A Further Study of the Preferred Mode of Cumulus Convection in a Conditionally Unstable Atmosphere

Transcription

1 February 1982 T. Asai and I. Nakasuji 425 A Further Study of the Preferred Mode of Cumulus Convection in a Conditionally Unstable Atmosphere By Tomio Asai Ocean Research Institute, University of Tokyo, Tokyo 164 and Isao Nakasuji Orient Man Power System, K.K. (Manuscript received 30 July 1981) Abstract 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. 1. Introduction The 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. 2. Model and formulation of problem 2.1 Basic equations Consider a horizontally uniform atmospheric layer of the depth, h, which has a conditionally unstable stratification. Convective motions are restricted in the layer between two horizontall planes fixed at z = 0 and z = h, respectively. Horizontally averaged mean vertical motion does not necessarily vanish and the air can go through these two boundaries. The convective motion is confined to the vertical (x, z) plane. A pseudoadiabatic process is assumed, i.e., condensed water falls out of the system immediately. Ice phase of water is not considered. Then the conservation equations of momentum, heat energy, water vapor and mass for the shallow convection under the Boussinesq approximation can be ob-

2 426 Journal of the Meteorological Society of Japan Vol. 60, No. 1 tamed as follows (Ogura and Phillips, 1962). where where *0, q0, *0, *, q and * are the initial basic values of potential temperature, specific humidity, pressure equivalent, *(p/ p0)(cp-c*)/cp and their departures from the respective initial basic values. m, L, C and qs are the constant mean potential * temperature, the latent heat of condensation, the condensation rate and the saturation specific humidity. The other symbols are the same as in paper AN and used customarily. 2.2 Boundary conditions Both the upper and the lower boundaries are fixed and smooth for the convective motion while a uniform mean vertical motion can go through both the boundaries. The potential temperature and the specific humidity are assumed constant at the lower boundary, while constant fluxes of heat and water vapor are assumed at the upper boundary. The symmetrical conditions are adopted with respect to the lateral boundaries x = 0 and x = d. In summary, where * is a uniform mean vertical velocity and d is the horizontal width of the domain. 2.3 Initial conditions The initial basic field is set up as follows, where T0 and p0 are the initial basic temperature and pressure, respectively. T00 and P00 are the temperature and pressure at the lower boundary, respectively. * is the constant temperature lapserate, R the gas constant of dry air and r the constant relative humidity, *0 and * 0 are obtained from T0 and p0 following their definitions. The following potential temperature disturbance is superimposed on the initial basic field. (x, z) = ar(x, z) (2.13) * or *(x, z)=a(x)sin2(*z/h) (2.14) where a is an amplitude of the initial potential temperature disturbance, b is its horizontal scale and R (x, z) is the two-dimensional random function which ranges from -0.5 to 0.5. (2.13) is adopted for determination of the preferred mode of convection while (2.14) for the other cases. 3. Computational procedure Introducing a stream function * and eliminating * from (2.1) and (2.2), we obtain a complete set of equations for *, * (= -* 2*), * and q. These differential equations are approximated by a set of finite-difference equations and solved numerically by the same method as paper AN. The variables are allocated to the respective grid points as shown in Fig. 1. The condensation rate C is computed by the same method as Asai (1965). We first estimate (t+ *t), q (t+ *t) and qs (t+ *t) by assuming * C=0 in (2.3) and (2.4). These tentatively estimated values of *, q and qs are denoted by **, q* and qs* respectively. When q0+q*<qs*, the air does not saturate with water vapor during the time interval from t to t + *t. Thus C = 0, 0(t+*t)=**, q(t+*t)=q* and qs(t+*t)= qs*. When q0 + q* * qs*, the air has become saturated during the period of time *t from t to t+ *t. We can adopt the following approximation,

3 February 1982 T. Asai and I. Nakasuji 427 where Kc, Pc and P are the kinetic energy of convection, the available potential energy and the potential energy which are defined respectively as Fig. 1 Grid system used for numerical experiment. C = (q0 + q* - qs*) because the restriction q0 + q * qs must be satisfied. Test computations were performed by varying the horizontal grid size *x, the number of subdivided layers n and the time increment *t. When *x *400m, n * 8 and * t * 30 sec, sufficiently accurate solutions were obtained. In the following numerical experiments we adopt n = 8 and t20 = sec, and 80m* x *340m. 4. Energy equations The equations of the convective kinetic energy, the available potential energy and the potential energy are derived in the same way as in paper AN as follows, respectively. * and Here *A*, A and A' denote the domain average, the horizontal average and the deviation from the horizontal average of A, respectively. [A]h0 denotes the value of A at z = h subtracted the one at z = 0. The left-hand sides of (4.1), (4,2) and (4.3) are the time change of the convective kinetic energy, the available potential energy and the potential energy, respectively. The first terms on the right-hand side of (4.1), (4.2) and (4.3) are the conversion rates of the respective energies due to vertical heat transport. The second term of (4.3) is the conversion rate of the potential energy due to the mean vertical motion. The second term of (4.2) and the third term of (4.3) are the conversion rates of the available potential energy and the potential energy due to the latent heat release, respectively. The last two terms of (4.1), (4.2) and (4.3) are the diffusional dissipation of the energies and the vertical flux divergence of the energies through the upper and lower boundaries due to the mean vertical motion, respectively. 5. Results A preferred mode of cumulus convection is obtained as the steady convection cell attained eventually after the random potential temperature disturbance denoted by (2.13) is imposed initially. An amplitude of the initial disturbance, a, is taken to be as small as possible to avoid a serious influence of the imposed initial disturbance on properties of the final steady convection (e.g., Ogura, 1971). We adopt a=0.001k for the cases with mean ascending currents or initially saturated basic field without mean descending currents. However, a=1k is adopted for the cases with an initially saturated basic field accompanying mean descending currents and a=

4 428 Journal of the Meteorological Society of Japan Vol. 60, No. 1 10K for the cases with an initially unsaturated basic field without mean ascending currents, since larger amplitudes of the initial disturbance are required to set up convection. A horizontal scale, d, of the domain is wide enough to contain many convection cells. We adopt 30km* d * 60km. We set T00 = 20*, p00= 900mb, h= 1km, *=100m2 s-1 for which Rayleigh parameter defined as g(* d- * m)h4/ (T00*2) is about 1.8 * 104 and the same order of magnitude as paper AN. Here r d is the dry adiabatic lapse-rate and m is the moist adiabatic lapse-rate. * Fig. 2 shows variations of the horizontal scale, will be scattered and then disappear as the mean descending motion becomes dominant. A decrease of l and an increase of * of preferred convection with * are similar to those in paper AN, though the preferred cumulus cloud is much smaller due mainly to lack of water vapor supply. In either case a preferred mode of cumulus convection in a conditionally unstable layer may appear to be much flatter than that of dry convection in an absolutely unstable layer. Fig. 3 shows variations of l/ h and * of the Fig. 2 Variations of the horizontal scale, l/h, and the cloud coverage, *, of the preferred cumulus convection cell with * for *=7Kkm-1 and =1. * Fig. 3 Variations of l/h and * of the preferred l/ h, non-dimensionalized by the depth of the convection layer, h, and the cloud coverage, *, of convection cell with * for two different values of w=0 and w=10cm s-1 at the fixed value the preferred cumulus convection cell with a of r=1. value of * for * =7Kkm-1 and r=1. This value of * corresponds to *(r - preferred convection cell with a value of * for r m) = 0.5. The horizontal scale, 1, is defined as two different values of w = 0 and w=10cm s-1 a spacing between neighboring maximum and at the fixed value of *=1. As * decreases, that minimum of vertical velocity at the mid-level, is, the stratification is less unstable, l increases while a is defined as a ratio of a saturated area and a decreases for 10=0, while for w=10cm to the entire one in each cell at the mid-level s-1 l decreases and a increases as * decreases. of the cloud layer. The preferred scale of convection These variations of l and * of the preferred con- cell decreases from 5 km to 2.7 km and vection cell with r are similar to those in paper. the preferred cloud coverage increases from 0.08 AN. It differs from the previous results, however, to 0.2 when the mean ascending current increases that a maximum of l and a minimum of from 0 to 10 cm s-1 which corresponds to unity for w=10cm s-1 are not found. It should * of dimensionless value in paper AN. On the be noted here that the preferred mode of cumulus other hand the preferred scale of convection cell convection depends heavily upon the mean increases rapidly and the preferred cloud coverage vertical motion in the less unstable atmospheric tends to decrease to zero as the mean descending layer. current increases. In other words cumulus clouds Fig. 4 shows variations of l/ h and * of the

5 February 1982 T. Asai and I. Nakasuji 429 Fig. 4 Variations of l/h and o- of the preferred convection cell with a value of r for two different cases of w=0 and w =10 cm s-i at the fixed value of y=9 K km-1. preferred convection cell with a value of r for the two different cases 113=0 and w=10 cm s-1 at the fixed value of r =9 K km-1. As r decreases, the preferred size increases and the preferred cloud coverage decreases in a convective layer for W0, while these variations are very small in the layer with the mean ascending current. As r decreases, a thin cloud layer is confined in an upper layer and eventually cumulus convection vanishes for r_0.8. Now we examine a steady convection cell in a domain attained from an initial disturbance given by (2.14) by investigating variations of magnitude of each term of the energy equation (4.1), (4.2) and (4.3) with a horizontal scale of convection cell. We can obtain a number of steady convections which may differ from the preferred convection by varying size of the domain. Fig. 5 shows variation of magnitude of each term of the potential energy equation (4.3) with a horizontal scale of convection cell 1/ h for w=10 cm s-1, r =7 K km-1 and r=1. Fig. 5 also shows numbers of occurrence of cell size, 1/h, for a steady state attained eventually after the random potential temperature disturbance is imposed initially. The potential energy (solid line) becomes minimum at the preferred scale 1/h = 2.7, and it is seen from the dotted line that the mean temperature lapse-rate <'> defined as r - [iro8]o/h also becomes minimum at the preferred scale. The horizontal scale of convection cell which maximizes the total vertical heat transport (dashed line) is larger than the preferred scale. All terms of the kinetic energy equation (3.1) and the available potential energy Fig. 5 Variations of magnitude of each term of the potential energy equation (4.3) with a horizontal scale of convection cell and number of occurrence of cell size, 1/h, for w=10 cm s-1, y=7 K km-1 and r=1. The scale for the potential energy is indicated on the right and for the other terms of (4.3) on the left. equation (3.2) have no maximum in the same range of the h rizontal scale of convection cell as in Fig. 5, though they are not shown here. Thus the selection hypothesis that the preferred convection maximizes any term of them is not supported. Fig. 6 is the same as Fig. 5 except for w=0, Fig. 6 Same as Fig. 4 except for w=0, y=9 K km-1 and r=0.9.

6 430 Journal of the Meteorological Society of Japan Vol. 60, No. 1 ' = 9 K km- i and r= 0.9. In this case the convective layer is supplied with water vapor only by diffusive flux through the lower boundary. The frequency distribution of convection cell obtained for five different random disturbances with the same magnitude of the amplitude are contained in Fig. 6. A peak in the frequency distribution nearly coincides with the minimum potential energy and the minimum mean temperature lapse-rate, though the frequency distribution spreads to larger cell size. Thus the conclusion that the preferred mode of a cumulus convection cell is the one for which the potential energy of the convective layer is at the lowest is valid in the present model in which water vapor is introduced explicitly. 6. Conclusions The model of the previous paper (Asai and Nakasuji, 1977) is revised to deal with water vapor explicitly. A preferred mode of cumulus convection is obtained by the same method as the previous paper. The results obtained are as follows. The preferred scale of convection cell decreases and the preferred cloud coverage increases as the mean vertical velocity increases. The preferred scale increases and the preferred cloud coverage decreases as the initial basic temperature lapse-rate ' decreases in the atmospheric layer without a mean ascending current, while in the layer with a mean ascending current this tendency of the variation is opposite so that the preferred scale decreases and the preferred cloud coverage increases. As the initial basic relative humidity decreases, the preferred scale increases and the preferred cloud coverage decreases in an atmospheric layer without a mean ascending current, while the preferred convection cell sligjitly changes its mode in the layer with a mean ascending current. The preferred convection realizes so as to minimize the potential energy and hence the mean temperature lapserate in the convective layer. The other selection hypotheses are not supported. These results are almost the same as those obtained by the previous paper. Acknowledgements The present work was financially supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. Most of the numerical calculations were made with the use of the FACOM M-1605 at the Ocean Research Institute, University of Tokyo. The authors are indebted to Miss M. Sobukawa for assisting computation and to Miss N. Takusagawa for typing the manuscript. References Asai, T., 1965: A numerical study of the air-mass transformation over the Japan Sea in winter. J. Meteor. Soc. Japan, 43, Asai, T. and I. Nakasuji, 1977: On the preferred mode of cumulus convection in a conditionally unstable atmosphere. J. Meteor. Soc. Japan, 55, Ogura, Y., 1971: A numerical study of wavenumber selection in finite-amplitude Rayleigh convection. J. Atmos. Sci., 28, Ogura, Y. and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sc., 19,

7 February 1982 T. Asai and I. Nakasuji 431