a. The Kuo-type cumulus parameterization scheme (Kuo 1974; Anthes 1977)

Here the basic condition for deep convection to occur is a conditionally unstable layer and the presence of large-scale moisture convergence. The former condition makes it possible for huge cumuli to penetrate into the upper troposphere, and the latter condition provides a lifting mechanism to trigger the convection. When the above two conditions are met, cumulus clouds are assumed to form, and the cumulus-scale heating and moistening are computed. In the NCMRWF model, several other checks are also applied to impose realistic limits on the kinds of columns that are capable of supporting deep convection. They are, for example, (i) the temperature at the second-lowest layer T(2) > 5°C, (ii) T(2) > T(3), (iii) the total moisture convergence in the layer (σ = 1) to (σ = 0.688) is sufficient to produce a rainfall rate of at least 2 mm day⁻¹, and (iv) the depth of the buoyant layer must exceed a thickness of 0.3 times the surface pressure. The cloud base is considered at the lowest unstable layer for which the relative humidity exceeds a critical value of 80%. The temperatures and moisture inside the cloud are calculated by a moist adiabat starting from the cloud base. The moistening parameter is computed based on the vertical distribution of the relative humidity using the Anthes (1977) method.

The cumulus clouds that form in this way are assumed to dissolve immediately by mixing with the environment and to impart heat and moisture into it. The heating and moistening are proportional to the difference in temperature and mixing ratio between cloud and environment.

b. The simplified Arakawa–Schubert scheme (Grell 1993; Pan and Wu 1995)

In the full A–S scheme, a spectrum of clouds of different sizes is considered. The clouds are supposed to be in different stages of their life cycle. Entrainment is considered to take place at all levels below the cloud top; detrainment is assumed to occur only from the top of the clouds. Modification of the environment takes place by warming and drying due to subsidence between clouds and by cooling and moistening due to evaporation of detrained liquid water from the cloud top. The closure of the model is based on a quasi-equilibrium hypothesis that states that the production of moist convective instability by large-scale processes and their destruction by clouds are in a state of balance over the timescale of the large-scale synoptic systems. It is this quasi-equilibrium hypothesis that makes the A–S scheme computationally very expensive.

In the simplified Arakawa–Schubert scheme, the cloud spectrum is reduced to one simple cloud (an updraft and downdraft couplet). The cloud size is prescribed to be the largest in the spectrum. Thus the mass flux distribution equation has an exact solution. This makes the scheme very efficient. All terms dependent on entrainment and detrainment are set to 0. The normalized mass flux is constant with height and is equal to 1. Thus, the cloud work functions for the updraft and downdraft simply become the available buoyant energy for up- and downdrafts, respectively.

In the current scheme, the parcels start below 700 hPa. The cloud base is considered to be the level of free convection (LFC). From the LFC, the parcel is assumed to be nonentraining up to cloud top. Ice phase has been neglected. All convective mass flux detrains at cloud top. A downdraft is included, but its starting level is set near 400 hPa just above the level of minimum moist static energy (LMSE), unlike in Grell (1993), where it is set exactly at the LMSE.

c. The relaxed Arakawa–Schubert scheme (Moorthi and Suarez 1992)

RAS modifies the entrainment relation and assumes that the normalized mass flux is a linear function of height (rather than being exponential as in the original A–S). This avoids the costly calculation that is necessary to find the entrainment parameter of clouds. One of the other simplifications made in RAS is that, rather than requiring that “quasi equilibrium” of the cloud ensemble be achieved each time the scheme is invoked, it only relaxes the state toward equilibrium. RAS considers one cloud type at a time. It computes the mass flux required for maintaining the invariance of the work function as if no other clouds were present. A fraction of this mass flux is used to change the environment and goes on to do the same for another cloud type. Thus, each step is in a process of single cloud equilibrium but, over time, all cloud types affect one another by modifying the environment. The standard A–S assumes that a quasi-static balance occurs between the entire cloud spectrum and the large-scale forcing. It is this assumption that results in an ill-posed problem.

In the following sections, we describe the medium-range forecast skill scores of KUO, RAS, and SAS during an active phase of the Indian summer monsoon.

a. Spatial distribution

Figures 1 and 2 illustrate observed total rainfall (cm) for August of 1999 over the Indian region and its corresponding 72- and 120-h forecasts by the KUO, RAS, and SAS schemes summed for August. The shaded area indicates rainfall of 1–10 cm. Contours are drawn at 10, 25, 50, 75, and 100 cm.

The observed precipitation rates are obtained by combining precipitation estimates from Indian National Satellite IR data and the rain gauge observations (Mitra et al. 1997). Observed precipitation rates show the two typical maxima of precipitation over the Western Ghat and the Bay of Bengal off the Arakan coast. Intense rainfall is also observed over the Himalayan region. The semiarid regions of northwest India and the rain-shadow areas of the eastern coast of southern peninsular India have much less precipitation. The rainfall amount simulated by KUO is reasonable as compared with observations. However, it underestimates the pockets of observed heavy rainfall over the Western Ghat, the Indo-Gangetic plain, and the Himalayan region. It also fails to predict the rain-shadow effect observed over southern peninsular India east of the Western Ghat both in 72- and 120-h forecasts. Underestimation of rainfall is a typical feature of the Kuo-type schemes (Krishnamurti et al. 1983; Das et al. 1988). RAS and SAS produce fairly good forecasts of the typical observed distributions of the rainfall over the Indian region. The rain-shadow effect is predicted very well by RAS as compared with the other two schemes. However, it increases the rainfall over the Bay of Bengal as compared with observations. Note that the rainfall values derived from geostationary satellites are underestimated using the Geostationary Operational Environmental Satellite Precipitation Index technique. Rainfall is generally overestimated in the 120-h forecast over the Bay of Bengal off the Arakan coast by all three schemes. Results also indicate that the Bay of Bengal rainfall maximum has been shifted much to the south and is close to the east coast of India, particularly in RAS. Higher rainfall obtained from RAS and SAS over the Bay of Bengal may be attributed to a stronger low-level jet, an upper-tropospheric easterly jet, and the associated divergent flow pattern at 200 hPa produced by these schemes (Das et al. 2001) over the region. In general, KUO is able to simulate the rainfall fairly well when the intensity of convection is weaker, but it underestimates the rainfall when the convection is intense. Correct simulation of rainfall at the right locations also depends upon the resolution of the model, with proper description of the moisture and divergence fields (Krishnamurti 1995). Some of the current deficiencies in rainfall forecasts at T80 resolution may be overcome by a higher-resolution model. The subjective comparison of the rainfall forecasts made here has not given us enough ground to say which scheme performs best. In the following section, we shall discuss the skill scores of the rainfall forecasts.

b. Skill scores

There are several ways to examine the skill scores of precipitation forecasts, such as bias, threat score, Brier score, Heidke skill score, Hanssen–Kuipers skill statistics, and true skill-score statistics (Hughes 1979; Mason 1982; Junker et al. 1989; Doswell et al. 1990; McBride and Ebert 2000). Wilks (1995) describes different methods of forecast verification. To examine the skills of rainfall forecasts by the three schemes, we have computed the bias and equitable threat scores for six different categories of rainfall: 0.1–1, 1–2, 2–4, 4–6, 6–8, and 8–10 cm. The bias and equitable threat scores (ETS) have been computed using the following formula (Mesinger and Black 1992):

View Expanded

where F is the number of forecast points within a threshold; O is the number of observed points within a threshold; H is the number of correctly forecast points above a threshold; CH is the expected number of randomly correct forecasts within a threshold, approximated by (FO)/N; and N is the number of points within a threshold. If the bias is less (greater) than 1, it indicates that the model is under- (over-) forecasting. If the ETS = 1, it indicates that there is no error in the forecasting. ETS = 0 indicates that none of the grid points are correctly predicted.

Figure 3 shows the rainfall bias in the day-3 and day-4 forecasts for different categories of precipitation obtained from the three schemes. The day-5 bias has not been shown, because the rain gauge stations in India report the 24-h accumulated rainfall at 0300 UTC, whereas the 120-h forecast from the NCMRWF model ends at 0000 UTC. Results indicate that KUO underestimates the rainfall categories of 1–6 cm but overestimates the light (0.1–1 cm) and heavy (6–8 cm) rainfall categories, particularly in the day-4 forecasts. Note that the KUO scheme does produce heavy rainfall some times, but it may not be at the right locations. The best forecast is produced by SAS in most of the rainfall categories for day 3 and day 4. The bias score, however, does not reflect whether a particular scheme has been able to produce the right amount of rainfall at the right locations. An answer to this point can be obtained from the equitable threat scores, which are shown in Fig. 4. The diagrams indicate that, although none of the schemes are able to produce a perfect forecast (ETS = 1), SAS generally produces the best forecasts in most of the rainfall categories. It also indicates that, although the scores are low in the heavy rainfall categories for all schemes, SAS is still better than the others.

a. Global domain

Figures 5 and 6 present the rmse of the wind (W850 and W200) and temperature (T850 and T200), representing the 850- and 200-hPa levels for the globe and the Indian region as obtained from the KUO, RAS, and SAS schemes. Results indicate that, over the global domain, there is no significant difference among the rmse of the three schemes for either temperature or the wind fields. However, the rmse of the temperature obtained from RAS is relatively lower at the 850-hPa level on all days. At the upper level, however, SAS is relatively better, particularly after 96 h. Results of the seasonal simulations of the global distribution of wind fields obtained from the three schemes by Das et al. (2001) have shown that the middle-latitude upper-level westerly jet streams of the two hemispheres are simulated fairly well by KUO, RAS, and SAS. A close comparison among the three schemes indicated that the strength of the jet stream in the Southern Hemisphere is better simulated by SAS than by the other two schemes. In the Northern Hemisphere, all three convection schemes indicate slight underestimation of the westerlies as compared with analysis (ANA). The tropical easterlies are slightly underestimated by RAS, and they are overestimated by KUO and SAS as compared with ANA. Results of Das et al. (2001) showed that the strength of the upper-level northerlies and lower-level southerly winds simulated by RAS was close to ANA. The strength of the tropical upper-level meridional circulation was overestimated by SAS and underestimated by KUO, and RAS was comparable to ANA. The anomaly correlation coefficients of the wind and temperature fields are not shown here for brevity, but they showed trends similar to those of the rmse for all of the 5-day forecasts.

b. Indian window

The previous discussion indicated that there were no significant differences among the rmse of the wind and temperature obtained by the three schemes over the global domain. However, results indicate that there are considerable differences in the rmse obtained from the three schemes over the Indian domain (Fig. 6). For example, results indicate that both RAS and SAS produce higher rmse of wind over the Indian region. Note here that the analysis was produced from the global data assimilation system, which uses the KUO scheme at present. Another reason for the higher rmse may be the overestimation of the strength of the low-level westerly jet (LLWJ), the upper-level tropical easterly Jet (TEJ), and the monsoon trough over the Indian region as shown by Das et al. (2001). The LLWJ is related to the intense pressure gradient produced by synoptic-scale forcing, which in turn is associated with the development and evolution of deep cumulus convection over the region. The strong LLWJ produced by RAS and SAS was due to enhanced convection produced by these two schemes. The TEJ is a part of the southern periphery of an upper-tropospheric anticyclone. Intense deep cumulus convection results in strong upper-level divergence, thereby strengthening the intensity of the TEJ.

Results indicate that, although the rmse of wind is higher in RAS and SAS over the Indian region, errors in temperature forecasts have been considerably reduced by the two schemes, particularly at 200 hPa on all the days. The reduction of rmse by RAS and SAS is due to proper redistribution of heat by deep cumulus clouds over the region. Convection can redistribute the heating through mixing in the vertical and can warm the atmospheric column by release of latent heat.

a. Wind fields

Figures 7 and 8 illustrate the systematic errors in the wind field at 850 and 200 hPa over the Indian region. Results indicate that KUO produces an easterly bias over the southwest Arabian Sea at 850 hPa and thus tries to weaken the normal southwesterly flow and the low-level jet during the active phase of the monsoon. Both RAS and SAS try to accelerate the low-level jet over the Arabian Sea, which is similar to the results of Das et al. (2001). The two schemes also try to intensify the easterlies over the Indo-Gangetic plain and thus strengthen the monsoon trough. Strengthening of the southwesterlies over the Arabian Sea and easterlies over the Indo-Gangetic plains indicates that both RAS and SAS enhance the intensity of convection.

Upper-level flow at 200 hPa (Fig. 8) indicates that KUO produces a weak westerly bias over southern peninsular India. The westerly bias becomes stronger in RAS and SAS. Strong easterlies are seen south of Sri Lanka over the Indian Ocean in SAS and RAS, which indicate that these two schemes shift the core of the TEJ stream to south of its normal position. These results are consistent with those found earlier by Das et al. (2001). In general, the upper-level flow is simulated better by KUO than it is by the other two schemes over the Indian region, which is also reflected in the rmse values shown in Fig. 6. The strong TEJ complements well the strong low-level monsoon flow, which is a result of enhanced deep convection produced by SAS. Intense deep cumulus convection results in strong upper-level divergence and velocity potential by RAS and SAS as seen in Das et al. (2001). The latent heat release in cumulus convection plays a major role in the location of the centers of divergent circulation, which in turn is related to the location of the warmest SSTs. Because the monsoon circulation is very sensitive to the location and intensity of the heating rates, use of different cumulus parameterization schemes can cause major differences in the simulated monsoon circulation.

b. Temperature fields

Figures 9 and 10 show systematic errors in the temperature field at 850 and 200 hPa over the Indian region. The shaded regions indicate the cold bias. Results show that KUO produces a weak warm bias of less than 1 K over some parts of northwestern India in the lower level. It produces cooling of about 1–2 K over the rest of India and the adjoining sea. The simulation by RAS indicates a nearly similar pattern, except that it has reduced the warm bias over northwestern India. The simulated temperature by SAS shows a cold bias over parts of central and south India, and an increased warm bias over northwestern India.

At 200 hPa (Fig. 10), KUO produced cooling of about 1 K almost everywhere over the Indian region. The cooling is slightly increased, particularly over northeast and south India, by RAS. This results in a higher rmse over the Indian window as seen earlier in Fig. 6. The cold bias is reduced by SAS. The improved simulation of temperature by this scheme results in the reduction of rmse as seen earlier in Fig. 6. The reduction of the cold bias by SAS indicates a proper redistribution of heat by deep convective clouds over the region by this scheme. Convection can redistribute the heating through mixing in the vertical and warm the atmospheric column by latent heat release.

a. Prediction of genesis

Prediction of the genesis of lows and monsoon depressions is a challenging task for numerical modelers. Large-scale numerical models usually are not able to predict the precise location and time of genesis of such systems. The prediction of such systems is normally done by high-resolution mesoscale or cyclone models. The large-scale models, however, are able to predict the genesis or at least a signature of the cyclogenesis despite their limitations. Because convection plays an important role in the formation of such systems, it is interesting to investigate the impact of different parameterizations of deep moist convection in the prediction of cyclogenesis.

As shown in Table 2, the two most important systems in August of 1999 were the movement of a deep depression over the Indian region during 27 July–1 August and formation of another low on 2 August over the Bay of Bengal, which subsequently also intensified into a depression and moved across the land. The second system formed over the bay on 2 August when the first system dissipated over Rajasthan in northwest India on 1 August. The last system that formed during this month was a low, which formed over the Bay of Bengal on 24 August and moved across the land during the next 11 days.

We have used KUO, RAS, and SAS to study the genesis of all the systems that formed during the month of August. Figure 11 shows the genesis of the low over the head of the Bay of Bengal on 2 August in the 850-hPa analysis and prediction of the genesis 72 h in advance with the three schemes. Results indicate that, although all the schemes produced a signature of the genesis, the best forecast of the location of the genesis was produced by RAS. Both KUO and SAS forecast the low to be much farther inland on that day. Diagrams of other cases are not shown here fore brevity, but results indicated that in the majority of the cases RAS was able to predict the genesis better than the other two schemes did.

b. Prediction of tracks

Prediction of the track of lows and the monsoon depressions using a large-scale global model is another challenging task. In most of the cases, tracks depend upon the initial position in the analyzed input of the model. Moreover, the maintenance of the system and the consistency of the predicted positions from different initial conditions in an operational environment are important. In this study, we have chosen the case of the only deep depression that occurred during August of 1999.

Figure 12 illustrates the tracks of the deep depression predicted in the 24- and 72-h forecasts based on several initial conditions from 24 to 31 July using the KUO, RAS, and SAS schemes. The observed track obtained from the NCMRWF analysis is also shown in the figure. As mentioned earlier, the depression had originated over the Bay of Bengal on 27 July and subsequently moved across the land and dissipated over Rajasthan on 1 August. Results indicate that the direction of movement and the positions of the depression in the 24-h forecasts are close to the observed track by all three schemes. However, the predicted positions are almost always north of the observed track in all three schemes. The root-mean-square errors between the observed and the predicted positions of the tracks were about 2° latitude–longitude in the 24-h forecasts. The SAS scheme produced the smallest errors. Figure 12b shows the predicted positions of the depression in the 72-h forecasts obtained from different initial conditions using the three schemes. Results indicate that, in most of the cases, KUO and RAS weakened the depression and showed it either as a feeble low or just as a trough. On the other hand, SAS was able to maintain the system on most of the days based on 72-h forecasts. However, none of the schemes was able to maintain the system properly in the 72-h forecasts after 30 July when it reached the central part of India. RAS and KUO generated the system abruptly in the 72-h forecast valid on 1 August over Rajasthan just before its dissipation. The results indicate that, although the genesis of the systems predicted is marginally better by RAS, the maintenance of the systems and the track of their movement are predicted better by SAS. However, more case studies may be required to arrive at a final conclusion.