Numerical experiments are made to determine a preferred mode of cumulus convection in a conditionally unstable atmosphere.The model developed in the previous study (Asai and Nakasuji, 1977) is extended to deal with water vapor explicitly. The preferred mode of cumulus convection is regarded as the steady convection cell attained eventually after a random potential temperature disturbance is imposed initially. It is shown that the preferred scale of the convection cell and the preferred cloud coverage depend on mean vertical velocity, static stability and relative humidity. It is confirmed that the preferred cumulus convection minimizes the potential energy and consequently the mean temperature lapse-rate in the convective layer. IntroductionThe cumulus convection in a conditionally unstable atmosphere was studied by Asai and Nakasuji (1977), which hereafter will be referred to as paper AN, to see how the preferred mode depends on a mean vertical motion and a static stability in the atmospheric layer and the magnitude of each term of the energy equations depends on the cell size of cumulus convection. The results obtained are that the preferred cell size decreases and the preferred area ratio of the ascending region to the descending one increases as the mean vertical velocity increases. Without a mean upward motion the preferred cell size increases and the preferred area ratio decreases as the static instability decreases. While with a mean upward motion the preferred cell size decreases and the preferred area ratio increases as the static instability decreases smaller than a certain value. The preferred mode of a cumulus convection cell is the one for which the potential energy of the convective layer is at the lowest so that the mean temperature lapse-rate beccmes minimum.It was assumed in the previous paper that the ascending motion was always saturated with water vapor while the descending motion was always unsaturated.This assumption may be allowed only when the convective layer is supplied with water vapor sufficiently.In the present paper water vapor is introduced into the model in an explicit form to examine the results obtained in the previous paper AN.

An investigation is made of the stability of the Ekman boundary layer flow with a thermally unstable stratification based on perturbation analysis. Three different types of the instability associated with the flow may be found, preference of which depends on Reynolds number, and Richardson number. The thermal instability, which is of gravitational origin modified by the shear flow, is dominant for Ri*10-3, while the other two, which are of inertial origin modified by the thermal stratification, are found for Ri*10-3 where the Richardson number Ri is taken to be positive for unstable stratification. One of the inertial instabilities may occur for low Reynolds number ranging 55*Re*150 and the other predominates for Re* 150. Stability properties of the three different types and characteristics of the associated unstable perturbations are examined in detail by means of numerical solution of the perturbation equations. It is noticed that both the types of the inertial instability are slightly influenced by thermally unstable stratification in the range of Ri*10-3.

A dynamical study is made of properties of cumulus convection and its preferred mode in a conditionally unstable atmosphere based on numerical experiments. The preferred mode of cumulus convection is determined as the steady convection cell which consists of an ascending saturated region and a descending unsaturated one attained eventually after random temperature disturbances are imposed initially. It is shown that the preferred cell size and the area ratio of the ascending region to the descending one depend on the mean vertical velocity and the static stability of the atmospheric layer in which the connections are imbedded. Inspection of each term of energy equations indicates that the preferred convection is of the mode for which the potential energy of the layer is at the lowest. Since the potential energy represents the static stability of the layer, the minimum value of the mean temperature lapse-rate can be observed when the preferred convection is realized.

Stability properties of an unstably stratified parallel flow with vertical and horizontal shears are investigated by solving numerically a system of perturbation equations. Horizontal shear as well as vertical shear tends to suppress thermal instability. Stabilizing influence of the shear on perturbations is more efficient as the wavelength in the longitudinal direction is shorter, while the longitudinal roll perturbation is free from the stabilizing effect of the shear. A longitudinal roll convection may thus be most favorable for development. Conversion between the kinetic energy of the basic flow and that of the perturbation is found besides conversion of potential to kinetic energy. A longitudinal p erturbation which is of the preferred mode is associated with horizontal and vertical transports of momentum to decrease the shear which result in transforming the kinetic energy of the flow into that of the perturbation.