Lau et al. (1993) and Sui et al. (1994) studied tropical water and energy cycles, and their roles in the tropical convective systems, by integrating the two-dimensional cloud-resolving model to the climate equilibrium states. The model is imposed with a time-invariant, horizontally uniform, large-scale vertical velocity and a fixed SST at 28° C, in which the simulated atmosphere is conditionally unstable below the freezing level and close to neutral above the freezing level. After the adjustment, in about 20 days, the simulations reach the quasi-equilibrium states. In the convective-radiative equilibrium conditions, two-thirds and one-third of surface rainfall come from convective and stratiform clouds, respectively. The vertically integrated moisture budget shows that three-fourths of the total moisture supply is from the moisture advec-tion associated with the imposed large-scale vertical velocity, whereas one-third of the total moisture supply is from the surface evaporation flux. The total moisture supply is completely converted into surface rainfall. The heat budget displays that the cooling from radiation, and temperature advection associated with the imposed large-scale vertical velocity, are mainly balanced by the latent heat release associated with the precipitation processes.
Ramanathan and Collins (1991) conducted an observational analysis of measurements from NASA's Earth Radiation Budget Experiment (ERBE) during the 1987 El Nino and proposed a cirrus cloud thermostat effect. They proclaimed that cloud-radiative cooling by cirrus counteracts the super-greenhouse warming, and limits SST over the western Pacific warm pool to a rather uniform distribution between 29°C and 30° C. Lau et al. (1994a) further used the cloud-resolving model to assess the cirrus-cloud thermostat effect for tropical SST by analyzing the net radiation flux at the top of the atmosphere and the net heat exchanges at the ocean-atmosphere interface. The model is integrated with the SST's of 28°C and 30°C, and with and without the large-scale forcing, respectively. The net radiation flux at the top of the atmosphere comprises the net absorbed solar radiation averaged over clear sky regions, the longwave radiation emitted by the ocean surface, atmospheric greenhouse effect, and longwave and shortwave cloud forcings. The net heat flux at the ocean-atmosphere interface consists of solar and longwave radiation, sensible and latent heat fluxes. The comparison of the experiments with the same large-scale forcing but different SST's shows that the largest changes in the components contributing to the net radiation flux at the top of atmosphere are due to the emission of the surface longwave radiation, and greenhouse warming by the increase in water vapor, which to a large degree offset each other. The magnitude of the emission of the surface longwave radiation is smaller than the greenhouse warming, suggesting no apparent "super-greenhouse" effect. The changes in longwave and shortwave cloud forcings are small, and are insensitive to the changes in the SST's, because the change in the SST induces the change in low and mid-tropospheric clouds, but does not have an impact on upper tropo-spheric clouds. The change in the net heat flux at the ocean-atmosphere interface is due mainly to the change in the surface latent heat flux. The increase in the SST induces surface cooling by increasing surface evaporation. The increase in the SST produces a 13% increase in surface precipitation. The comparison of the experiments with the same SST's but different large-scale forcings (with and without the forcing) shows that the largest changes in the budget at the top of the atmosphere occur in the shortwave and longwave cloud forcings, which in large part cancel each other out. The experiment without the forcing undergoes a large reduction of the greenhouse effect by decreasing the moisture. At the ocean-atmosphere interface, the largest change appears in the surface radiative flux as a result of the largest difference of clouds between the experiments with and without large-scale forcing. More discussions on the cloud-resolving modeling assessment of the cirrus cloud thermostat hypothesis can be found in Ramanathan et al. (1994) and Lau et al. (1994b).
2.2. Cumulus ensemble responses to radiative and microphysical processes
Li et al. (1999) conducted two experiments to study cloud-radiation interaction. The cloud single scattering albedo and asymmetry factor varied with clouds and environmental thermo-dynamic conditions in one experiment, whereas they were fixed at 0.99 and 0.843, respectively, in the other experiment. A comparison of solar radiation calculations between the two experiments showed that the experiment with the varying single scattering albedo and asymmetry factor had stronger solar radiation absorption by ice clouds in the upper troposphere than did the experiment with the constant single scattering albedo and asymmetry factor. The difference in temperatures between the two experiments further showed that the temperature around 200 mb was 2°C warmer in the experiment with the variable single scattering albedo and asymmetry factor than in the experiment with constant values.
A statistical analysis of the clouds and surface rain rates revealed that stratiform (convec-tive) clouds contributed to 33 (67) % of the total rain in the experiment with the variable cloud optical properties and 40 (60) % in the experiment with the constant cloud optical properties. The fractional cover by stratiform clouds increased from 64% in the experiment with the variations to 70% in the experiment with the constants. These sensitivity tests show the cloud-radiation interaction processes for stabilizing the atmosphere, in which the change in the vertical heating gradient by solar radiation due to variations of cloud optical properties stabilizes the middle and upper troposphere and contributes to the reduction of stratiform clouds, which further stabilizes the cloud system by reducing infrared cloud top cooling and cloud base warming.
Li et al. (2005) carried out two experiments to investigate the role of precipitation-radiation interaction in thermodynamics. One experiment includes the precipitation-radiation interaction, while the other excludes it. The experiment excluding the interaction produces 1-2° C colder and 1-1.5 g kg-1 drier than the experiment including the interaction. The comparison of the heat budget between the two experiments shows that the experiment excluding the interaction exhibits a more stable upper troposphere (above 500 mb) and a more unstable lower troposphere (below 500 mb) compared to the experiment including the interaction. The more stable upper troposphere suppresses the development of ice clouds that are responsible for the cooling bias, whereas more radiative cooling accounts directly for a cooling bias in the mid- and lower troposphere in the experiment excluding the interaction. The analysis of moisture budgets shows that the suppression of rain evaporation as a result of a less stable mid- and lower troposphere induces a drying bias when the experiment excludes the precipitation-radiation processes.
Cloud-microphysical processes determine conversion between environmental moisture and cloud hydrometeors. The microphysical parametrization of cloud ice and snow proposed by Hsie et al. (1980) was originally used in the cloud-resolving model, which produced a relatively small amount of cloud ice and snow compared to the observations. Hsie et al. (1980) modified the work of Orville and Kopp (1977) based on the equation of the rate of growth of ice crystals by deposition proposed by Koenig (1971), and formulated the depositional growth of cloud ice in mass and size to become snow by the mixing ratio divided by a time scale that is needed for an ice crystal to grow from a radius of 40 ¡m to 50 ¡m. Based on the aircraft observations, Krueger et al. (1995) suggested that the time scale in the depositional growth of snow from cloud ice should be for a crystal to grow from a radius of 40 ¡m to 100 ¡m, which increases the amount of cloud ice and snow significantly, as indicated by Li et al. (1999). More cloud ice leads to more infrared cooling at the cloud top, and less heating below the cloud top. Li et al. (1999) conducted additional experiments in which the cloud-radiation interaction is excluded, and found that the exclusion of cloud-radiation interaction and the reduction of ice clouds have a similar thermal effect, whereas the two experiments have different impacts on moisture. The simulation excluding cloud-radiation interaction causes drying by enhancing condensation, whereas the simulation with reduced ice clouds by the microphysics scheme induces moistening by suppressing condensation.
Li et al. (2005) further examined the role of the depositional growth of snow from cloud ice by conducting the comparison study with two experiments: one with the snow depositional growth, and the other without. The results show that the experiment without the snow deposi-tional growth produces a much larger amount of cloud ice than the experiment with the snow depositional growth. The budget of cloud ice further reveals that the vapor deposition rate is balanced by the conversion from cloud ice to snow and the depositional growth of snow from cloud ice. When the growth of snow from cloud ice is absent, the cloud ice could be accumulated and its amount becomes anomalously large. The analysis of the heat budget indicates that the anomalous amount of cloud ice reflects a large amount of solar radiation, and the upper tropospheric atmosphere becomes anomalously cold whereas it traps a large amount of infrared radiation and the lower tropospheric atmosphere becomes anomalously warm. Thus, the deposi-tional growth of snow from cloud ice is an important sink for cloud ice.
Tropical convection (surface rainfall) occurs as a result of instability (convective available potential energy, CAPE) in the environment. Since the environmental time scales (a few days and longer) are much longer than the con-vective time scales (a few hours or less), the "quasi-equilibrium" between the rate of production of available potential energy by the large-scale processes and the rate of consumption of the available potential energy by the convection is the basic assumption which Arakawa and Schubert (1974) used to develop their cumulus parametrization scheme. A decrease in the CAPE often coincides with the development of convection so that the CAPE and rain rate are negatively correlated (e.g. Thompson et al., 1979; Cheng and Yanai, 1989; Wang and Randall, 1994; Xu and Randall, 1998). The phase relation between the CAPE and rainfall is due to the coupling between the environmental dynamic and thermodynamic fields (Cheng and Yanai, 1989). The phases of the CAPE and rainfall could be different, because it takes time for clouds to develop. This phase difference can be included with relaxation of the quasi-equilibrium assumption in cumulus parametrization (e.g. Betts and Miller, 1986; Randall and Pan, 1993). The minimum CAPE typically occurs a few hours after the maximum rainfall, as indicated by the observational analysis. Xu and Randall (1998) interpreted the maximum phase lag as the adjustment time scale from disequilibrium to equilibrium states in the presence of time-varying large-scale forcing. Since the CAPE is calculated in a Lagrangian framework and the relevant equations cannot be derived, the physical processes responsible for the phase difference between the CAPE and the surface rain rate cannot be examined. However, an alternative for studying the phase difference has been developed in an Eulerian framework by Li et al. (2002b), in which potential and kinetic energy in an Eulerian framework represent the CAPE and surface rain rate in a Lagrangian framework, respectively.
Lorenz (1955) introduced the concept of available potential energy for a dry atmosphere that represents the portion of the potential energy that can be transferred into kinetic energy. He defined the available potential energy for a dry atmosphere as the difference between the actual total enthalpy and the minimum total enthalpy that could be achieved by rearranging the mass under adiabatic flow. The dry enthalpy per unit mass is defined as the product of the temperature and the specific heat at constant pressure. In the absence of energy sources and sinks, the total kinetic energy and total enthalpy are conserved during adiabatic expansion. In a moist atmosphere, latent heat energy should be included in the energy conservation. The latent heat energy per unit mass is defined as the product of the specific humidity and the latent heat of vaporization at 0°C. In the absence of energy sources and sinks, the total kinetic energy, enthalpy and latent heat energy are conserved during dry and subsequent saturated adiabatic expansion. Therefore, the moist available potential energy is defined as the difference between the actual moist potential energy (sum of the enthalpy and latent heat energy) and the minimum moist potential energy that could be achieved by rearranging the mass under moist adiabatic processes. Li et al. (2002b) derived a set of equations for conversions between the moist available potential energy and kinetic energy in an Eulerian framework. Their equations were demonstrated to be the same as those derived by Lorenz (1955) in the absence of moisture.
Lag correlation analysis by Li et al. (2002b) showed that the maximum perturbation kinetic energy associated with the simulated convective systems and its maximum growth rate lags and leads the maximum imposed large-scale upward motion by about 1-2 hours respectively, indicating that the convection is phase-locked with the imposed large-scale forcing. Their imposed large-scale vertical velocity had time scales longer than the diurnal cycle, whereas the simulated convective systems had an average lifetime of about 9 hours. The imposed large-scale upward motion decreases the horizontal-mean moist available potential energy by the associated vertical advective cooling, providing a favorable environment for the development of convection.
They further showed that the maximum latent heating and vertical heat transport by perturbation circulations cause maximum growth of perturbation kinetic energy to lead maximum loss of perturbation available potential energy by about 3 hours. The maximum vertical advec-tive cooling, the horizontal-mean cloud-related heating, and perturbation radiative processes cause maximum loss of perturbation moist available potential energy to lead maximum loss of the horizontal-mean moist available potential energy by about 1 hour. Consequently, the maximum gain of perturbation kinetic energy leads the maximum loss of horizontal-mean moist available potential energy by about 4-5 hours (about half of the lifetime of the simulated convection).
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