As outlined by Krueger and Cederwall (1997), the rationale for conducting an SCM intercomparison study was to coordinate and unify the disparate efforts of several Single-Column modelers into a standardized approach with respect to both the datasets and methods used. Previously, members of the ARM SCM community shared their results by means of workshops conducted periodically at Lawrence Livermore National Laboratories (LLNL). However, in the interest of identifying potential individual model biases, it became evident that it would be beneficial to everyone involved with single-column modeling if identical datasets and prescription/forcing methods were used. Therefore, it was resolved that the July 1995 SGP CART IOP would serve as a suitable dataset for the study due to the quality of the data and the incidences of deep convection accompanied with precipitation, and that the three forcing methods mentioned above would provide a representative spectrum of the various modes in which an SCM may be forced. Full details on Intercomparison Study may be obtained from Cederwall et al. Additionally, the CSU SCM was run with three different "alpha parameter" settings, which controls the quasi-equilibrium parameterization. Below, we merely present a small sampling of the CSU SCM results compared to those of other participants as a means of setting the stage for the quasi-equilibrium discussion that follows. As can be seen in Figures 1 and 2, the CSU SCM relaxation forcing mode provided the best results compared to the observations while the revealed forcing mode generally showed the least amount of agreement with the observations.

---

As a means of approximating the intensity of cumulus convection in numerical computations, Arakawa and Schubert (1974; hereafter AS) proposed a cumulus parameterization featuring "quasi-equilibrium" of the cloud work function as a closure assumption. The cloud work function is defined as the vertical integral of the buoyancy of cloud air with respect to the large-scale environment, for a particular cumulus cloud subensemble. Thus, a positive value of the cloud work function indicates that potential energy of the mean state is available for conversion into convective kinetic energy. The cloud work function is hypothesized to be quasi-invariant as non-convective ("large-scale") processes which destabilize the troposphere, such as horizontal and vertical advection of heat and moisture, or radiative heating and cooling, tend to be balanced by convective processes that restore atmospheric stability. AS found this to be especially true when the forcing time-scale of the non-convective processes is considerably longer than the "adjustment time" over which cumulus convection acts. The quasi-equilibrium hypothesis has been tested and found to be valid by several people, including Lord and Arakawa (1980), Wang and Randall (1994), and Cripe (1994) using data from both the tropics and the mid-latitudes. For further discussion of quasi-equilibrium, its strengths, weaknesses, and related studies, the reader is directed to Randall et al. (1997).
 

An examination of the results by Xu (1991) using a cumulus ensemble model (CEM) with GATE data suggests a setting of alpha ~ 10⁸ m⁴ kg^-1 or larger. Indeed, statistical analyses of various fields such as temperature, as shown in our results below (Figures 3, 4 and 6, 7), indicate that a setting of alpha ~ 10⁹ m⁴ kg^-1 appears to give the most satisfactory results using mid-latitude ARM CART IOP data, with the relaxation mode giving the closest agreement to observed conditions. Additionally, in an alpha parameter sensitivity study with the full CSU General Circulation Model (GCM), Randall et al. (1997) found that there was a general decrease in the cumulus precipitation rate accompanied by an increase in large-scale precipitation as the alpha parameter was increased from 10⁷ m⁴ kg^-1 to 10⁹ m⁴ kg^-1, with a general decrease in overall precipitation. We obtained similar results with respect to cumulus and large-scale precipitation rates; only the combined precipitation results are presented here (Figures 5 and 8) in which the general decrease may be observed as the alpha parameter is increased. Note the improved agreement between model runs and observations in the alpha parameter setting of 10⁹ m⁴ kg^-1, as demonstrated in the statistical analyses.

---

Figure 3. Time-height plot of observed temperature field and results from all 3 SCM forcing modes, for alpha parameter setting of 10⁷ m⁴ kg^-1. Horizontal temperature advective tendency results shown in lower panels.

---

Figure 4. Statistical analysis (time-averaged) of results shown in Figure 3 above.

---

Figure 5. Plot of precipitable water and precipitation results (with statistical analysis in margin) from all 3 forcing modes, for alpha parameter setting of 10⁷ m⁴ kg^-1.

---

Figure 6. Time-height plot of observed temperature field and results from all 3 SCM forcing modes, for alpha parameter setting of 10⁹ m⁴ kg^-1. Horizontal temperature advective tendency results shown in lower panels.

---

Figure 7. Statistical analysis (time-averaged) of results shown in Figure 6. Note the improved agreement with observations as the alpha parameter is set to 10⁹ m⁴ kg^-1, compared to the 10⁷ m⁴ kg^-1 case (Figure 4).

---

Figure 8. Plot of precipitable water and precipitation results (with statistical analysis in margin) from all 3 forcing modes, for alpha parameter setting of 10⁹ m⁴ kg^-1. Again, note improvements over the 10⁷ m⁴ kg^-1 case (Figure 5).

---

 
 

• 1) The CSU SCM performed well relative to other participants in the SCM Intercomparison Study conducted earlier this year.

 
• 2) The SCM Intercomparison Study served to illustrate, among other things, the improved performance across all participating SCMs using relaxation forcing. Depending on the intended use of an SCM, other forcing modes may be desirable.

 
• 3) As Xu (1991) discovered with tropical data, we found that an alpha parameter setting of at least 10⁸ m⁴ kg^-1 gives the most realistic results using mid-latitude ARM data. Indeed, a setting of 10⁹ m⁴ kg^-1 gave even better results, indicating that, although treated as a constant, the alpha parameter may take different values depending on the type of atmospheric conditions and cumulus convection regimes being modeled.

 
• 4) We also note the general decrease in cumulus precipitation and increase in large-scale precipitation, and combined precipitation rate decrease, that Randall et al. (1997) found with the full GCM as the alpha parameter went from lower to higher values.

Arakawa, A., and W. H. Schubert (1974) Interaction of a cumulus cloud ensemble with the large-scale environment, Part I, J. Atmos. Sci., 31 674-701