As a point of reference, we first impose the following forcings to our CRM at 6-hourly resolution, as in many previous studies: (1) large-scale vertical motion to advect domain-averaged potential temperature and moisture, (2) horizontal advective tendencies of moisture and temperature, (3) time-varying uniform sea surface temperature (SST) as the lower boundary condition, and (4) relaxation of the horizontal domain mean horizontal winds to the observed IFA mean profiles with a relaxation time scale of 1 h.
 Of these forcings, large-scale vertical velocity is by far the most important for controlling surface rainfall. Horizontal advection terms are included in the above reference simulation, but not in the simulations with parameterized large-scale dynamics.
 Horizontal advection can be treated in a number of ways: the horizontal advection term—dot product of a large-scale horizontal velocity with a large-scale horizontal moisture gradient—can be imposed directly (as is typically done in simulations with imposed large-scale vertical motion or vertical advection); one can parameterize the horizontal advection term by applying a specified relaxation of the horizontally averaged moisture profile in the CRM toward a reference profile of moisture, representing advection by a specified rotational large-scale velocity on a specified length scale [e.g., Sobel and Bellon, 2009; Wang and Sobel, 2012]; or one can parameterize the advection as “lateral entrainment,” as defined in Raymond and Zeng , representing the drawing of the reference profile air into the CRM domain by a divergent horizontal velocity diagnosed from the vertical WTG mass flux.
 Contrasting results with and without horizontal advection terms in our reference simulations suggests that their impact is small for surface rainfall for the reference simulation (not shown), and their proper inclusion in simulations with parameterized large-scale dynamics is a subtle matter. Including them as fixed forcings neglects their actual dependence on the local state and can allow bad behavior if the model is biased (for example, if the advective forcing on humidity is negative and the humidity becomes zero, there is nothing to prevent it from becoming negative). We have represented horizontal moisture advection as a relaxation to an upstream value in idealized studies [e.g., Sobel and Bellon, 2009; Wang and Sobel, 2012], but determining the appropriate upstream value and relaxation time is more complex in the present observation-based case studies. The lateral entrainment approach neglects advection by the rotational flow, which is often larger than the divergent component. Because of these limitations, we defer inclusion of horizontal advection to future work, while acknowledging its potential importance for tropical intraseasonal variability.
 We have used both the WTG and damped gravity wave methods to parameterize large-scale dynamics. In both methods, large-scale vertical motion is dynamically derived as part of the model solution and used for advecting domain-averaged temperature and moisture in the vertical. In the WTG method, large-scale vertical velocity W in the free troposphere is derived as follows:
where θ is potential temperature horizontally averaged over the CRM domain, and θB is the target potential temperature. Within the boundary layer, W is linearly interpolated between its value at the top of boundary layer obtained from equation (1) and its surface value W = 0. Here, we simply take the boundary layer height to be 1.5 km and apply equation (1) from 1.5 km to 17 km (~100 hPa). θB is the observed value to which the potential temperature is relaxed at a time scale of τ = 4 h. Following Raymond and Zeng , we also place a lower bound on the value of , replacing the observed value by 1 K/km if it becomes smaller than that bound.
 In the damped gravity wave method [Kuang, 2008, 2011, Blossey et al., 2009, Romps, 2012a, 2012b], the large-scale vertical velocity is obtained using an equation that relates it to virtual temperature anomalies [see derivations in Blossey et al., 2009 or Kuang, 2011]:
where p is the pressure, ω is the pressure velocity, ε is the inverse of the time scale of momentum damping, k is the wave number, Rd is the dry gas constant, Tv is the domain-averaged virtual temperature, and is the target virtual temperature against which linearized wave perturbations are defined. In idealized simulations, is taken constant in time, while here it is set to the observed time-varying virtual temperature profile. For the experiment below, ε = 1 day−1 and k = 10−6 m−1. The elliptic equation (2) is solved with boundary conditions ω = 0 at the surface and 100 hPa. Both equations ((1)) and ((2)) are solved at every time step of the model integrations.
 In one set of integrations, the target potential temperature, θB , and virtual temperature, , are taken from the observations directly. However, because of model biases, observational errors, or other factors, the observed time-mean state may differ from the model's own equilibrium, and this may generate biases when using equation (1) or ((2)) to derive W. An alternative method is to derive only the time-varying perturbations in the target potential temperature or virtual temperature from the observations and impose those perturbations on top of time-mean profiles taken from a “no large-scale circulation” integration, which is nearly identical to the reference simulations except that the large-scale vertical motion is not imposed. In other words, we may add to θB a “correction” term, θC, which equal to the difference between the time mean of the model's profile in the no large-scale circulation integration and the time mean of the observed potential temperature.
 While the time-mean large-scale vertical velocity did not vanish over the IFA during TOGA-COARE, we hypothesize that using the no large-scale circulation profile as a reference may reduce any bias which is due to differences between the model's natural convectively adjusted state and that observed. This hypothesis appears to be partly correct; the correction improves WTG simulations, as shown below, though it does not improve the Damped-wave simulation.
 The model is initialized with the sounding on 00Z 1 November 1992. We discuss multiple experiments using the above-mentioned two methods. The first one, with imposed large-scale vertical velocity, will be referred to as “Imposed-W.” The experiments using the WTG and damped gravity wave methods will be referred to as “WTG” and “Damped-wave,” respectively. Uniformly distributed random noise of magnitude 1 K is added to the initial potential temperature field. In the WTG simulation without the correction term, θC, as described above, we find that the model atmosphere settles into a persistently dry, nonprecipitating state. This behavior is presumably directly related to the existence of multiple equilibria under steady forcings [Sobel et al., 2007; Sessions et al., 2010]. We prevent the occurrence of this dry solution by setting the initial relative humidity to 85% over the whole troposphere, and so use this ad hoc step in the WTG simulations which do not use the correction of the time-mean temperature profile to the “zero large-scale circulation” profile.
 We do not compute radiative fluxes interactively in either the WTG or the Damped-wave experiments. Instead, we impose the time-dependent areal-mean radiative heating obtained from the Imposed-W experiment. We do this to avoid complications resulting from cloud-radiative feedbacks. These feedbacks are much more important with parameterized dynamics than in the standard approach and can cause large errors. We leave the detailed investigation of the role of radiation to future work and specify the radiative heating in the parameterized dynamics experiments in order to better control them. In all experiments, we specify the SST and relax the horizontal mean profile of horizontal wind toward that observed.
 In both sets of simulations, observations influence the model through four pathways: the zonal and meridional winds, the model-derived radiation from the Imposed-W experiment (in which the convection is closely constrained by the imposed vertical motion, and the clouds and water vapor strongly influence the radiation), the observed SST, and the free-tropospheric temperature. The first three forcings are directly related to moist static energy sources; surface winds and SST control the surface turbulent fluxes, and radiation is a direct forcing on moist static energy. The free-tropospheric temperature, on the other hand, is a state variable and can influence the moist static energy budget only indirectly through its coupling with large-scale vertical motion and the other interactive processes.
 To further clarify the relative importance of the four forcing factors to the model-simulated surface rainfall, sensitivity experiments are performed in which we replace one time-dependent forcing at a time with its time-mean value. These experiments are named Fix-winds, Fix-radiation, Fix-SST, and Fix-temperature, respectively. Fix-wind and Fix-SST both reduce variability in surface turbulent fluxes; to further examine the role of these fluxes, another experiment (named Fix-SST-winds) is also performed in which both are replaced by time-mean values. All these experiments are done using both the Damped-wave and WTG methods.