefited from National Science Foundation Grant ATM-8920508. The authors thank Mark Whipple (CIMSS) and Arthur Garder (Sun City, Arizona) for suggestions that improved the readability of the manuscript. Kerry Emanuel is thanked for making software copies of his cumulus parameterization scheme available to the general public. The authors also thank Leo Donner of GFDL for useful discussions on this topic.
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APPENDIX
Effects of Horizontal Smoothing on the Moisture Field
Results from two numerical forecasts will illustrate how horizontal smoothing impacts a moisture forecast. Figure A1a shows the total precipitable water predicted from a 24-h forecast that uses the sixth-order tangent filter to smooth the temperature, horizontal wind components, and mixing ratio fields at each time step in the CRAS (80-km horizontal grid resolution) forecast. This precipitable water field can be contrasted with that shown in Fig. A1b, which is produced by a parallel forecast that used second-order diffusion (K = 6.25 × 105 m2 s−1) in place of the filter. The two approaches lead to large differences in the gradients of the total precipitable water in the Gulf of Mexico and Atlantic Ocean. These forecasts are verified using analyzed GOES-8 retrievals of the total precipitable water, which are available at locations displayed in Fig. A1c. A measure of the accuracy of the gradients (Anthes 1983) in the filtered forecast yields an S1 score of 40.97 while the forecast containing diffusion produces a larger or poorer S1 value of 49.28. It is quite clear that the discriminating power of the sixth-order tangent filter allows more horizontal structure and tighter gradients in the total precipitable water. These very factors allow the filter to produce improved S1 scores. The differences shown in Fig. A1 will eventually lead to significant differences in the forecasted precipitation. Using the selective sixth-order filter in our forecast model with a filter factor of 0.25 resulted in localized excessive precipitation, which necessitated the development of the Rayleigh drag procedure.5
(a) The 12-h model-forecasted vertical profiles of omega (hPa h−1), W × Ps (hPa h−1), and the liquid water (g kg−1) on the sigma coordinate for a location in mountainous Central America valid at 1200 UTC 12 July 1995. For the same location, (b) vertical profiles of the drag on the acceleration estimated from the Rayleigh drag, as defined in Eq. (5), and from the full precipitation drag term defined in Eq. (4).
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
(a) The 12-h model-forecasted vertical profiles of omega (hPa h−1), W × Ps (hPa h−1), and the liquid water (g kg−1) on the sigma coordinate for a location in mountainous Central America valid at 1200 UTC 12 July 1995. For the same location, (b) vertical profiles of the drag on the acceleration estimated from the Rayleigh drag, as defined in Eq. (5), and from the full precipitation drag term defined in Eq. (4).
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
(a) The 12-h model-forecasted vertical profiles of omega (hPa h−1), W × Ps (hPa h−1), and the liquid water (g kg−1) on the sigma coordinate for a location in mountainous Central America valid at 1200 UTC 12 July 1995. For the same location, (b) vertical profiles of the drag on the acceleration estimated from the Rayleigh drag, as defined in Eq. (5), and from the full precipitation drag term defined in Eq. (4).
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Precipitation observations (shaded) for the 24-h period ending 1200 UTC 26 April 1994.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Precipitation observations (shaded) for the 24-h period ending 1200 UTC 26 April 1994.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Precipitation observations (shaded) for the 24-h period ending 1200 UTC 26 April 1994.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
The 36-h control forecast (no drag) showing the last 24-h of accumulated precipitation predicted using (a) the Kuo scheme and (b) Emanuel’s scheme, valid for the period ending 1200 UTC 26 April 1994. The forecast’s horizontal resolution is 150 km.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
The 36-h control forecast (no drag) showing the last 24-h of accumulated precipitation predicted using (a) the Kuo scheme and (b) Emanuel’s scheme, valid for the period ending 1200 UTC 26 April 1994. The forecast’s horizontal resolution is 150 km.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
The 36-h control forecast (no drag) showing the last 24-h of accumulated precipitation predicted using (a) the Kuo scheme and (b) Emanuel’s scheme, valid for the period ending 1200 UTC 26 April 1994. The forecast’s horizontal resolution is 150 km.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
(Continued)
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
(Continued)
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
(Continued)
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 3a except the proposed drag parameterization is used with a qcrit = 0.001 g g−1.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 3a except the proposed drag parameterization is used with a qcrit = 0.001 g g−1.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 3a except the proposed drag parameterization is used with a qcrit = 0.001 g g−1.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 3a except diffusion is used in place of the sixth-order tangent filter.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 3a except diffusion is used in place of the sixth-order tangent filter.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 3a except diffusion is used in place of the sixth-order tangent filter.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Rain gauge reports are compared against model results to verify the CRAS precipitation forecast over consecutive 24-h period. Verification statistics are presented for a range of different drag (qcrit) coefficients and for the two types of horizontal smoothing. The latter includes the sixth-order tangent filter with a filter factor of ε = 0.25 or 0.01, and second-order diffusion.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Rain gauge reports are compared against model results to verify the CRAS precipitation forecast over consecutive 24-h period. Verification statistics are presented for a range of different drag (qcrit) coefficients and for the two types of horizontal smoothing. The latter includes the sixth-order tangent filter with a filter factor of ε = 0.25 or 0.01, and second-order diffusion.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Rain gauge reports are compared against model results to verify the CRAS precipitation forecast over consecutive 24-h period. Verification statistics are presented for a range of different drag (qcrit) coefficients and for the two types of horizontal smoothing. The latter includes the sixth-order tangent filter with a filter factor of ε = 0.25 or 0.01, and second-order diffusion.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 6 except showing the maximum rainfall (mm) during 24-h periods. Reported rain gauge maxima are also shown. Note that the proposed drag is able to reduce the forecasted rainfall on 150-km horizontal grid spacing to be less than the rain gauge maxima.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 6 except showing the maximum rainfall (mm) during 24-h periods. Reported rain gauge maxima are also shown. Note that the proposed drag is able to reduce the forecasted rainfall on 150-km horizontal grid spacing to be less than the rain gauge maxima.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Same as Fig. 6 except showing the maximum rainfall (mm) during 24-h periods. Reported rain gauge maxima are also shown. Note that the proposed drag is able to reduce the forecasted rainfall on 150-km horizontal grid spacing to be less than the rain gauge maxima.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the forecast precipitation rms errors (mm) computed only when both observation and model calculations experience surface accumulation. Note that the drag using qcrit = 0.001 g g−1 improves this statistic.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the forecast precipitation rms errors (mm) computed only when both observation and model calculations experience surface accumulation. Note that the drag using qcrit = 0.001 g g−1 improves this statistic.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the forecast precipitation rms errors (mm) computed only when both observation and model calculations experience surface accumulation. Note that the drag using qcrit = 0.001 g g−1 improves this statistic.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
The 500-mb temperature rms errors are shown as a function of the drag coefficient and for second-order diffusion. All the drag cases use the sixth-order tangent filter with filter parameter ε = 0.25. Note that a drag using qcrit = 0.001 g g−1 gives slight improvement over the no-drag case. Second-order diffusion produces the poorest results.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
The 500-mb temperature rms errors are shown as a function of the drag coefficient and for second-order diffusion. All the drag cases use the sixth-order tangent filter with filter parameter ε = 0.25. Note that a drag using qcrit = 0.001 g g−1 gives slight improvement over the no-drag case. Second-order diffusion produces the poorest results.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
The 500-mb temperature rms errors are shown as a function of the drag coefficient and for second-order diffusion. All the drag cases use the sixth-order tangent filter with filter parameter ε = 0.25. Note that a drag using qcrit = 0.001 g g−1 gives slight improvement over the no-drag case. Second-order diffusion produces the poorest results.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the mean sea level pressure fields from different 48-h forecasts of the “great blizzard of 1993,” with initial conditions at 1200 UTC 12 Mar 1993. The control forecast (a) predicts a central pressure of 968 mb, while (b) the optimal drag (qcrit = 0.005 g g−1) retains that central pressure. However, excessive drag (qcrit = 0.001 g g−1) as applied in (c) will alter the storm track. On the other hand, second-order diffusion, because it is not scale-selective enough, will fill in the surface low (d). Contour interval is 4 mb.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the mean sea level pressure fields from different 48-h forecasts of the “great blizzard of 1993,” with initial conditions at 1200 UTC 12 Mar 1993. The control forecast (a) predicts a central pressure of 968 mb, while (b) the optimal drag (qcrit = 0.005 g g−1) retains that central pressure. However, excessive drag (qcrit = 0.001 g g−1) as applied in (c) will alter the storm track. On the other hand, second-order diffusion, because it is not scale-selective enough, will fill in the surface low (d). Contour interval is 4 mb.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the mean sea level pressure fields from different 48-h forecasts of the “great blizzard of 1993,” with initial conditions at 1200 UTC 12 Mar 1993. The control forecast (a) predicts a central pressure of 968 mb, while (b) the optimal drag (qcrit = 0.005 g g−1) retains that central pressure. However, excessive drag (qcrit = 0.001 g g−1) as applied in (c) will alter the storm track. On the other hand, second-order diffusion, because it is not scale-selective enough, will fill in the surface low (d). Contour interval is 4 mb.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the predicted total accumulation of precipitation from 48-h forecasts. The control is given in (a), while the forecast using the proposed drag scheme with qcrit =0.005 g g−1 presented in (b) reduces the heavier precipitation while leaving the lighter amounts unaltered. Using excessive drag (qcrit = 0.001 g g−1) can reduce the precipitation production too severely as shown in (c). Contour interval is 12 mm.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the predicted total accumulation of precipitation from 48-h forecasts. The control is given in (a), while the forecast using the proposed drag scheme with qcrit =0.005 g g−1 presented in (b) reduces the heavier precipitation while leaving the lighter amounts unaltered. Using excessive drag (qcrit = 0.001 g g−1) can reduce the precipitation production too severely as shown in (c). Contour interval is 12 mm.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are the predicted total accumulation of precipitation from 48-h forecasts. The control is given in (a), while the forecast using the proposed drag scheme with qcrit =0.005 g g−1 presented in (b) reduces the heavier precipitation while leaving the lighter amounts unaltered. Using excessive drag (qcrit = 0.001 g g−1) can reduce the precipitation production too severely as shown in (c). Contour interval is 12 mm.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are scatterplots comparing the predicted precipitation (ordinate) with observations (abscissa) for the control (a) and when the proposed drag (qcrit = 0.005 g g−1) is used (b).
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are scatterplots comparing the predicted precipitation (ordinate) with observations (abscissa) for the control (a) and when the proposed drag (qcrit = 0.005 g g−1) is used (b).
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Shown are scatterplots comparing the predicted precipitation (ordinate) with observations (abscissa) for the control (a) and when the proposed drag (qcrit = 0.005 g g−1) is used (b).
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Impact of the drag parameterization for six CRAS forecasts. Forecasts without the drag term are plotted with triangles, the drag term forecasts with squares, and observations with dots. The plots show (a) the maximum vertical motion in microbars per second, (b) the rms error (mm) of the 24-h forecasted 24-h accumulated precipitation verified against surface rain gauge reports, and (c) the maximum 24-h accumulated (mm) for each forecast.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Impact of the drag parameterization for six CRAS forecasts. Forecasts without the drag term are plotted with triangles, the drag term forecasts with squares, and observations with dots. The plots show (a) the maximum vertical motion in microbars per second, (b) the rms error (mm) of the 24-h forecasted 24-h accumulated precipitation verified against surface rain gauge reports, and (c) the maximum 24-h accumulated (mm) for each forecast.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Impact of the drag parameterization for six CRAS forecasts. Forecasts without the drag term are plotted with triangles, the drag term forecasts with squares, and observations with dots. The plots show (a) the maximum vertical motion in microbars per second, (b) the rms error (mm) of the 24-h forecasted 24-h accumulated precipitation verified against surface rain gauge reports, and (c) the maximum 24-h accumulated (mm) for each forecast.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Fig. A1. The total precipitable water field from the 24-h CRAS forecast computed (a) using the sixth-order tangent filter, and (b) when the filter is replaced by second-order horizontal diffusion. Contour interval is 3 mm. (c) Location of the GOES observations at 1200 UTC 25 April 1996 is indicated.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Fig. A1. The total precipitable water field from the 24-h CRAS forecast computed (a) using the sixth-order tangent filter, and (b) when the filter is replaced by second-order horizontal diffusion. Contour interval is 3 mm. (c) Location of the GOES observations at 1200 UTC 25 April 1996 is indicated.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Fig. A1. The total precipitable water field from the 24-h CRAS forecast computed (a) using the sixth-order tangent filter, and (b) when the filter is replaced by second-order horizontal diffusion. Contour interval is 3 mm. (c) Location of the GOES observations at 1200 UTC 25 April 1996 is indicated.
Citation: Monthly Weather Review 126, 3; 10.1175/1520-0493(1998)126<0693:IPFUPP>2.0.CO;2
Twenty-four-hour precipitation statistics.
The 500-mb temperature statistics.
High horizontal resolution in nonhydrostatic models actually resolves both convective and nonconvective mesoscale precipitation simultaneously.
Instabilities are nevertheless not removed immediately in the atmosphere, as pointed out by Krishnamurti et al. (1980), who clearly showed in observations the existence of a 1–2-day delay between maximum conditional instability and rainfall. To get the correct timing, convective parameterizations must rely on several environmental controls of the convection.
In the derivation of Eq. (2), the assumption of an incompressible atmosphere implies that d(ρgw)/dt = ρgdw/dt since dρ/dt = 0.
Emanuel’s scheme does contain some representation for moist downdrafts.
Rain drag is one nonhydrostatic quantity that remains important in a nonhydrostatic model even when used at hydrostatic scales.
