Simple Cumulus Models in One-Dimensional Radiative Convective Equilibrium Problems

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  • 1 Department of Earth and Planetary Physics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo, Japan
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Abstract

The cumulus model presented by Lindzen et al. for calculating one-dimensional radiative convective equilibria is examined. When only the balance of moist static energy is considered, the value of the convective mass flux Mc is required to be externally specified. Dependency of equilibrium solutions on Mc shows that an upper limit of the value of Mc exists above which the temperature in the region of upward motion is lower than that in the region of downward motion; that is, the buoyancy is negative. Lindzen et al. tried to specify the value of Mc by introducing the surface heat fluxes. However, it is found that the buoyancy of their solution is negative.

In order to obtain an appropriate equilibrium solution where the buoyancy is positive, the balance of kinetic energy, especially the dissipative process, should be considered. It is found that the value of Mc, which gives a realistic value of the dissipation rate, is close to the upper limit. In order to have a solution with a more realistic temperature profile, the model assumption that Mc is independent of time and height should be released.

Calculations on the greenhouse effect show that dependency of Mc on the total optical thickness changes sign within the range of the observed dissipation rate. The water vapor content at the tropopause becomes larger as the total optical thickness increases.

Abstract

The cumulus model presented by Lindzen et al. for calculating one-dimensional radiative convective equilibria is examined. When only the balance of moist static energy is considered, the value of the convective mass flux Mc is required to be externally specified. Dependency of equilibrium solutions on Mc shows that an upper limit of the value of Mc exists above which the temperature in the region of upward motion is lower than that in the region of downward motion; that is, the buoyancy is negative. Lindzen et al. tried to specify the value of Mc by introducing the surface heat fluxes. However, it is found that the buoyancy of their solution is negative.

In order to obtain an appropriate equilibrium solution where the buoyancy is positive, the balance of kinetic energy, especially the dissipative process, should be considered. It is found that the value of Mc, which gives a realistic value of the dissipation rate, is close to the upper limit. In order to have a solution with a more realistic temperature profile, the model assumption that Mc is independent of time and height should be released.

Calculations on the greenhouse effect show that dependency of Mc on the total optical thickness changes sign within the range of the observed dissipation rate. The water vapor content at the tropopause becomes larger as the total optical thickness increases.

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