The Vertical Profile of Entrainment Rate Simulated by a Cloud

Journal of the Meteorological Society of Japan, Vol. 83, No. 5, pp. 745--770, 2005 745

The Vertical Proﬁle of Entrainment Rate Simulated by a Cloud-Resolving Model and Application to a Cumulus Parameterization

Akihiko MURATA and Mitsuru UENO

Typhoon Research Department, Meteorological Research Institute, Tsukuba, Japan

(Manuscript received 8 June 2004, in ﬁnal form 2 June 2005)

Abstract

To investigate the vertical profiles of fractional entrainment rate as to cumulus convection, numerical simulations of a tropical cyclone rainband are conducted, using a high-resolution three-dimensional cloud-resolving model (CRM), with the 200-m horizontal resolution. On the basis of the results of CRM simulations, vertically variable entrainment rate is applied to the Arakawa-Schubert (AS) cumulus parameterization. Fractional entrainment rate, derived from the calculation based on the vertical gradient of cloud mass flux, clearly shows larger near cloud base and top. Between the heights of cloud base and top, entrainment rate is smaller, and even negative in many cases, suggesting laterally detrained air from a cumulus into the environment. From the analyses where entrainment rate is divided into three terms, it is found that the contributions of updraft is relatively large near the cloud base and top, compared to that in between. The cloud amount contribution depends on whether cloudy areas expand or shrink accompanying cloud growth or decay, respectively. On the basis of the result of the CRM simulations, vertically variable entrainment rate is applied to the AS scheme. For investigating the effect of the modifications of the scheme, simulations of typhoon Saomai (2000) are conducted. The simulations show significant improvements: underestimates of moisture in the mid- to upper troposphere are reduced. The result is predominantly attributed to cloud mass flux, greatly influenced by lateral detrainment. The peak height of the mass flux corresponds to that in the moisture tendency.

1. Introduction tropical cyclone, with some degree of realism, was rather insensitive to such a treatment. It Precipitation in a tropical cyclone appears to was pointed out on the other hand that numer- be sensitive to the formulation for cumulus pa- ical simulations for tropical cyclones provided a rameterization, because latent heat release is useful test for the depth of knowledge about a key factor for tropical cyclone intensification. cumulus parameterization (Ueno 2000). Smith (2000) pointed out that the details of The sensitivity of cumulus parameterization tropical cyclone evolution seemed to be sensi- to tropical cyclone simulations therefore has tive to the treatment of convection in numerical been reported by several studies. For example, models, although the simulation of a mature Baik et al. (1991) indicated a sensitivity of the time of rapid storm deepening to the choice of different convection schemes used in their nu- Corresponding author: Akihiko Murata, Meteoro- merical model. We also have investigated ef- logical Research Institute, Nagamine 1-1, Tsu- fects of a couple of cumulus parameterizations kuba, Ibaraki 305-0052, Japan. E-mail: [email protected] on tropical cyclone simulations. Murata and ( 2005, Meteorological Society of Japan Ueno (2000) investigated the effect of two major 746 Journal of the Meteorological Society of Japan Vol. 83, No. 5 cumulus parameterizations on the intensity of It has been recognized that direct observa- typhoon Flo (1990). The central pressure of the tions of the vertical profile of cloud mass flux is storm in the simulation, with a version of the useful as the truth data for cumulus parame- Arakawa-Schubert (AS) scheme (Arakawa and terization (Ooyama 1971). Yuter and Houze Schubert 1974; Kuma et al. 1993), was more (1995) determined vertical mass fluxes in Flor- realistic than that in the simulation with a ida cumulonimbus by analysis of ground-based moist convective adjustment scheme (Gadd and dual-Doppler radar data. They indicated that Keers 1970). Precipitation averaged within the entrainment of environmental air occurred core region of the storm in the former was sig- along trajectories of parcels in a cumulus and nificantly larger than that in the latter. The thereby most of the updraft parcels did not result was explained by difference in the degree reach the top of the storm, suggesting lateral that the schemes eliminated vertical static in- detrainment from the cumulus into the envi- stability. The degree of the elimination was ronment. Grinnel et al. (1996) examined the also a key factor for representation of a promi- vertical mass flux profile of Hawaiian trade cu- nent rainband in the tropical cyclone (Murata muli, using two ground-based Doppler radars et al. 2003). The vertical instability was exces- and an instrumented aircraft. The profile, sively eliminated when the moist convective which underwent the characteristic evolution adjustment was used, which suppressed the that was relatively independent of cloud size band formation. As a consequence it was found and organization, were compared to those pro- that the AS scheme assured relatively good duced by cloud models included in a couple of performance in tropical cyclone simulations. cumulus parameterizations. They mentioned Fractional entrainment rate is a key param- that typical conceptual models for convective eter in the AS cumulus parameterization. It is clouds, such as the Raymond and Blyth (1992) not only true for the AS scheme, but also for buoyancy sorting model, or entraining plume any other cumulus parameterizations that use models, predicted cloud mass flux profiles simi- the cloud model, in which the concept of mass lar to the early phase of their observations. Al- flux is included (e.g., Tiedtke 1989; Grell 1993). though the vertical mass flux has recently been In these schemes, except for those in which the determined quantitatively by means of Doppler concept of buoyancy sorting is included (e.g., radars, it is difficult to accurately obtain the Kain and Fritsch 1990; Emanuel 1991), en- property from the observations. trainment rate, governing the vertical profile A cloud-resolving model (CRM), a numerical of cloud mass flux, is vertically constant. The model that resolves cloud-scale circulations, same applies to the AS scheme, although en- can be used for the development of cumulus trainment rate depends on the cloud spectrum parameterization. Moncrieff et al. (1997) (i.e., cloud-top height). However, there is no ev- pointed out that, for a larger-scale model, a idence suggesting that entrainment rate is ver- CRM was able to determine the collective tically constant. effects of subgrid-scale processes on the large- Entrainment profiles in real clouds have scale field. They mentioned that bulk proper- been investigated. Reuter (1986) reviewed the ties, such as cloud mass flux, were readily cal- history of ideas related to the dynamical as- culated from the results produced by a CRM. pects of cumulus entrainment. According to his CRMs have recently been used to evaluate review, it had been thought that entrainment and develop physically-based cumulus parame- took place laterally, in a manner similar to terizations. Liu et al. (2001a) examined squall- plumes modeled in laboratory tank experi- type cloud systems in the Global Atmospheric ments. Moreover, he provided evidence that dry Research Program (GARP) Atlantic Tropical air entered through cloud top in addition to the Experiment (GATE), using a two-dimensional side of the cloud. Blyth (1993) also reviewed CRM, and compared the result to that produced pieces of information gathered over more than by the simulation with the Kain-Fritsch con- 40 years, and mentioned that entrainment oc- vective parameterization. In the same way, Liu curred at all levels of a cumulus cloud. He in- et al. (2001b) investigated convective cloud sys- dicated that entrainment occurred at ascending tems in the Tropical Ocean Global Atmosphere cloud top, at least in a growing cloud. Coupled Ocean-Atmosphere Response Experi- October 2005 A. MURATA and M. UENO 747 ment (TOGA /COARE) and pointed out defi- files in numerical simulations for a tropical cy- ciencies of the Kain-Fritsch scheme used on clone. Following it, we discuss the performance the 15-km horizontal grid. The deficiencies in- in greater detail in section 6. Finally, in section cluded, excessive detrainment of water vapor 7, we summarize and conclude the results. and condensate in the upper troposphere, and were attributed to the single-plume model that 2. Numerical model and experimental represented the whole convective updrafts in a design horizontal grid. 2.1 Numerical model It has been pointed out that the vertical pro- The Meteorological Research Institute/ files of entrainment rate in cumulus parame- Numerical Prediction Division Nonhydrostatic terizations did not agree with results simulated Model (MRI/NPD-NHM) (Saito et al. 2001) is by CRMs. Lin and Arakawa (1997), and Lin used as the CRM with which the vertical profile (1999), examined entrainment profiles using of cumulus entrainment rate is investigated. simulated data from a two-dimensional CRM. The model has fully compressible equations They found that entrainment for each cloud with a map factor, and employs a semi-implicit type, which was categorized in terms of cloud- time integration scheme. The vertical coordi- top height, tended to become larger near cloud nate is terrain-following and contains 76 levels, base and top. Cohen (2000) developed a method with variable grid intervals of Dz ¼ 40 m to in which numerical tracers were utilized to 1480 m. The lowest level is located at 20 m diagnose entrainment rates. He applied the from ground surface, while the highest level is method to two-dimensional nonhydrostatic at 28 km. We use the model of 200-m horizon- simulations, and found that the entrainment tal resolution (200 m-NHM). The time-step in- rates in updrafts were large near, and above tervals are Dt ¼ 1s. cloud base. Gregory (2001) discussed entrain- The numerical model includes bulk cloud mi- ment rate linked to the rate of vertical kinetic crophysics, introduced by Ikawa et al. (1991). energy produced by buoyancy. The vertical pro- The scheme predicts the mixing ratios of six files of entrainment rate, obtained from his water species (water vapor, cloud water, rain, formulation, were consistent with those of Lin cloud ice, snow, and graupel), and the number (1999). Swann (2001) calculated entrainment concentration of cloud ice. The size distribu- and detrainment from CRM simulations of tions of the water substances are assumed to be deep convection in a variety of environmental inverse exponential for rain, snow, and graupel, conditions. The analysis confirmed that the and assumed to be mono-disperse for cloud mass-flux approached in bulk-updraft convec- water and cloud ice. The treatment is based tion parameterization schemes could represent on Lin et al. (1983), Murakami (1990), and net heating and moistening due to convection. Murakami et al. (1994). A box-Lagrangian rain- One of our goals is to investigate the verti- drop scheme (Kato 1995) is incorporated for cal profiles of fractional entrainment rate in calculating rain fall-out. cumulus convection in a simulated tropical cyclone rainband using a high-resolution three- 2.2 Experimental design dimensional CRM. The other is to propose a A domain, near the Okinawa islands, covers new approach to calculating entrainment rate a part of a spiral rainband of typhoon Saomai, profiles for a cumulus parameterization using assigned the serial typhoon number 14 in 2000 the findings obtained from the CRM simula- by the Regional Specialized Meteorological tions. The rest of the paper is organized as fol- Center (RSMC)-Tokyo. The horizontal grid con- lows: Section 2 describes the numerical model sists of 280 280 grid points in x and y on and design of experiments. In section 3, we in- a Lambert conformal map projection, with a vestigate the vertical profile of entrainment grid spacing of 200 m. To conduct the high- rate in cumulus convection using the CRM. In resolution model integrations, we adopt a grid- section 4, we explain the application of the nesting strategy for the lateral boundary con- findings, obtained from the CRM results, to a ditions: triply nested MRI/NPD-NHM (Fig. 1a). cumulus parameterization. Section 5 describes The nested quantities include the six water the performance of the new entrainment pro- species, in addition to standard meteorological 748 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 1. Design of (a) model domains and (b) model integration. One-hour accumulated rainfall amounts at 7 hr for 5 km-NHM and at 1 hr for 1 km-NHM are shown. Contours are drawn at 1, 10, 20 mm hr1. Domain of 1 km-NHM is shown by the solid square box in the domain of 5 km- NHM. Domain of 200 m-NHM is also shown in the domain of 1 km-NHM.

quantities, such as wind components and po- boundary are produced by the Regional Spec- tential temperature (Mashiko et al. 2001). The tral Model (RSM) (NPD/JMA 1997), one of the inclusion of the six water species in the initial operational models of the Japan Meteorological and boundary ﬁelds enables the model to spin Agency (JMA). The RSM domain is located on up quickly. The nesting procedure (Fig. 1b) a part of eastern Asia (6 N–35 N, 116 E–155 E), is as follows: The initial (1100 UTC 12 Septem- with 20-km grid distance. The RSM initial ber 2000) ﬁeld and lateral boundary data for includes the typhoon bogus that contains both 200 m-NHM are obtained from forecasts pro- symmetric and asymmetric components of duced by MRI/NPD-NHM (280 280 grid wind and geopotential height (Ueno 1995). points), with the horizontal resolution of 1 km The time-step interval for 1 km-NHM is 5 s, (1 km-NHM). Similarly, those data for 1 km- and for 5 km-NHM is 15 s. The bulk cloud NHM are provided by MRI/NPD-NHM microphysical scheme used in 1 km-NHM, and (280 280 grid points), with 5-km horizontal 5 km-NHM, is the same one as the 200 m- resolution (5 km-NHM), whose initial and NHM. October 2005 A. MURATA and M. UENO 749

3. Vertical profile of entrainment rate updraft averaged over the cumulus grids. The interval between the two times is set to 5 s. It 3.1 Method for entrainment rate estimation should be noted that the effect of the turbu- Fractional entrainment rate, l, in a cumulus lence, whose spatial scale is less than horizon- is calculated using output from the CRM tal grid interval, 200 m in this case, is also (200 m-NHM) for the estimate of the vertical included in (3.3), because the calculation is profile of the rate. Before the calculation, a cu- conducted in Lagrangian manner. The calcula- mulus area, where a single cumulus is con- tion enables us to estimate the sum of the ef- tained, is extracted from the CRM output. fects of turbulence and advection, dubbed as The entrainment rate calculation is con- turbulent entrainment and dynamic entrain- ducted on the basis of the vertical differentia- ment, respectively, by Houghton and Cramer tion of cloud mass flux as follows: (1951). 1 dM l ¼ ; ð3:1Þ 3.2 Results M dz Three cases are chosen to analyze entrain- where M is cloud mass flux and z is height. ment rate profiles for the present study. The Cloud mass flux, M, is estimated at each model negative values in the first profile (Fig. 2a) is level as follows: not so noticeable as those in the second one (Fig. 2b). The cloud in the last case (Fig. 2c) M ¼ Arw; ð3:2Þ differs from the others in its top height: rela- where A is cloud amount: the rate of model tively high (8 km) in Figs. 2a, b, and low (5 km) grids judged as being cumulus grids within the in Fig. 2c. In these cases, the cloud top is as- extracted cumulus area, r is moist air density sumed to be located just above the levels where that includes the effect of water substances, entrainment rate rapidly increases. The rapid and w is vertical velocity. Both r and w are increase in the rate indicates the growing cloud quantities averaged over the cumulus grids. top. The criterion for the judgment whether a grid The results of the calculation clearly show belongs to cumulus or not, is based on vertical large entrainment rate just above cloud base, velocity and mixing ratio of cloud water and and just below cloud top. Figure 2a shows the cloud ice. If vertical velocity is positive, denot- characteristic entrainment profiles that have ing updraft, and the sum of mixing ratios of larger rates below 1.5-km high, and above 6-km cloud water and cloud ice is larger than a high. The layers in between are marked by threshold, set to 0.1 g kg1, the grid point is relatively small entrainment rate. Figures 2b judged as a cumulus grid. The inclusion of ver- and c also show larger entrainment rates near tical velocity in the criterion intends to esti- cloud base and top, and smaller in between, mate quantities only in the updraft part of a suggesting that the characteristic features are convective cloud. The criterion enables us to independent of cloud-top height. Negative make a comparison between the results of the values in the middle layers (i.e., between cloud updraft part, produced by the CRM, and by the base and top), which are noticeable particularly coarse model, with the AS cumulus parameter- in Fig. 2b, indicate that the air in the cumulus ization. emanates laterally there. It should be noted Vertical differentiation in (3.1) is performed that we analyzed other sampled clouds, and in the following manner: found that most of them had the features similar to those shown in Fig. 2. 1 dM 2ðMt M0Þ In order to investigate the cause of the height ¼ ; ð3:3Þ M dz ðMt þ M0Þðzt z0Þ dependency, we divide entrainment rate into three terms, thereby estimating the contribu- where M and z denote quantities at an arbi- 0 0 tion of each term to the whole. Substituting trary model level and an arbitrary time, re- (3.2) into (3.1), we can write entrainment rate spectively. As for M and z , quantities at a t t as the sum of the three terms: higher level and a later time are used because the left-hand side of (3.3) shows Lagrangian 1 dM 1 dw 1 dA 1 dr l ¼ ¼ þ þ : ð3:4Þ differentiation. zt is determined using the mean M dz w dz A dz r dz 750 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 2. Vertical proﬁles of entrainment rate with respect to three convective clouds extracted from the 200 m-NHM output. The proﬁles are drawn every 20 sec within (a) 19–22 min, (b) 16– 19 min, (c) 16–19 min.

The vertical profile of each component (Fig. interval Dt (5 s in this case), can be written as 3) demonstrates that larger entrainment rates follows: near cloud base and top are attributed to the w ¼ Bt þ w ; ð3:5Þ first term, referred to as ‘‘updraft entrainment 0 rate’’, and the second term, referred to as ‘‘cloud where w0 is w at t ¼ 0, and B is the sum of the amount entrainment rate’’, of the right-hand acceleration due to vertical forcing, such as side in (3.4). The degrees of both contributions buoyancy and pressure gradient, and is defined are comparable. The figure also shows that the as last term of the right-hand side, referred to as dw ‘‘density entrainment rate’’, has a negligible ef- B 1 : ð3:6Þ fect on the total entrainment rate, dM/Mdz. dt From Fig. 3a, it is found that both updraft The updraft entrainment rate therefore can be and cloud amount entrainment rates consis- written as follows: tently maintain the characteristic profiles that 1 dw 1 dt dB 1 B dB feature larger values near cloud base and top, ¼ B þ t ¼ þ t and smaller at the layers in between. This is w dz w dz dz w w dz roughly true, but does not always hold for all B t dB cases. In Fig. 3c, the updraft entrainment rates ¼ þ : ð3:7Þ w2 w dz are not so large near cloud top. Vertical velocity, w, assuming vertical accel- The updraft entrainment rate is proportional to eration does not change during the small time the sum of the forcing, and inversely propor- October 2005 A. MURATA and M. UENO 751

Fig. 3. Vertical profiles of components of entrainment rate. The profiles are drawn at (a) 21 min, (b) 18 min, (c) 17 min with respect to the clouds of Fig. 2a, b, and c, respectively. tional to the square of vertical velocity, because panying cloud growth, seems to be responsible careful examination shows that the last term of for the larger rates. Since cloud amount, A in the right-hand side in (3.7) is negligible. Since (3.4), is relatively small near cloud base and the updraft horizontally-averaged within cu- top, the effect remarkably appears there (Fig. mulus grids tends to have a minimum at cloud 5). base and top, in addition B does not change Lateral detrainment is examined in detail vertically (Fig. 4a) so much as the square of using (3.4). From Fig. 2b, it is found that the the updraft (Fig. 4b), the updraft entrainment negative values of entrainment rate (i.e., de- rate is larger there (Fig. 4c). In contrast, the trainment) are remarkable at middle parts of rate is lower between cloud base and top, the cloud layers (i.e., around heights of 2 and due to relatively strong updraft there. The 5 km). The negative rates in the upper layer feature of the simulated updraft profiles is are attributed to both main components of the qualitatively consistent with that in real con- rate, the updraft entrainment rate, and the vective clouds. For example, Braun and Houze cloud amount entrainment rate (Fig. 3b). In the (1994) extracted a convective region from an lower layer, the updraft entrainment rate is observed storm, and obtained positive values of responsible for the negative rates. vertical velocity for all heights, with a maxi- The negative updraft entrainment rates are mum at the middle level between the cloud explained by vertical deceleration [i.e., negative base and top. value of B in (3.7)]. The vertical deceleration As for the cloud amount entrainment rate, manifests in the form of rapid decrease in up- the effect of the increasing cloudy area accom- draft velocity with time around 2 and 5 km 752 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 4. Vertical proﬁles of (a) vertical acceleration, (b) square of updraft velocity, and (c) updraft entrainment rate, with respect to the cloud of Fig. 2a.

high (Fig. 6a), in contrast to the aforemen- 2 and 5 km, corresponding with the heights of tioned case where the decrease in updraft ve- the negative updraft entrainment rates (Fig. locity with time is not seen. To examine the 3b). In contrast to that, negative buoyancy does cause of the vertical deceleration, we estimate not appear above a height of 4 km in Fig. 7a, buoyancy in the cumulus, B1, from calculating where no lateral detrainment occurs (Fig. 3a), the difference in density, including the effect of suggesting that the cloud is in the stage of water substances, between the cumulus and growing. the environment as follows: The negative cloud amount entrainment rates can be explained by the local differential r ½r B ¼g ; ð3:8Þ of cloud amount, both in time and height. As in 1 ½r Fig. 6b, cloud amount decreases with time where r is density in the cumulus, ½r is density and height in the layer above a height of 3 km in the environment, defined as the quantity in general, in contrast to lower levels. The for- averaged over 100 100 horizontal grids mer effect, however, seems to be essential, con- equivalent to a 20 20 km area, and g is grav- sidering the aforementioned case where the ity acceleration. It is noted that only the effect increase in cloud amount with time above a of buoyancy is considered in B1, whereas B in height of 4.5 km, suggests the stage of cloud (3.7) formally includes all components of verti- growing (Fig. 5). The rapid decrease in cloud cal acceleration. The vertical profiles of the es- amount with time in the present case suggests timated buoyancy are shown in Fig. 7b. The the stage of cloud decaying. In fact, it is found profiles have negative values around heights of from Fig. 8 that the cloud is decaying consider- October 2005 A. MURATA and M. UENO 753

rate to the steady-state cloud model in a cumulus parameterization. As for the steady-state cloud model, temporally averaged cloud mass flux should be used. In this case, entrainment rate is equivalent to that estimated by the local differentiation of mass flux in a vertical direction. We analyze the same clouds as used in the preceding section. The results of the calculation clearly show large entrainment rate just above cloud base (Fig. 9), as in the results of the non-steady case (Fig. 2), where the method for calculating the rate is mentioned in the preceding section, although the quantity is not so large just below cloud top. For instance, Fig. 9a shows larger rates below 2-km high and around 4-km high. Between the levels, entrainment rates are rel- Fig. 5. Same as in Fig. 4, but number atively small; even negative at some levels. of grid points judged as a part of the The rates are also negative above a height cumulus. of 4.5 km, indicating cloud top detrainment. These features are also shown in Fig. 9b. The profile of Fig. 9c, however, has no remarkably ably. In particular, the upper part (i.e., above higher values near cloud top, although a weak 4-km high) of the cloud becomes compressed peak is found in each profile. remarkably, resulting in the negative cloud One major difference in the profiles, between amount entrainment rates. steady and non-steady cases, is the sign of en- 3.3 Situation supposed steady state trainment rate near cloud top. The positive A different procedure is required for the ap- values in the former change the negative one in plication of the vertically variable entrainment the latter, explained by the difference in cloud

Fig. 6. Vertical proﬁles of (a) updraft velocity and (b) number of cumulus grid points with respect to the cloud of Fig. 2b. 754 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 7. Vertical proﬁles of buoyancy acceleration with respect to the clouds of Fig. 2.

mass flux differencing, between Lagrangian kawa and Schubert 1974). To maintain the and Eulerian views. The negative value in the quasi-equilibrium, the original AS scheme di- steady case is attributed to the feature that agnostically calculates cloud-base mass flux for cloud mass flux decreases with height near all cloud types simultaneously, where clouds cloud top. In contrast, the positive value in the are classified according to their top altitudes. non-steady case is explained by cloud mass flux Instead, on the basis of the theory proposed by increasing with time due to growing cloud top. Randall and Pan (1993), cloud-base mass flux The effect counteracts, and overcomes the in- in J-AS is prognostically determined by the fluence of the mass flux decreasing with height. cloud-type-dependent calculation as follows: 4. Application of vertically variable qM ðiÞ AðiÞA ðiÞ M ðiÞ entrainment rate to the Arakawa- B ¼ 0 B ; ð4:1Þ qt 2a i 2t Schubert scheme ð Þ D

4.1 Formulation of the Arakawa-Schubert where i is the cloud type, MB ð>0Þ is the cloud- scheme at the JMA base mass ﬂux, A is the cloud work function, The Arakawa-Schubert scheme, currently in A0 is the empirically-determined threshold for operational use at the JMA (J-AS), is a sim- A (Load and Arakawa 1980), a, and tD are pliﬁed version with a prognostic closure as- the prescribed parameters of convective adjust- sumption (Kuma et al. 1993), as opposed to ment time and dissipation time, respectively. a quasi-equilibrium closure assumption (Ara- A0 is introduced so that the cloud work function October 2005 A. MURATA and M. UENO 755

Fig. 8. Vertical cross sections of vertical wind velocity along a line of (a) east-west and (b) north- south direction at 19 and 22 min. Contours are drawn at 200 cm s1 intervals. Solid (broken) con- tour lines denote positive (negative) values. Tick marks on each abscissa are at 200 m intervals.

ð zT maintains equilibrium. If the predicted value of TcðiÞT M is negative, M is set to be zero. AðiÞ¼g hðiÞ dz; ð4:2Þ B B z T The cloud work function, A, is a expression of B moist convective instability for a given cloud- where Tc is the temperature of the cumulus, T top height, zT, and is written as: is the temperature of the environment, g is the

Fig. 9. Same as in Fig. 2, but for the situation supposed steady state. 756 Journal of the Meteorological Society of Japan Vol. 83, No. 5 acceleration of gravity, zB is the height of cloud 2c1z þ c2 l ¼ 2 ; when zB < z < zDB: base and is set to 740 m, and h is the normal- c1z þ c2z þ c3 ized mass flux defined as ð4:7Þ MðiÞ Above the base of the detrainment layer, en- hðiÞ¼ ; ð4:3Þ MBðiÞ trainment rate is assumed as follows: where M is cloud mass flux that depends on l ¼ lD; when zDB < z < zDT; height. ð4:8Þ lT; when zDT < z < zT; 4.2 Part of modification We modify J-AS in order to investigate the where lD ð<0Þ denotes the effect of lateral de- impact of entrainment rate on the simulated trainment. tropical cyclone. The difference between J-AS, The unknown constants, zDB, zDT, lB, and lD, and the modified scheme (M-AS), is in their are determined on the basis of the results of the vertical profiles of fractional entrainment rate, 200 m-NHM simulation, although the cases of l, which is defined, for the steady-state cloud convective clouds are limited. The former three model, as of them are prescribed as follows:

1 qM zDB ¼ zB þ cDBðzT zBÞ; l ¼ : ð4:4Þ M qz zDT ¼ zB þ cDTðzT zBÞ; ð4:9Þ

Cloud mass flux, M, therefore becomes as fol- lB ¼ s þ a expðz/bÞ; lows: 4 where cDB ¼ 0:4, cDT ¼ 0:6, s ¼ 1:0 10 lðzzBÞ 1 2 1 M ¼ MBe G MBf1 þ lðz zBÞg; ð4:5Þ [m ], a ¼ 2:0 10 [m ], and b ¼ 5000 [m]. The last equation describes that the cloud-base where the mass flux is assumed to be a linear entrainment rate decreases with increasing function of height. This simplification was in- cloud-top height, consistent with the concept troduced by Moorthi and Suarez (1992). The of the original Arakawa-Schubert scheme. The entrainment rate for each spectrum of model coefficients, s, a, and b, of the assumed expo- clouds is vertically constant in J-AS. nential curve are determined considering the In contrast to that, M-AS uses the vertically results of the 200 m-NHM simulation, suggest- variable entrainment rate that represents ing that l is 5:0 103 m1, and 2:5 103 larger values near cloud base and top. The rate B m1 in the case of the 4, and 8 km cloud-top has negative values, representing lateral de- height, respectively (Fig. 9). The other un- trainment, in a part of the cloud layer. De- known constant, lD, is prescribed in a manner trainment at cloud top, same as J-AS, is also that the following condition is satisfied: included in M-AS. Firstly, for each cloud spectrum, we prescribe the lateral detrainment MDT ¼ð1 kÞMDB; when 0 e k e 1; ð4:10Þ layer and set the heights of the base, z , and DB where M and M are cloud mass flux at top, z , of the layer. Below the base, the verti- DB DT DT z ¼ z ; z , respectively. The proportionality cal profile of entrainment rate should increase DB DT constant, k, is set to 0.5, describing that the with height. For simplicity’s sake, we assume mass flux at the bottom of the detrainment an equation of cloud mass flux as follows: layer reduces in half at the top of the layer. The remaining constant, l , is automatically deter- M T ¼ c z2 þ c z þ c ; when z < z < z ; mined so that the following condition is sat- M 1 2 3 B DB B isfied: cloud top height is located where buoy- 4 6 ð : Þ ancy in the cumulus disappears, in other where c1, c2, and c3 are constants determined words, where moist static energy in the cumu- by the following conditions: l ¼ lB at z ¼ zB and lus is equal to saturated moist static energy of l ¼ 0atz ¼ zDB, where lB is the entrainment the environment. This is the same condition rate at cloud base. Entrainment rate, l, there- as that adopted in J-AS, and the original AS fore takes the profile as follows: scheme. October 2005 A. MURATA and M. UENO 757

Noteworthy advantages of M-AS include, large-scale condensation scheme are included. that the cloud-base entrainment rate in each Two simulations (i.e., with J-AS or M-AS) are spectrum of clouds is allowed to be corrected to considerably different in the horizontal distri- fit into a more reliable value, such as the result bution of precipitation. The patterns of 3-hour of a CRM, because there is a freedom for de- accumulated rainfall at 24 hr are shown in Fig. termining the cloud-base entrainment, as men- 11. The scheme modification reduces the area tioned before. In contrast to that, entrainment coverage of precipitation, particularly an area rate in J-AS has to be constant vertically, along the meridian of 135 E. As for the rain- thereby changes of the rate at cloud base are fall accompanying Saomai, the experiment with not permitted. It should be also noted that M-AS produces several rainbands, whereas the there is still room for improvement in the arbi- J-AS one shows wide-spread, light precipitation trary formulation (4.6), (4.8)–(4.10). surrounding the core of the storm. The light precipitation in the J-AS suggests that vertical 5. Simulation with the new scheme static instability has already been eliminated for a tropical cyclone excessively. In contrast, the sustained instabil- Simulations of typhoon Saomai are con- ity near the rainband seems to be responsible ducted using MRI-NPD/NHM with 20-km hori- for the concentration of precipitation in the zontal grid spacing (20 km-NHM). The model M-AS. has 140 140 horizontal grid points and 38 The degree of the concentration of precipita- vertical levels. Data for the initial, 1200 UTC tion seems to affect the intensity of the tropical 11 September, are obtained from the 12-hour cyclone. The storm in the J-AS has a tendency forecast of RSM (Fig. 1b). The boundary data of intensification, whereas the M-AS one almost are also produced by RSM at 3-hour intervals. keeps its intensity (not shown). In the latter The integration time of the 20 km-NHM is experiment, the concentration of rainfall, rela- 36 hr (Fig. 10). The model does not contain tively far from the storm center, seems to in- cloud microphysical schemes. Instead, a cumu- hibit the intensification of the storm. The result lus parameterization (J-AS or M-AS), and a suggests the importance of the horizontal distribution of precipitation for tropical cyclone intensification. One of disadvantages in the results using J-AS is excessive drying over the troposphere. Before showing that, we first calculate moist static energy, h, defined as follows:

h ¼ gz þ cpT þ Lq; ð5:1Þ where g is gravity acceleration, z is geopotential height, cp is specific heat capacity, T is temperature, L is latent heat release, constant from water vapor to liquid water, and q is specific humidity. The quantities derived from the results of the 20 km-NHM with J-AS and Fig. 10. The track of typhoon Saomai by 200 m-NHM are compared (Fig. 12a). It should RSMC Tokyo Typhoon Center during be noted that the former is raw values, whereas September 2000. The position shown by the latter is horizontally averaged (100 100 an open circle is at 0000 UTC and by a grid points) values, corresponding to 20-km closed circle is at 1200 UTC. A plotted horizontal resolution. We should also note that numbers denote dates (on the left of ten profiles in the 20 km-NHM result are the position symbol) and central pres- sures [hPa] (on the right of the position shown considering the discrepancy in the rain- symbol). The double line describes the band location, between the simulations using period used for the numerical experi- 200 m- and 20 km-NHM. The location of the ments using 20 km-NHM (J-AS and M- profiles chosen in the 20 km-NHM result, is AS) mentioned in the text. shown in Fig. 11a. 758 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 11. Horizontal distribution of three-hour accumulated rainfall amount from 0900 UTC to 1200 UTC 12 September for the experiment with (a) J-AS and (b) M-AS. Contours are drawn at 1, 5, 10, 20 mm 3 hr1.

Figure 12a clearly shows underestimates in is situated at the same location against the moist static energy all over the troposphere in storm center as the line for the J-AS result in the 20 km-NHM result. The division of moist Fig. 11a. In Fig. 13, the profile of the 200 m- static energy into three components, as shown NHM result is also shown in addition to the in (5.1), enables us to separately estimate the results of J-AS, and M-AS. From the figures, contributions of temperature and moisture. The it is found that both moist static energy and vertical profiles of the components are shown specific humidity profiles in the M-AS result in Figs. 12b, c, where the differences in the approach those of the 200 m-NHM result, in- quantities between the results of the two simu- dicating the reduction of the underestimates in lations are displayed. The abscissas of the fig- the J-AS one. The noticeable improvement is ures are scaled so that the contributions of the evident from middle to upper troposphere than two components to the difference in moist static below. The result suggests that cloud mass energy are equal. From the figures, we find fluxes at those levels are suppressed, due to most of the whole difference is accounted for by lateral detrainment, which reduces the mass the contribution of specific humidity. That is, fluxes above. The improvement, on the other specific humidity in the result with J-AS is sig- hand, is insufficient in the lower troposphere, nificantly underestimated all over the tropo- implying that the cloud mass flux there is still sphere, suggesting that the cumulus parame- too large. It is possible that the mass flux be- terization brings about excessive drying. In comes smaller by means of enhancing lateral addition, temperature is overestimated above a detrainment. The effects are described in the height of 10 km, implying that cloud mass flux following section, where we conduct sensitivity is larger than expected in those levels. experiments of this influence. The moisture field, and also moist static en- The marked difference in the amount of ergy, is improved when we use M-AS as cumu- moisture between the two experiments is visi- lus parameterization. The vertical profiles of ble over an outer area (i.e., between the 250 both quantities are shown in Fig. 13 as the av- and 500 km radius) of Saomai. The feature is erage of ten soundings along the line shown in seen in the radial profile of specific humidity, Fig. 11. The line for the M-AS result in Fig. 11b particularly from middle to upper troposphere October 2005 A. MURATA and M. UENO 759

Fig. 12. Vertical profiles of (a) moist static energy, (b) the deviation of temperature from the 200 m- NHM result, and (c) the deviation of specific humidity from the 200 m-NHM result at the targeted rainband at 23 hr (1100 UTC 12 September) for the experiment with J-AS. In (a), quantity obtained from the 200 m-NHM simulation is also plotted (dot). Raw values of ten profiles, along the bold line denoted in Fig. 11a, are drawn with respect to the J-AS experiment. As for the 200 m- NHM result, values averaged over 100 100 horizontal grid points are plotted so that the horizontal scale is matched to that in the J-AS experiment.

(Fig. 14). The figures show the large difference simulation than that in the M-AS one. In addi- between the two at radii from 250 to 500 km, tion, greater lateral moisture flux from the although a large difference is also noted in a outer area (e.g., outside of the 500-km radius, further eastern area (over the 700 km radius) not shown) enables the former simulation to of the center (Fig. 11). From Fig. 14, it is found provide water vapor sufficient for continuous, that the discrepancy at the radii between 250 widespread precipitation. and 500 km has already arisen by 6 hr. The It is desirable to compare the present 200 m- next section explores the cause of the discrep- NHM simulations against observations for ancy in greater depth. demonstrating the reliability of the CRM re- More widespread rainfall and a drier bias in sults. As the observational counterpart, we se- the J-AS simulation, shown in the precipitation lect a set of sonde data near the corresponding pattern (Fig. 11) and the moisture field (Fig. rainband of Saomai. The rainband passed by 14), suggest that the degree of the conversion of the island of South-Daito (26 N, 131 E) at about water vapor into rainfall is larger in the J-AS 0000 UTC 12 September. Using the sounding 760 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 13. Vertical proﬁles of (a) moist static energy and (b) speciﬁc humidity at the targeted rainband at 23 hr (1100 UTC 12 September) for the experiment with J-AS and M-AS. Quantities obtained from the 200 m-NHM simulation and the observation at the island of South-Daito (the location is shown in Fig. 10) are also plotted.

Fig. 14. Radial proﬁles of axisymmetric speciﬁc humidity at a height of 4820 m at 6, 12, 18, 24 hr for the experiment with J-AS and M-AS.

above the island at that time, moist static en- the middle and upper troposphere. The 200 m- ergy and speciﬁc humidity are calculated. The NHM result therefore is reliable, at least from quantities, superimposed in Fig. 13, are in close the middle to upper troposphere, where the agreement with those simulated by the CRM in difference in each quantity is unremarkable, October 2005 A. MURATA and M. UENO 761

Fig. 15. Time series of the sum of vertically accumulated cloud water and cloud ice obtained from the 200 m-NHM simulations with horizontal grid spacing stepped from 200 to 400 m in increments of 50 m. as mentioned above. At lower levels, however, 6.2 Sensitivity of CRM-simulated cumulus to moist static energy is smaller, and specific hu- criterion of the cumulus judgment midity is larger compared to that produced by As mentioned before, we employ a criterion the CRM. That is, the observed air is relatively for the judgment whether a grid of the CRM cool and moist. It is possible that the discrep- belongs to cumulus or not. The definition of cu- ancies in the quantities between the observa- mulus is as follows: the sum of cloud water and tion and the CRM are attributed to the differ- cloud ice mixing ratios is larger than 0.1 g kg1, ence in data averaging. The observational data and vertical velocity is larger than 0 m s1.To are obtained at a location, whereas the CRM demonstrate the validity of the definition, we data are horizontally averaged. conduct sensitivity tests. The sensitivity of the simulated cloud to the thresholds of the mixing 6. Discussion ratio, 0.01, 0.1, and 1 g kg1, is assessed by 6.1 Sensitivity of the CRM results to runs Q001, Q01, and Q1, respectively. In these horizontal grid spacing runs, the threshold of vertical velocity is not We have been employed the simulations with changed. The runs W0, W2, and W4 test the 200-m grid spacing as ‘‘truth’’ data. To demon- sensitivity of the model cumulus to variation in strate the validity of using the data, we per- the thresholds of the vertical velocity, 0, 2, and form sensitivity experiments on horizontal grid 4m s1, respectively, without changing the spacing; stepped from 200 to 400 m in incre- mixing ratio threshold. ments of 50 m. Figure 15 shows the time evo- Figure 16 shows the vertical profiles of buoy- lution of the sum of vertically accumulated ancy estimated from (3.8). The analyzed case cloud water and ice. It is found from the figure corresponds to that of Fig. 7a. From Fig. 16a, that the evolutions in the 200- and 250-m grid it is found that cloud base in the run Q1 is length are nearly equal, compared to those in located at a height of 1 km, indicating that be- coarser-mesh simulations. The convergence of low the height there are no grids where mixing the result allows us to regard the 200-m spac- ratio of cloud water is more than 1 g kg1. The ing as a control. 1-km height of cloud base is probably higher 762 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 16. Vertical proﬁles of buoyancy acceleration, with respect to the cloud of Fig. 2a, showing the sensitivity to the thresholds of (a) the sum of cloud water and cloud ice mixing ratios, 0.01, 0.1, and 1gkg1, and (b) vertical velocity, 0, 2, and 4 m s1.

than that in real convective clouds in tropical First the vertical profiles of cloud mass fluxes cyclone rainbands. The runs Q001 and Q01 are are compared (Fig. 17a). It should be noted not so different in the profile, except for near that the mass flux in each level has been nor- cloud base. The threshold of 0.01 g kg1 is malized by the quantity at cloud base. Thereby probably too small, considering previous stud- vertical profiles, not the absolute values, are ies (e.g., Tao et al. 1993). The threshold of ver- focused. The profile of the normalized cloud tical velocity, on the other hand, does not af- mass flux in the CRM has the maximum at a fect the profile so much, compared to the other height of 4.5 km. Another feature of the CRM threshold, although the profiles are slightly mass flux is its rapid increase from cloud base different below a height of 2 km. The threshold to a height of 2 km. The features are also rep- of 0 m s1 is adopted in the present study, so resented in the mass flux in the M-AS result, that relatively weak updrafts, which probably although the peak value is larger than that in appear near the cloud base and top, can be the CRM one, and the values above a height of captured. 5.5 km are overestimated. The mass flux in the J-AS result, on the other hand, increases line- 6.3 Properties in convective updrafts arly from cloud base to top, which is completely In order to support the modified Arakawa- different from the profile provided by the CRM. Schubert formulation, we compare updraft Thermodynamic properties in the cumulus properties obtained from the coarser-mesh updrafts of the M-AS experiment are more con- model with the parameterizations (i.e., J-AS sistent with those in the CRM result. From a and M-AS), to those from the CRM. The prop- comparison of moist static energy (Fig. 17b), erties compared are cloud mass flux, moist it is found that the rapid decrease in the quan- static energy and specific humidity. We use the tity from cloud base to a height of 3 km, and data of the cumulus that has the 8-km cloud vertical uniformity between 3- and 6-km high, top in the targeted rainband described pre- are represented well in the M-AS simulation, viously. The cloud corresponds to the case of although the absolute values are larger than Fig. 2a in the results of the CRM. the CRM one. Underestimates of specific hu- October 2005 A. MURATA and M. UENO 763

Fig. 17. Vertical profiles of (a) normalized cloud mass flux, (b) moist static energy, and (c) specific humidity, within a convective cloud in the targeted rainband at 23 hr (1100 UTC 12 September) for the experiment with J-AS and M-AS. Quantities obtained from a cloud simulated by 200 m-NHM, corresponding to Fig. 2a, are also plotted. midity above a height of 5 km in the J-AS re- bounded by circles with 250- and 500-km radii sult is slightly improved in the M-AS one (Fig. centered on the forecast storm location, are 17c). shown in Fig. 18. From Fig. 18a, it is clearly found that, except for the lowest 2-km levels, 6.4 Components of temperature and moisture the temperature changes are less in the experi- tendencies in parameterization ment with M-AS compared to those in the J-AS The difference in the amount of moisture, experiment. Specific humidity changes (nega- which is evident over the outer area, between tive values) are also greatly different in the the two experiments (i.e., J-AS and M-AS), has middle and upper troposphere (Fig. 18b). The already arisen by the 6 hr as seen in Fig. 14. To peak is located at an upper level (i.e., 7-km explore the cause of the discrepancy, we evalu- high) in J-AS, whereas the M-AS peak is not so ate the changes of temperature and moisture remarkable. derived from cumulus parameterization. The In order to investigate the cause of the dif- vertical profiles of the changes between 1 and ference in the changes of temperature, T, and 6 hr, as the average within the annular area specific humidity, q, between the two experi- 764 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 18. Vertical profiles of the change of (a) temperature and (b) specific humidity due to cumulus parameterization between the 1 and 6 hr for the experiment with J-AS and M-AS. ments, we divide each change into several com- The effect of cloud mass flux is the greatest ponents, as follows: among the three terms of the right-hand side in each equation. The vertical profiles of those X X X qT M qT D Le components in (6.2) are shown in Fig. 19. The ¼ i þ i ðT TÞ i ; qt r qz r i c mass flux term brings about negative changes i i i P in specific humidity in both experiments, ð6:1Þ whereas other two terms prevent the negative qq X M qq X D X changes. The difference in the magnitude of ¼ i þ i ðq qÞþ e ; ð6:2Þ qt r qz r i i the mass flux term between J-AS and M-AS i i i predominantly accounts for that in the total change. The remarkable difference in the peak qMi Di ¼ ; ð6:3Þ (negative value) height of the mass flux term qz between the two corresponds to that in the where M is cloud mass flux, r is density, z is total moisture change. height, L and cp are the constants of latent heat The difference in the peak height of the mass release and specific heat at constant pressure, flux term is mainly explained by that in the and e is evaporation rate. In the equations, over profile of mass flux itself. The net mass flux, bar denotes grid-scale (non-parameterized) which is almost accounted for by upward mass quantities and subscript i denotes cloud type, flux, has the maximum at a height of 9 km in categorized in terms of cloud-top height. The J-AS, whereas 2 km in M-AS (Fig. 20). In the first, second, and last terms of the right-hand latter case, upward mass flux decreases with side in each equation, (6.1) and (6.2), represent height above the peak, located at relatively low the effects of cloud mass flux, detrainment, and altitude (i.e., 2 km). In particular, the rapid de- evaporation, respectively. The second term, in crease is noticeable between 3- and 7-km high. the M-AS case, includes the effect of lateral de- This rapid decrease is primarily attributed to trainment in addition to cloud-top detrainment. the lateral detrainment occurred at the levels. Note that all the terms in each equation stand In the former, on the other hand, the peak of for the quantities that represent the cloud en- upward mass flux is located in higher levels semble (i.e., ensemble of all cloud types). because no lateral detrainment is included October 2005 A. MURATA and M. UENO 765

Fig. 19. Vertical proﬁles of components of speciﬁc humidity change due to cumulus parameterization between the 1 and 6 hr for the experiment with (a) J-AS and (b) M-AS.

Fig. 20. Same as in Fig. 19, but components of cumulus mass ﬂux.

in the scheme. It should be noted that, in the hand side in (6.2) are smaller. However, the present case, the areally averaged cloud-base two experiments are noticeably different in the mass flux does not vary so much with the two effect of detrainment, the second term, whereas experiments (Fig. 20). The modified entrain- the profiles of the evaporation effect, the last ment rate brings about the weaker mass fluxes term, are similar. The detrainment term tends above the cloud base, thereby the magnitude of to prevent drying, because the term enables the drying term in (6.2) becomes smaller. water vapor to be released laterally from the The magnitudes of other terms of the right- cloud into the environment. 766 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Fig. 21. Vertical profiles of cumulus mass flux for the experiment of (a) J-AS and M-AS at 23 hr, and (b) 200 m-NHM result averaged from 16 min to 21 min 30 sec. Profiles for describing a plus or minus sigma (standard deviation) are also shown in (b).

The height of the peak mass flux in M-AS (4.10) in the experiments D3, D5, and D7 are is consistent with that represented by the 0.3, 0.5, and 0.7, respectively. CRM. Both results exhibit the peak height in The drying over the troposphere is reduced, lower troposphere (i.e., 2-km high) (Fig. 21b), with increasing the magnitude of the lateral whereas in upper troposphere (i.e., 8-km high) detrainment (Fig. 22a). The result, however, is in J-AS (Fig. 21a). Above the altitude, the mass not primarily attributed to the direct effect of flux in the 200 m-NHM simulation decreases detrained moist air, but rather to the magni- with height except for few levels, suggesting tude of cloud mass flux [i.e., the first term of the that lateral detrainment from cumulus occurs right-hand side of (6.2)]. From the vertical pro- in several layers. The feature is reproduced files of cloud mass flux (Fig. 22b) it is found only by the M-AS result. The variation of cloud that the quantity decreases as the detrained air mass fluxes by a factor of 100 between Fig. 21a increases. The peak height of the mass flux is and b is attributed to the difference in the areas located at 2 km. Above the layer, the mass flux where the averaged mass fluxes are calculated. decreases with height ranging from 2- to 7-km The rate of the occupation of the area by con- high, suggesting the effect of lateral detrain- vective clouds is much lower in the former than ment. The rate of the decreasing mass flux the latter, leading to smaller mass flux in the is the highest in the experiment D7 and the former. lowest in D3. 7. Summary and conclusions 6.5 Sensitivity of temperature and moisture tendencies to parameters in new Numerical simulations of a tropical cy- entrainment profile clone rainband are conducted using a high- We conduct sensitivity experiments to con- resolution, three-dimensional cloud-resolving firm that the degree of drying induced by the model (CRM), for investigating the vertical cumulus scheme decreases with the amount of profiles of fractional entrainment rate as to lateral detrainment. The magnitude of the de- cumulus convection. On the basis of the find- trainment, defined in (4.10), varies depending ings obtained from the CRM simulations, a new on the experiments as follows: the values of k in method for the treatment of entrainment rate October 2005 A. MURATA and M. UENO 767

Fig. 22. Vertical profiles of (a) specific humidity change and (b) mass flux due to cumulus parameterization in the sensitivity experiments.

profiles is proposed for use in a cumulus pa- of cloud amount contribution, the effect of rameterization. In the present simulations, the cloudy area increasing with the growth of a Meteorological Research Institute/Numerical cumulus seems to be responsible for larger en- Prediction Division Nonhydrostatic Model trainment rate. Since cloud amount is rela- (MRI/NPD-NHM) (Saito et al. 2001) with the tively small near cloud base and top, the effect 200-m horizontal resolution (200 m-NHM) is is enhanced there. employed as the CRM. Negative values of entrainment rate shown Fractional entrainment rate, derived from in levels between cloud base and top indicate the calculation based on the vertical gradient of lateral detrainment from cumulus. The nega- cloud mass flux, is estimated using the output tive entrainment rates are attributed to both of 200 m-NHM. The results of the calculation updraft and cloud amount contributions. The clearly show larger entrainment rates just heights where the former term is negative cor- above cloud base, and just below cloud top, and respond to those where buoyancy is negative. show smaller entrainment rates in between. The latter term is explained by the local differ- The characteristic features seem to be indepen- ential of cloud amount in time and height. The dent of cloud-top height. result implies the cloud is during the stage of Entrainment rate is divided into three terms decaying. for analyzing it in detail. The term of updraft For the application of the vertically variable contribution is proportional to the sum of verti- entrainment rate, to the Arakawa-Schubert cu- cal forcing, and is inversely proportional to the mulus parameterization, the vertical profiles of square of vertical velocity. Since convective up- the entrainment rate for the steady-state cloud draft, horizontally averaged over a cumulus, model are estimated. The results of the calcu- tends to have a minimum at cloud base and top, lation show qualitatively the same features as in addition the vertical forcing does not change in the non-steady cases. That is, the character- vertically so much as the square of the updraft istic features include larger entrainment rates velocity, the updraft term is larger there. In near cloud base and top, although the peak between, the term is smaller because the up- near cloud top is not so remarkable. draft is relatively strong there. As for the term The Arakawa-Schubert scheme used opera- 768 Journal of the Meteorological Society of Japan Vol. 83, No. 5 tionally at the Japan Meteorological Agency Updraft properties, obtained from the (J-AS) is modified on the basis of the findings coarser-mesh model with the parameteriza- about the fractional entrainment rate men- tions and from the CRM, are also compared. tioned above. The difference between J-AS and The profile of the normalized cloud mass flux in the modified scheme (M-AS) is in their vertical the CRM has the maximum at a height of profiles of entrainment rate. M-AS employs the 4.5 km. As compared with the result with J-AS, vertically variable entrainment rates, repre- cloud mass flux, moist static energy, and spe- senting larger near cloud base and top, for each cific humidity provided by the experiment with spectrum (categorized in terms of cloud-top M-AS are more consistent with those in the height) of cumulus. The entrainment rates are CRM result. smaller between cloud base and top, and are It is evident that the CRM is a significantly even negative, intended as lateral detrainment, useful tool for evaluating the vertical profile of in a part of the cloud layer. The features are fractional entrainment rate. The use of CRMs assumed to be represented by the combination is expected to be effective in the verifications of of linear functions for simplicity’s sake. other parts of cumulus parameterizations, such For investigating the effect of M-AS, simu- as trigger functions. It is also necessary that lations of typhoon Saomai (2000) are conducted arbitrary parameters in the present scheme using MRI-NPD/NHM with the 20-km horizon- are verified. Since CRMs enable us to simulate tal resolution (20 km-NHM). In the simula- various cumuli realistically, the physical mech- tion with M-AS, the moisture field simulated anisms of entrainment and detrainment in the around the storm is improved as compared clouds can be investigated. If the mechanisms with that in the J-AS simulation, showing that are fully understood, we may find the relation- the vertical profiles of specific humidity ap- ship between the arbitrary parameters and proach those produced by the CRM (200 m- cloud-scale or grid-scale variables, thereby de- NHM). The underestimate of moisture, a dis- termining the values of the parameters. In the advantage of simulations with J-AS, is reduced present study, convective clouds in a particu- from the middle to upper troposphere. lar phenomenon, that is, a tropical cyclone The tendency of specific humidity, due to cu- rainband, are investigated exclusively. Cumu- mulus parameterization, is estimated for ex- lus within various phenomena and situations ploring the cause of the difference in moisture should be treated for the generalization of the between the two results (i.e., with J-AS and obtained results. M-AS). The tendency is divided into four components, effects of cloud mass flux, cloud-top Acknowledgements detrainment, lateral detrainment, and evapo- The authors acknowledge M. Nakagawa and ration. The vertical profiles of the components A. Shinpo for supplying computer codes and for demonstrate that the effect of the mass flux is their useful comments. The authors are grate- the greatest, and the peak height of the term ful to anonymous reviewers and the editor for corresponds to that in the moisture tendency. their valuable comments on the manuscript. The difference in the peak height between the The numerical experiments are performed us- two simulations is predominantly explained by ing the HITAC SR8000 computer system at that in the profile of the mass flux, which is Meteorological Research Institute. greatly influenced by lateral detrainment. To examine the effect of mass flux on the References moisture tendency, we conduct sensitivity experiments changing the degree of lateral de- Alexander, G.D. and W.R. Cotton, 1998: The use of trainment. With increasing the magnitude of cloud-resolving simulations of mesoscale convective systems to build a mesoscale parame- the detrainment, the drying over the tropo- terization scheme. J. Atmos. Sci., 55, 2137– sphere is suppressed. The result is primarily 2161. attributed to cloud mass flux reduced above the Arakawa, A. and W.H. Schubert, 1974: Interaction of levels of lateral detrainment, not to the direct a cumulus cloud ensemble with the large-scale effect of detrained moist air from inside the environment, Part I. J. Atmos. Sci., 31, 674– clouds. 701. October 2005 A. MURATA and M. UENO 769

Baik, J.-J., M. DeMaria, and S. Raman, 1991: trainment processes of simulated cumulus en- Tropical cyclone simulations with the Betts semble. Part II: Testing the entraining-plume convective adjustment scheme. Part III: Com- model. J. Atmos. Sci., 54, 1044–1053. parisons with the Kuo convective parameter- ———, 1999: Some bulk properties of cumulus en- ization. Mon. Wea. Rev., 119, 2889–2899. sembles by a cloud-resolving model. Part II: Blyth, A.M., 1993: Entrainment in cumulus clouds. Entrainment profiles. J. Atmos. Sci., 56, 3736– J. Appl. Meteor., 32, 626–641. 3748. Braun, S.A. and R.A. Houze Jr, 1994: The transition Lin, Y.H., R.D. Farley, and H.D. Orville, 1983: Bulk zone and secondary maximum of radar reflec- parameterization of the snow field in a cloud tivity behind a midlatitude squall line: Results model. J. Climate Appl. Meteor., 22, 1065– retrieved from Doppler radar data. J. Atmos. 1092. Sci., 51, 2733–2755. Liu, C., M.W. Moncrieff, and W.W. Grabowski, Cohen, C., 2000: A quantitative investigation of en- 2001a: Hierarchical modelling of tropical con- trainment and detrainment in numerically si- vective systems using explicit and para- mulated cumulonimbus clouds. J. Atmos. Sci., metrized approaches. Quart. J. Roy. Meteor. 57, 1657–1674. Soc., 127, 493–515. Emanuel, K.A., 1991: A scheme for representing cu- ———, ———, and ———, 2001b: Explicit and mulus convection in large-scale model. J. At- parameterized realizations of convective cloud mos. Sci., 48, 2313–2335. systems in TOGA COARE. Mon. Wea. Rev., Gadd, A.J. and J.F. Keers, 1970: Surface exchanges 129, 1689–1703. of sensible and latent heat in a 10-level model Load, S.J. and A. Arakawa, 1980: Interaction of a atmosphere. Quart. J. Roy. Meteor. Soc., 96, cumulus cloud ensemble with the large-scale 297–308. environment, Part II. J. Atmos. Sci., 37, 2677– Gregory, D., 2001: Estimation of entrainment rate in 2692. simple models of convective clouds. Quart. J. Mashiko, W., K. Saito, and M. Ueno, 2001: Develop- Roy. Meteor. Soc., 127, 53–72. ment of a typhoon model using MRI-NHM Grell, G.A., 1993: Prognostic evaluation of assump- (Part II). Proc. the Spring Conference of Japan tions used by cumulus parameterizations. Meteor. Soc., 79, P270 (in Japanese). Mon. Wea. Rev., 121, 764–787. Moncrieff, M.W., S.K. Krueger, D. Gregory, J.-L. Grinnell, S.A., C.S. Bretherton, D.E. Stevens, and Redelsperger, and W.-K. Tao, 1997: GEWEX A.M. Fraser, 1996: Vertical mass flux calcu- Cloud System Study (GCSS) Working Group lations in Hawaii trade cumulus clouds from 4: Precipitating convective cloud systems. Bull. dual-Doppler radar. J. Atmos. Sci., 53, 1870– Amer. Meteor. Soc., 78, 831–845. 1886. Moorthi, S. and M.J. Suarez, 1992: Relaxed Houghton, H.G. and H.E. Cramer, 1951: A theory of Arakawa-Schubert: A parameterization of entrainment in convective currents. J. Meteor., moist convection for general circulation 8, 95–102. models. Mon. Wea. Rev., 120, 978–1002. Ikawa, M., H. Mizuno, T. Matsuo, M. Murakami, Y. Murakami, M., 1990: Numerical modeling of dynam- Yamada, and K. Saito, 1991: Numerical mod- ical and microphysical evolution of an isolated eling of the convective snow cloud over the Sea convective cloud.—The 19 July 1981 CCOPE of Japan.—Precipitation mechanism and sen- cloud—. J. Meteor. Soc. Japan, 68, 107–128. sitivity to ice crystal nucleation rates—. J. ———, T.L. Clark, and W.D. Hall, 1994: Numerical Meteor. Soc. Japan, 69, 641–667. simulation of convective snow clouds over the Kain, J.S. and J.M. Fritsch, 1990: A one-dimensional Sea of Japan; Two-dimensional simulation of entrainment/detrainment plume model and its mixed layer development and convective snow application in convective parameterization. J. cloud formation. J. Meteor. Soc. Japan, 72,43– Atmos. Sci., 47, 2784–2802. 62. Kato, T., 1995: A box-Lagrangian rain-drop scheme. Murata, A. and M. Ueno, 2000: The Effects of dif- J. Meteor. Soc. Japan, 73, 241–245. ferent cumulus parameterization schemes on Kuma, K., C.-H. Cho, C. Muroi, and M. Ueno, 1993: the intensity forecast of typhoon Flo (1990). J. Improvement of typhoon track forecast by the Meteor. Soc. Japan, 78, 819–833. JMA global model: An impact of cumulus pa- ———, K. Saito, and M. Ueno, 2003: The Effects of rameterization. Papers presented at the third Precipitation Schemes and Horizontal Resolu- Technical Conference on SPECTRUM. Report tion on the Major Rainband in Typhoon Flo No. TCP-33, WMO/TD-No. 595. World Meteor. (1990) Predicted by the MRI Mesoscale Non- Org., Geneva, Switzerland, IV1–8. hydrostatic Model. Meteor. Atmos. Phy., 82, Lin, C. and A. Arakawa, 1997: The macroscopic en- 55–73. 770 Journal of the Meteorological Society of Japan Vol. 83, No. 5

Numerical Prediction Division/Japan Meteorological Swann, H., 2001: Evaluation of the mass-flux Agency, 1997: Outline of the operational nu- approach to parametrizing deep convection. merical weather prediction of the Japan Mete- Quart. J. Roy. Meteor. Soc., 127, 1239–1260. orological Agency. 126 pp. [Available from Tao, W.-K., J. Simpson, C.-H. Sui, B. Ferrier, S. JMA, 1-3-4 Otemachi, Chiyoda-ku, Tokyo 100- Lang, J. Scala, M.-D. Chou, and K. Pickering, 8122, Japan.] 1993: Heating, moisture, and water budgets of Ooyama, K., 1971: A Theory on parameterization of tropical and midlatitude squall lines: Compar- cumulus convection. J. Meteor. Soc. Japan, 49, isons and sensitivity to longwave radiation. J. 744–756. Atmos. Sci., 50, 673–690. Randall, D.A. and D.M. Pan, 1993: Implementation Tiedtke, M., 1989: A comprehensive mass flux of the Arakawa-Schubert cumulus parameter- scheme for cumulus parameterization in large- ization with a prognostic closure. Meteor. Mon- scale models. Mon. Wea. Rev., 117, 1779– ogr., 46, 137–144. 1800. Raymond, D.J. and A.M. Blyth, 1992: Extension of Ueno, M., 1995: A study on the impact of asymmetric the stochastic mixing model to cumulonimbus components around tropical cyclone center on clouds. J. Atmos. Sci., 49, 1968–1983. the accuracy of bogus data and the track fore- Reuter, G.W., 1986: A historical review of cumulus cast. Meteor. Atmos. Phy., 56, 125–134. entrainment studies. Bull. Amer. Meteor. Soc., ———, 2000: Numerical prediction for tropical cy- 67, 151–154. clones. Meteor. Res. Notes, 197, 131–286 (in Saito, K., T. Kato, H. Eito, and C. Muroi, 2001: Doc- Japanese). umentation of the Meteorological Research Yuter, S.E. and R.A. Houze Jr, 1995: Three- Institute/Numerical Prediction Division Uni- dimensional kinematic and microphysical fied Nonhydrostatic Model. Tech. Rep. of MRI, evolution of Florida cumulonimbus. Part III: 42, p133. Vertical mass transport, mass divergence, Smith, R.K., 2000: The role of cumulus convection in and synthesis. Mon. Wea. Rev., 123, 1964– hurricanes and its representation in hurricane 1983. models. Rev. Geophys., 38, 465–489.