Structure of the Atmosphere in Radiative Convective Equilibrium

Transcription

1 15 JULY 2002 IWASA ET AL Structure of the Atosphere in Radiative Convective Equilibriu YOSHIHARU IWASA Frontier Research Syste for Global Change, Yokohaa, Japan YUTAKA ABE Departent of Earth and Planetary Science, University of Tokyo, Tokyo, Japan HIROSHI TANAKA* Institute of Hydrospheric Atospheric Sciences, Nagoya University, Nagoya, Japan (Manuscript received 3 August 2000, in final for 14 May 2001) ABSTRACT To investigate water vapor transport in an atosphere in radiative convective equilibriu, a siplified dynaical convection odel (DCM) was constructed that explicitly odels oist convection and longwave radiation in a gray atosphere. In the subsidence region of the equilibriu, atosphere is predicted by the DCM, dynaical heating, and radiative cooling balance. Subsidence that satisfies local therodynaical balance includes detrainent fro adjacent cuulus updraft at all levels in the free troposphere, fro high levels with a sall absolute huidity to low levels with a large absolute huidity. In this subsidence region, absolute huidity increases downward, but relative huidity is approxiately constant with height. This contrasts sharply with results fro a cuulus chiney odel (CCM) that liits detrainent to near the tropopause and produces drying in the free troposphere. To deonstrate the accuracy of the transport echanis iplied by the DCM, results fro a kineatic circulation odel (KCM) were exained. The DCM and the KCM both produced an atosphere far oister than predicted by the CCM. The feature of the detrainent at all levels of the free troposphere under a noral atospheric situation does not depend on the radiation schees used in the odels. Furtherore, an analytic solution of the huidity fields, obtained using a few additional assuptions on atospheric properties, agrees with the huidity fields in the DCM and KCM. The relative huidity in the subsidence region in the free troposphere has a ostly unifor vertical profile and the ean value in the horizontal is independent of the horizontal scale. Water vapor transport oistens the atosphere, preventing the excess drying that occurs in the CCM. 1. Introduction When describing the vertical distributions of teperature and huidity in the atosphere, one ust consider tropospheric radiation and convection to be in global equilibriu. Several paraeterizations of convective transfer of heat and oisture have been proposed for general circulation odels of the atosphere. Early paraeterizations were based on the concept of convective adjustent (Manabe and Strickler 1964; Manabe and Wetherald 1967; Huel and Kuhn 1981). How- *Current affiliation: Division of Earth and Environental Sciences, Nagoya University, Nagoya, Japan. Corresponding author address: Yoshiharu Iwasa, Frontier Research Syste for Global Change, Yokohaa Institute, Japan Marine Science and Technology Center, , Showa-achi, Kanazawaku, Yokohaa, Kangawa , Japan. E-ail: iwaasa@jastec.go.jp ever, the convective adjustent odel (CAM) a priori assigns final properties to the convection, including the lapse rate and the tropospheric relative huidity, and it does not explicitly paraeterize the physical process of individual cuulus convection. Subsequent paraeterizations have attepted to introduce the effects of individual cuulus convections deterinistically (e.g., Kuo 1965, 1974; Arakawa and Schubert 1974; Tiedtke 1989; Gregory and Rowntree 1990; Eanuel 1991; Moorthi and Suarez 1992; Hu 1997). The cuulus chiney odel (CCM, Lindzen et al. 1982; Lindzen 1990) was developed to odel the vertical distribution of huidity aintained by cuulus convection. Ascending air parcels in a narrow cuulus updraft with constant upward ass flux are assued to retain oist static energy and to lose condensed water as precipitation when the air saturates. Parcels are detrained near the tropopause and then, copensating for the cuulus updraft, descend in a wide subsidence do Aerican Meteorological Society

2 2198 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 ain. Water vapor is conserved and a balance between radiative cooling and dynaical heating is aintained during the subsidence. For this case, the ixing ratio in the entire troposphere is siilar to that of the air at the detrainent level near the tropopause. Thus, the relative huidity in the lower troposphere is very sall. Using the CCM, Lindzen (1990) claied that as greenhouse waring occurs, cuulus convection becoes ore active and the tropopause rises, lowering the teperature at the level of detrainent and aking the surrounding area drier. If the content of water vapor decreases, the atosphere becoes less opaque to longwave radiation. In this way, water vapor feedback can be negative, aeliorating global waring. Even if the teperature of the detrained air increases less than the tropospheric teperature itself, positive water vapor feedback is still weakened (Held and Soden 2000). Using a CCM with individual regions of updrafts and downdrafts, Satoh and Hayashi (1992) obtained an atosphere at radiative convective equilibriu for different values of cuulus updraft ass flux. In their odel, the water vapor feedback effect was positive or negative depending on the value of the cuulus updraft ass flux, an external paraeter that ust be deterined in advance. Many general circulation odels, however, show that relative huidity reains unchanged despite waring, suggesting that there is ore water vapor in the odel atosphere (e.g., Manabe and Wetherald 1975). Satellite observations of the atosphere also suggest the tendency for increasing aounts of water vapor in a wared heisphere (Rind et al. 1991). An effective approach to avoid convective paraeterizations is to construct a odel with explicit treatent of cuulus-scale dynaics. Modern high-perforance coputers allow tie integrations of a two-diensional odel in which oist convective processes are treated explicitly (e.g., Nakajia and Matsuno 1988; Krueger 1988; Cohen 2000). Held et al. (1993) first obtained an atosphere in radiative convective equilibriu using a two-diensional cuulus odel with explicit treatents of detailed cloud icrophysical processes, as well as radiative and convective processes. Sui et al. (1994) analyzed the detailed water cycle and heat budgets in the tropical environent using the sae kind of dynaical cuulus odel. We begin with a dynaical convection odel (DCM), which explicitly odels both radiation and convection, as done in Held et al. and Sui et al. Our purpose is not to reproduce a realistic atosphere, but to see if the chiney-type circulation doinates in the equilibrated atosphere, and to understand why the atosphere does not dry as in the CCM. To siplify the analysis, the odel will be as siple as possible while still yielding equilibriu between radiation and convection. We assue a two-diensional atosphere with no large-scale forcing. Cloud icrophysics and interactions between the atosphere and solar radiation are ignored. The assuption of the gray atosphere for longwave radiation is justified by showing that results are unaffected by the radiation schee. The siplifications used are suitable for our purposes. They do ake the odeled atospheric state soewhat different fro observations. We will concentrate on the evolution of the tie ean circulation of ass and water vapor; this circulation is responsible for the final water vapor distribution at equilibriu. Coparison between the results of the DCM and a nuerical kineatic circulation odel (KCM), in which no explicit dynaical transports of ass and water vapor are assued outside the cuulus cloud, confirs that the circulation echanis constructed in this study deterines the water vapor distribution of the free troposphere. The KCM is constructed in an analytical for with a few liiting assuptions. The circulation and the echanis controlling the equilibrated atosphere are different fro those in the chiney type. This paper describes the circulation echanis; the waring feedback issue is the subject of a future report. Section 2 describes the siple DCM. Results fro the DCM are analyzed in section 3 to extract the circulation echanis responsible for water vapor distribution in the free troposphere. Based on that echanis, the KCM is constructed in section 4 and its accuracy in predicting water vapor transport is discussed in section 5. We derive an analytic solution for water vapor distribution sustained by the circulation echanis in section 6. This echanis is copared to the chineytype circulation echanis in section 7. Conclusions follow in section Description of the DCM a. Moentu equations This section describes the siplified two-diensional DCM. The odel includes the anelastic approxiation to exclude sound waves. The continuity equation is (u) (w) 0, (1) x z where u(x, z, t), w(x, z, t), and (z) are horizontal and vertical velocities, and horizontal ean atospheric density, respectively. The bar notation represents horizontal ean values. The density profile (z) is represented by values for an atosphere in radiative equilibriu (appendix A). Moentu equations are given by (u) u (u) w (u) uw t x z z 2 4 p (u) (u) x2 2 x4 4 K K x x x [ ] (u) Kz and (2) z z

3 15 JULY 2002 IWASA ET AL (w) u (w) w (w) w 2 t x z z 2 p (w) x2 2 g 0.608q K z x [ 4 (w) (w) x4 4 z K K (3) x z z ] in the horizontal and vertical directions, respectively, where g 9.8s 2 is the gravitational acceleration, and the constant is (R /R) 1 for the gas constants, R JK 1 for water vapor, and R JK 1 for dry air. Pressure is denoted by p, (x, z, t) is potential teperature defined below, and q (x, z, t) is water vapor ixing ratio. The constants K x s 1 and K x s 1 are the second- and fourth-order horizontal eddy diffusion coefficients, respectively. Here K z ( 2 s 1 )atz() is a vertical eddy diffusion coefficient given by z K z(z) 2.5 exp. (4) The diffusive ters are introduced to avoid nuerical instability. If we introduce a streafunction (x, z) and a vorticity (x, z, t) that satisfy u z (5) w and x (u) (w), (6) z x Eqs. (1), (2), and (3) can be transfored into a vorticity equation 1 D1 D2 t z x x z x z x 2 1 q g K x2 2 x x x 4 x4 4 z K K (7) x z z and a Poisson equation 2. (8) In Eq. (7) D 1 (z) and D 2 (z) are factors associated with the atospheric density stratification, given by 2 D1 and (9) 2 z [ ] D2, (10) 2 2 z z respectively. To reduce the reflections at the upper boundary, a daping factor exp(dt) (11) with D(s 1 )atz() given as D(z) {( ) exp[( )z]} (12) is applied to the vorticity at each tie step t. The daper is ost effective within the upperost 5 k. However, the daper forcibly weakens gravity waves, allowing unrealistic and undesirable ean wind oscillations. The oscillations are avoided by eliinating the 0-wavenuber coponent of the horizontal wind at every tie step, following the ethod of Held et al. (1993). b. Teperature equations Potential teperature (x, z, t) is defined as T, (13) where T(x, z, t) is atospheric teperature, (z) [p(z)/p ] R/c p 00 is the Exner function, in which c p 7R/2 is the specific heat of air at constant pressure, p(z) is the horizontal ean atospheric pressure, and p 00 is the atospheric surface pressure assued to be 1000 hpa. The change in potential teperature is 1 L T u w q c t x z c t p 2 4 Kx2 K x4, (14) 2 4 x x where q c (x, z) is the ixing ratio of condensed water, L Jkg 1 is the latent heat of condensation of water, and T/t R (x,z) is the radiative heating rate defined below. c. Water vapor equations The change in the ixing ratio q (x, z, t) is written as q q q u w q t x z c 2 4 q q Kx2 K x4. (15) 2 4 x x R

4 2200 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 Water vapor condenses to liquid only when the ixing ratio exceeds the saturated value. Thus, q ax(q q*, 0), c (16) where q* is the saturation ixing ratio. Tetens forula 380 T 273 q*(p, T) exp17.27 (17) p T 36 is used to obtain the saturation ixing ratio q* at pres- sure p(pa) and teperature T(K). For siplicity, we ake the pseudoadiabatic assuption that condensed liquid water is iediately reoved fro the atosphere as precipitation. d. Equations for longwave radiation The radiative equations are siilar to those of Nakajia et al. (1992). We assue that the atosphere interacts with longwave radiation only in the vertical direction in each horizontal grid. Moreover, we assue that the atosphere is gray (i.e., no wavelength dependence) for longwave transission. Water vapor and noncondensable greenhouse gases (NGG) are the absorptive constituents in this odel. For siplicity, NGG are assued to be a single constituent. NGG does not change phase in the atosphere. The ixing ratio q n is constant throughout the atosphere. Therefore, the optical depth at an arbitrary height z at a specified horizontal location x for longwave radiation easured downward fro the top of the atosphere is given by z ztop (x, z) (z)[k q(x, z, t) kq] dz, (18) n n where k and k n are the extinction coefficients for water vapor and NGG, respectively. Longwave radiation transfer follows Schwarzschild s law: di() I() B(), (19) d where, I(), and B() are optical depth, radiative intensity, and the Planck function, respectively, all integrated over the entire longwave radiation wavelength range. The Planck function is 4 B T, (20) where W 2 K 4 is the Stefan Boltzann constant. The transfer equation (19) is solved for a gray atosphere (Liou 1980; Goody and Yung 1989) to give the net upward longwave radiative heat flux F R at an arbitrary optical depth as FIG. 1. Scheatic of the configuration of the two-diensional dynaical convection odel. ] d[b()] 3 [ d 2 ] 0 d[b()] 3 [ d 2 ] [ 3 2 ] 3 surf bot bot [ F R () (B B ) exp ( ) 2 bot top exp ( ) d exp ( ) d top (B 0) exp ( 0), (21) where the superscripts surf, bot, and top indicate the values at the ground surface, and at the botto and top of the atosphere, respectively. In this paper, a superscript is generally used to represent the position at which the variable takes a value; a subscript is used to represent the kind of variable. The Planck functions B i (i surf, bot, and top) are calculated by (20) using the respective teperatures T i. The radiative heating rate is obtained fro the net upward longwave radiative heat flux F R as T 1 F R. (22) t cp z R e. Model paraeters and boundary and initial conditions The calculation doain is X 128 k wide and z upper 25 k high, and is shown in Fig. 1. Grid points are every x 1 k horizontally and every z 250 vertically within the calculation doain. Cyclic lateral boundaries are assued for all physical variables: (x 0) (x X), (23) where is an arbitrary physical variable. The upper (z

5 15 JULY 2002 IWASA ET AL z upper ) and surface (z 0) boundaries are free-slip rigid walls: u upper 0, w 0 at z 0 and z z. (24) z The boundary conditions for vorticity and streafunction are thus upper 0, 0 at z 0 and z z. (25) The upper boundary transits radiative heat fluxes, but prohibits dynaical heat fluxes. The atosphere in radiative equilibriu extends above the upper boundary (appendix A). The surface is assued to be saturated at teperature T surf (x): surf surf q (x) q*[p, T (x)], 00 (26) as calculated by (17). Since the incident shortwave solar radiation does not interact with the atosphere, the downward shortwave heat flux has a constant value F R0 throughout the atosphere. If the atosphere is in a radiative convective equilibriu, the total net upward heat flux ust also have the constant value F R0 throughout the atosphere to balance the constant downward shortwave heat flux. Thus, the surface boundary is set to have a total net upward heat flux, coposed of the surf surf sensible F (x), latent F (x), and net upward longwave S surf F R L radiative (x) heat fluxes, equal to the constant value F R0 at every horizontal grid point throughout the doain. This surface boundary condition together with the variable heat flux of outgoing longwave radiation (OLR) at the upper boundary allows us to define exactly the energy budget of the odel, and to deterine the equilibriu state when the net OLR heat flux approaches F R0. The surface boundary condition can be written as surf surf surf F S (x) F L (x) F R (x) F R0. (27) The respective heat fluxes at the surface on the left-hand side of (27) are written as surf bot surf F S Ksurf c p[t T(z 1)], (28) surf bot surf F L Ksurf L[q q (z 1)], and (29) surf surf bot F R (B B ) [ ] [ ] 3 2 top 0 d[b()] 3 exp ( bot ) d d 2 bot top bot (B 0) exp ( 0). (30) TABLE 1. Abbreviations for the experiental scenarios and the values of the controlling optical paraeters, together with the initial state properties used in the DCM. Scenario STD XTR k ( 2 kg 1 ) k n q n ( 2 kg 1 ) Tropospheric relative huidity (%) 75 Teperature at the botto of the atosphere (K) Tropopause height (k) Total optical depth The iddle level of the lowest atospheric layer is z 1, and K surf (x) is the surface transport coefficient deterined fro the absolute value u(x, z 1, t) of horizontal velocity at z 1 as K surf(x) ax[c D u(x, z 1, t), K surf in ], (31) where C D is a drag coefficient. The iniu transport coefficient K s 1 surfin is introduced to aintain the dynaical heat supply fro the surface even when convective otion vanishes. Surface teperatures T surf (x) are nuerically deterined by solving Eqs. (26)(31) siultaneously. The OLR heat flux at the upper boundary F OLR (x) F R (x, upper ) is then obtained fro Eq. (21). When the difference between F R0 and the horizontal ean OLR heat flux X 1 upper upper F F ( ) F (x, ) dx (32) OLR R R X 0 becoes negligible, we define the odel as being in equilibriu. To provide the current global ean condition, we assue F R0 F s (1 A)(S c /S s ) 240 W 2, where F s 1365 W 2, A 0.3, S c R 2, and S s 4R 2 are the solar constant, albedo, and areas of cross section and surface of the earth with radius R, respectively. The results for two different waring scenarios, standard (STD) and extra-enhanced (XTR), will be presented for convenience. Different scenarios have different values of k n q n, while the value of k is constant, as shown in Table 1. The values were chosen to provide the initial state of the atosphere for the STD scenario with appropriate properties (surface teperature, tropopause height, total optical depth of the atosphere, and waring sensitivity). The paraeter values chosen for the gray atosphere are validated in appendix B. The results fro a coparable one-diensional CAM (appendix A) provided the initial conditions. The CAM cannot deterine the water vapor distribution, so the relative huidity was set to zero above the tropopause and to a constant (75%) within the troposphere. Nuerical ethods used for odel integrations are suarized in appendix C. f. Transition to the equilibriu state Figure 2 shows the teporal evolution in the horizontal ean heat fluxes at the upper and lower boundaries for the STD scenario. The XTR scenario yielded siilar results. Convective cells initially appear at the botto of the atosphere, foring a convective bound-

6 2202 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 surf FIG. 2. Teporal change in the horizontal ean sensible ( F S, broken line), latent ( surf surf F L, dotted line), and radiative ( F R, lower solid line) heat fluxes at the surface boundary, and the horizontal ean OLR heat flux at the upper boundary ( F OLR, upper solid line), obtained by the DCM for the STD scenario fro the initial state to its equilibrated state. Each plot shows the running ean values averaged over 24 h every 6 h. ary layer (CBL). At first, the convective cells are sall and entirely dry. As tie elapses, the cells gradually enlarge and extend upward to the condensation level. The CBL persists at the botto of the odel atosphere fro this point throughout the tie integration. About 10 h into the integration, soe of the convective cells penetrate the top of the CBL and grow into the free atosphere. As one convective cell reaches the tropopause, the other cells are restricted to below the CBL height. Subsequent to the collapse of the first tall cuulus cell, other cuulus convective towers occasionally develop. The initial relative huidity in the troposphere is greater than the ean value at final equilibriu. Much water vapor is reoved fro the atosphere by condensation associated with cuulus convection. Latent heat release due to the condensation wars the atosphere in the first 4 days. Concurrently, the OLR heat flux at the upper boundary exceeds the incident solar heat flux F R0 and reaches a axiu value (260 W 2 ) on the fourth day. Since the OLR heat flux reains large, the atosphere gradually cools and reaches equilibriu after ore than 150 days in each experiental scenario. The tie integration was halted when the difference between the constant incident solar heat flux F R0 and the 10-day average of the horizontal ean OLR heat flux F OLR was less than 1 W 2 in each waring scenario. 3. Results of the DCM In this section, we describe a echanis responsible for the water vapor distribution in the free atosphere at equilibriu in the DCM. The XTR scenario is analyzed here because it produced a higher tropopause and presented a clearer structure of the troposphere than the STD scenario. The STD scenario does produce siilar structures, however. Section 3a describes the huidity distribution at equilibriu in the subsidence region of the free atosphere and shows that this distribution cannot be aintained by chiney-type convective echaniss. Then section 3b shows that the subsidence field is negligibly affected by the convection. The flow field associated with convection is localized in both tie and horizontal extent; furtherore, there is little buoyancy in the ean free atosphere. We also show that the subsidence field therodynaically balances the radiation field in section 3c. Based on these features, we then analyze water vapor transport at equilibriu in the subsidence region in section 3d fro the standpoint of a circulation regulated by radiative cooling. a. Huidity fields in the equilibrated free troposphere Figure 3 shows the two-diensional distribution of the tie ean huidity in the equilibriu state fro the DCM. Snapshot distributions, whether taken during a convective event or during a cal interval, are siilar to the ean shown. The distribution of ixing ratio q fors a triangle syetric about the cuulus updraft (at x 51 k in the figure) as shown in Fig. 3a. Held et al. (1993) present a siilar result. Water vapor in the free troposphere is ore abundant at low level and close to the cuulus updraft axis. Three regions of high relative huidities are present in the atosphere in Fig. 3b: at the top of the CBL, in the cuulus doain, and near the tropopause. The first two are aintained by shallow and deep convections, respectively. The last doain does not indicate uch water vapor, as shown in Fig. 3a, rather it is a anifestation of low teperatures (and low saturated ixing ratios) near the tropopause. The iddle troposphere shows a relative huidity distribution approxiately constant in the vertical that is dependent on the horizontal distance fro the cuulus updraft. A region of inial relative huidity in the lower free troposphere exists in the subsidence region (around x 111 k in the figure). The nonunifor distribution of the ixing ratio (Fig. 3a) in the subsidence region of the free troposphere suggests that water vapor enters the doain not only fro the tropopause as in the CCM, but also fro other regions. One possible path is upward fro the CBL. However, this odel contains no vertical diffusion coefficient of water vapor, and subsidence predoinates outside the cuulus updraft. Hence, the CBL is an unlikely direct source of water vapor. Another possible path is horizontal transport of water vapor across the flank of the cuulus updraft and into the subsidence doain. For nuerical stabilization purposes, the odel does include ters causing horizontal diffusion of water vapor. However, the horizontal diffusive flux convergence F x /x diff (kg 3 s 1 ) is too

7 15 JULY 2002 IWASA ET AL FIG. 3. Two-diensional distributions of the tie ean (a) water vapor ixing ratio and (b) relative huidity in the equilibrated state obtained by the DCM for the XTR scenario. sall copared to the downward flux divergence associated with subsidence F z /z subs (kg 3 s 1 ) to aintain the triangle-shaped distribution of ixing ratio around the cuulus tower. It is likely that convective otions associated with cuulus convection ove water vapor fro the cuulus updraft doain to the subsidence doain during convective events. As will be shown in the next section, however, the structure of a cuulus convective cell is neither siple nor steady. It was difficult to extract directly fro the turbulent convective odel flows trends in horizontal water vapor transport. b. Ineffective convective forcing in the subsidence region Figure 4 shows the tie evolution of the axiu upward and downward ass fluxes per unit area and

8 2204 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 4. Teporal variation in the axiu upward and downward ass fluxes per unit area and horizontal ean precipitation rate over 10 days at equilibriu obtained by the DCM for the XTR scenario. the horizontal ean precipitation rate for 10 days during a period of equilibriu. The fluxes and precipitation are synchronized, suggesting interittent cuulus developent. The typical duration of convective events is about 1 h. The cal periods between convective events typically last 10 h. Continuous shallow convection in the CBL aintains nonzero ass fluxes between the ain convective events. Figure 5 shows the evolution of streafunction during one convective event. As noted in previous studies (e.g., Held et al. 1993), successive convective developent recurs at the sae horizontal location where the first large cell appeared. A typical life cycle is as follows. In a cal atosphere (0 in), a sall convective cell grows (20 in) and reaches a ature phase (40 in). The cell then decays (50 and 60 in) and produces gravity waves (70 in) that ove rapidly through the entire atosphere in the final phase of the convective event. Figures 4 and 5 suggest both teporal and horizontal confineent to the convective effects. The total upward and downward ass fluxes in the odel ust balance the ass continuity. The horizontal width of the developing cuulus updraft is liited to a few grid points (1 4 k). If convectively forced downdrafts extend over the entire subsidence region ( k), the axiu value of the downward ass flux per unit area ust be saller (by a factor of 1/127 1/31) than the corresponding axiu value of the upward flux for each convective event. In each convective event, however, the axiu value of the downward flux is saller than the corresponding axiu value of the upward flux by a factor of only 1/2 1/3 (Fig. 4). Moreover, the axiu value of the downward flux appears iediately adjacent to the updraft doain in the developing phase (e.g., at the grid 3 k fro the updraft center at 40 in in Fig. 5). This suggests that the convective downdraft is essentially liited to be within a horizontal width two or three ties the cuulus updraft width; that is, it does not extend over the entire subsidence region. The horizontal localization of convective circulation is anifest as wedge-shaped streafunction distributions at 20 and 40 in in Fig. 5. At 40 in, the downdraft associated with the cuulus convection 10 k or ore in the horizontal fro the cuulus updraft center is as weak as the perturbations associated with the gravity waves shown at 70 in in Fig. 5. The tie ean streafunction is shown in Fig. 6. In FIG. 5. Typical changes in the streafunction during a convective event in the equilibrated state obtained by the DCM for the XTR scenario. Light shades show clockwise circulation, while dark shades show counterclockwise circulation. The contour interval is 1000 kg 2 s 1. Here 0 in: cal phase (when a cuulus begins to develop), 20 in: early phase of developent, 40 in: ature phase, 50 in: phase with coplex cell structure, 60 in: phase with vertically folded cells, and 70 in: phase radiating gravity waves.

9 15 JULY 2002 IWASA ET AL FIG. 6. Tie ean streafunction in the equilibrated state obtained by the DCM for the XTR scenario. The contour interval is 60 kg 2 s 1. addition to a shallow circulation within the CBL, a deep circulation extends through the whole troposphere. The deep circulation does not have the wedge-shaped distribution characteristic of the convective flow field as shown in Fig. 5, but has detrainent through the cuulus flank at ultiple levels below the tropopause height. The differences between Figs. 5 and 6 suggest that a circulation other than a convective one ay develop in the tie ean flow field. In the CCM proposed by Lindzen, the tropopause height rises and the teperature at the detrainent level decreases, reducing the content of water vapor outside the cuulus cloud, when the cuulus updraft ass flux becoes large. The large updraft-forced ass flux gives rise to a large downdraft-forced ass flux outside the cuulus, causing adiabatic waring outside the cuulus. Satoh and Hayashi (1992) pointed out that in order to give negative water vapor feedback, Lindzen s CCM requires a very large cuulus updraft ass flux for the cuulus eventually to becoe negatively buoyant (downdraft doain warer than the cuulus updraft doain at the sae level). They claied that such a large cuulus updraft ass flux is unreasonable when the cuulus buoyancy is positive. Figure 7 shows vertical profiles of virtual teperature T ( q )T (33) FIG. 7. Vertical profiles of the tie ean virtual teperature within the troposphere along the centers of the cuulus updraft (broken lines) and subsidence (solid lines) regions in the equilibrated state obtained by the DCM for the XTR scenario. calculated fro the tie ean profiles of the water vapor ixing ratio q and teperature T at the centers of the cuulus updraft and subsidence regions (the brackets denote the tie ean values). The cuulus updraft doain is only slightly warer than the subsidence doain. The largest difference in the free tro-

10 2206 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 posphere is 0.45 K at k. Although the vertical profile of the tie ean virtual teperature ( q )T can be ore perturbed, the difference between two centers at the sae level is usually less than 0.5 K. The cuulus updraft is colder than the subsidence in ters of teperature T by up to 3.0 K. Alost no buoyancy is generated fro the tie ean field; no convective forces are effective in the tie ean field of the free atosphere at equilibriu. c. Subsidence consistent with radiative cooling We placed 100 inertia-free tracers uniforly throughout the calculation doain in the equilibriu state in the XTR scenario to deterine the transport paths of air parcels. As shown in Fig. 8, when tracers (A D) initially located within or just above the CBL enter the cuulus updraft region, strong updrafts associated with cuulus convection rapidly lift the tracers to higher levels. Note that although the convective events ay be different for each tracer, the paths are plotted together in the figure. Once the tracers reach their highest levels, they do not descend to lower levels iediately. Rapid downward otion near the cuulus updraft, as inferred fro the streafunction (Fig. 5), is not clearly observed. Such strong downdrafts are liited to a narrow region that ascends as the cuulus develops. The probability of the tracers staying in the narrow region is sall, and tracers are rarely observed in the downdraft. The rearkable downward otion predicted by the CCM to occur concoitant with cuulus developent in regions far fro the cuulus updraft is not observed. Tracers D, H, and J approach the cuulus at certain ties and ove away fro the cuulus at other ties as they descend. Such coplex tracer otions betray no definite trends in either siple entraining or siple detraining due to the turbulent convective flows. The ost coon feature of tracer oveent is gradual descent in the free troposphere outside of cuulus region, independent of cuulus developent. The tracer paths agree with the tie ean streafunction shown in Fig. 6. Tracers ejected at higher levels (e.g., tracers A, B, E, and F) ove ore horizontally, while those at lower levels (tracers C, K, and L) ove downward. Nevertheless, the tracers shown in Fig. 8 have siilar vertical velocities in the subsidence region of the free troposphere, typically sinking about 5 k in 10 days. This yields subsidence velocities w subs of approxiately 0.6 c s 1 and gives a corresponding copressional heating rate in excess of the lapse rate ( K 1 ) of the abient air T g (w subs) t subs c p (6 10 )(K s ) (K s ) 2.2(K day ). (34) Radiative cooling rate is calculated as the negative of (22) and has a typical value of 2.5 K day 1 for the XTR scenario in Fig. 9, showing surprisingly good agreeent with the excess copressive heating rate (34). The tracer experient suggests that subsidence is consistent with the radiative cooling. Sherwood (1996), Salathé and Hartann (1997), and Dessler and Sherwood (2000) have all observed siilar trajectories in radiative cooling studies with horizontal scales larger than ours, in agreeent with our results. d. Radiatively regulated circulation The tie ean value of the horizontal vapor flux is divided into two parts: qu qu q(u), (35) where the brackets and the pries denote the tie ean and the perturbation coponents, respectively. If the net convective circulation is weak, as suggested in section 3b, the second ter q ( u) on the right-hand side of (35) can be ignored in the subsidence region far fro the cuulus for long tie periods. Since the subsidence sees to balance the radiative cooling, as shown in section 3c, we evaluate the first ter q u on the righthand side of (35) by assuing the flow field is controlled by the radiative cooling. Suppose there is an air parcel in the subsidence doain without any convective forcing in the equilibriu state. Radiative cooling and adiabatic heating are the only processes working on this descending parcel. They ust balance as T g wr, or (36) t R c p 1 g T c t p R w, (37) R where T/t R is the radiative cooling rate [the negative of (22)]. The subsidence velocity w R of the parcel is controlled by the radiative cooling. It is a radiatively regulated velocity. Such a velocity is proportional to the radiative cooling rate. Multiplying Eq. (37) by the air density gives the radiatively regulated subsidence ass flux 1 g T c t p R w. (38) R Figure 10a shows the distribution of w R in the free troposphere derived fro Eq. (38), which is based on the distribution of the tie ean radiative cooling rate shown in Fig. 9. Figure 10b shows the tie ean distribution of the downward ass flux w observed for the sae state. Although the distribution shown in Fig. 10a lacks the cuulus updraft doain with its upward ass flux and fluctuations, both of which are ob-

11 15 JULY 2002 IWASA ET AL FIG. 8. Typical tracer paths every 5 in for 10 days in the equilibrated state obtained by the DCM for the XTR scenario. For each tracer, the with a letter and with a pried letter indicate the initial and final points, respectively. served in the distribution shown in Fig. 10b, the contours agree well in the subsidence doain. This agreeent and the inefficiency of the convective forcing shown in section 3b together suggest that the subsidence field is forced priarily by radiative cooling. A radiatively regulated downward ass flux w R in the earth s free troposphere, such as the one shown in Fig. 10a, is in general divergent. The density on the right- FIG. 9. Two-diensional distribution of the tie ean radiative heating rate in the equilibrated state obtained by the DCM for the XTR scenario.

12 2208 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 10. (a) Two-diensional distribution of the tie ean downward subsidence ass flux expected fro the radiative cooling field (Fig. 9). (b) That observed in the equilibrated state obtained by the DCM for the XTR scenario. hand side of (38) increases downward (appendix A) uch ore rapidly than the rest {product of the factor [(g/c p ) ] 1 and the radiative cooling rate T/ t R } varies in the vertical direction. Mass flux divergence in the subsidence doain induces horizontal ass flux convergence by ass continuity. Fro Eq. (1), we can express the relationship as ur (wr), (39) x z where u R is the tie ean coponent of horizontal ass flux forced by the radiative process. Due to syetry, u R is assued zero at subsidence centers x S x subs nx(n 0, 1) within the cyclically expanded horizontal range (X, 2X) for a subsidence center x subs obtained in the calculation region. The tie ean ass flux u R at an arbitrary point (x, z) is obtained by horizontal integration of (39) fro a subsidence center x S to x. The starting point x S and the

13 15 JULY 2002 IWASA ET AL direction of the integration are chosen such that the integration range (x S, x) contains no centers in the cuulus updraft regions. Integrating (39) in this way, while substituting (38), we have [ ] x g ur(x, z) (z) (x, z) z xs c p 1 T (x, z) dx. (40) t R Note that such a copensating horizontal ass flux should be induced fro the cuulus updraft doain, the only doain that can be horizontally divergent in the free atosphere. Figure 11a shows the distribution of the first ter q u R on the right-hand side of Eq. (35), generated by the radiatively regulated flow and coputed as the product of the ixing ratio q (shown in Fig. 3a) and the induced horizontal ass flux u R [obtained by (40)]. On the other hand, Fig. 11b shows the distribution of q u derived fro the tie ean vertical vapor flux q w in a way siilar to Eq. (40) as x q u(x, z) [q w(x, z)] dx. (41) z x S This equation is not strictly correct, because water vapor, unlike ass, is nonconservative because of condensation. However, condensation in the subsidence region occurs only near the tropopause, and the aount of water vapor condensing at that height is negligible. Equation (41) and Fig. 11b include both the tie ean coponent q u and the perturbed coponent q ( u), but Fig. 11a only includes the ean. In spite of this difference, the distributions of the subsidence doain in Figs. 11a and 11b agree. The axiu horizontal vapor flux induced by the radiatively regulated circulation appears at z 6.5 k, at the center of the cuulus updraft region. If the cuulus updraft region is about 20 k wide, the level at which the actual axiu vapor detrainent through the cuulus flank occurs is lowered to z 5 k. This contrasts with the CCM assuption of detrainent being liited to near the tropopause. Agreeents between the subsidence ass flux fields (Fig. 10) and between the horizontal vapor flux fields (Fig. 11) suggest that the radiatively regulated circulation deterines the air otion and huidity distribution within an atosphere in radiative-convective equilibriu. 4. Description of the KCM If the circulation echanis described in the previous section controls the net transport of water vapor, it should be possible to drive a circulation and to build a water vapor distribution in the subsidence region siilar to that obtained by the DCM, without circulations driven by cuulus convection. Here, we construct an entirely kineatic circulation odel in which the circulation in the subsidence region is not driven by the cuulus convection (although we accept cuulus convection as a source of water vapor and as a teperature adjuster), but is driven by the radiation-controlling echanis presented in the previous section. a. Boundary conditions We consider a two-diensional doain as shown in Fig. 12. The doain has Nx width and z upper height, where N and x denote the nuber and the interval of the horizontal grids, respectively. The center of the cuulus region is located at the leftost grid x 0. The center of the subsidence region is located at the right end of the doain x x S (N 1/2) x. The calculation region is horizontally divided into cuulus (x/2 x x cu ) and subsidence (x cu x x S ) regions. All atospheric otions associated with cuulus convection are confined to within the cuulus region, and any direct convective transports of ass and water vapor are prohibited in the subsidence region. All of the physical fields are assued to be horizontally syetric about the fixed centers. Therefore, u(x) u(x) u(x S x) u(x S x) and (x) (x) (42) (x S x) (x S x) for the horizontal velocity u(x, z) of the atosphere and for a physical variable other than u, respectively. By syetry, air parcels cannot cross the centers: u(x 0) u(x x S ) 0. (43) The periodic horizontal length X is given as X 2xS (2N 1)x. (44) We assue a dry atosphere in radiative equilibriu extending above the upper boundary. If the upper or botto boundaries are rigid plates, a horizontal jet toward the cuulus region is artificially generated within the layer iediately adjacent to the boundary. External values of ass and vapor fluxes are extrapolated using second-order polynoials, so that air and water vapor can flow freely across the porous upper and lower boundaries. Teperature gaps do not exist between the botto of the atosphere and the surface boundary. b. Radiative process Radiative processes are treated the sae as in the DCM. Equations (18)(22) are the sae, except that (21) does not include the ter corresponding to the teperature gap at the surface boundary (T surf T bot ).

14 2210 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 11. (a) Two-diensional distribution of the tie ean horizontal vapor flux obtained fro the ixing ratio (Fig. 3a) and the distributions of horizontal ass flux induced by the radiatively regulated subsidence ass flux (Fig. 10a). (b) That expected fro the distribution of tie ean downward vapor flux obtained by the DCM for the XTR scenario. The doain of white color at the cuulus center includes values beyond the contour range. In a gray atosphere in radiative equilibriu with a net upward longwave radiative heat flux F R0, the surface surf ground teperature T rad and the botto teperature bot are required to have a teperature gap satisfying T rad F surf bot B(T ) B(T ) R0 rad rad, (45) where the Planck function B(T) is given by (20), to 2

15 15 JULY 2002 IWASA ET AL teperature profile T rad (z) there. In the troposphere, the relative huidity has a constant value h cu given as an external paraeter, and the vertical teperature profile is oist adiabatic T adiab (z): trop T rad(z) for z z T cu(z) (46) T (z) for z z trop and adiab FIG. 12. Scheatic of the two-diensional kineatic circulation odel. aintain the radiative heat flux near the surface. In addition, the DCM has a teperature gap at the surface boundary to provide a constant total net upward heat flux (27) there. Siilarly, the KCM requires a teperature gap T surf T bot 0 at the surface on each horizontal grid to aintain a natural profile of the radiative heat flux near the surface. Such a teperature gap is not included in the KCM, because the correct profile in this dynaics-free atosphere is unknown. Therefore, profiles of the radiative heat flux obtained by the KCM are distorted to a saller value near the surface. c. Convective adjustent The basic state of the atosphere is assued to be in radiative equilibriu as in the DCM. To incorporate convective effects (except for water vapor transport in the subsidence region), we apply a convective adjustent ethod (Manabe and Strickler 1964; Manabe and Wetherald 1967; Huel and Kuhn 1981) extended to the two-diensional doain. All of the variables (except for the huidity in the subsidence region) are deterined instantaneously at every tie step. In appendix C, we suarize the econoical ethod for aking calculations including the convective adjustent. We do not consider the detailed internal circulations within the cuulus doain, although their net fluxes will be diagnostically obtained as shown in appendix D. We expect the only to aintain the vertical profiles of teperature T cu (z) and ixing ratio q cu (z) ofthe cuulus cloud at appropriate values throughout the tie integration as follows. Vertical profiles in the cuulus region are horizontally unifor. The tropopause height z trop divides the atosphere of the cuulus region into the stratosphere (above z trop ) and the troposphere (below z trop ). The teperature profile is continuous at the tropopause. The cuulus region contains no water vapor in the stratosphere, and acquires radiatively equilibrated 0 for z trop z h q* (z) for z z trop cu cu, q (z) (47) cu where q* cu (z) is the saturation ixing ratio for the te- perature of the cuulus cloud. Here T adiab and q* cu are calculated by (A5) and (17), respectively. Section 3b showed that in the DCM, the equilibrated atosphere averaged over a long period generates little buoyancy. Accordingly, we can assue the virtual teperature T ( q )T (48) is horizontally unifor where q and T are the water vapor ixing ratio and teperature of the atosphere, respectively. Once the vertical profiles of teperature T cu and ixing ratio q cu are deterined in the cuulus region fro Eqs. (46) and (47), respectively, the teperature T in the subsidence region with water vapor ixing ratio q can be calculated as q cu(z) T(x, z) T cu(z). (49) q (x, z, t) The assuption of a horizontally unifor virtual teperature sets the atosphere in a dynaical equilibriu in which no buoyancy is present. Since both the vertical water vapor profile (47) in the cuulus region and the teperature distribution (49) for a given water vapor distribution q in the subsidence region depend on the vertical teperature profile (46) in the cuulus region, the botto teperature T cu (0) in the cuulus region can be an independent variable for the convective adjustent. Here T cu (0) is adjusted so as to balance the radiative budget, providing a horizontal ean OLR heat flux at the upper boundary X 1 upper upper F F (z ) F (x, z ) dx OLR R R X 0 x 1 S upper R F (x, z ) dx (50) x S 0 equal to a constant of net incident solar radiative heat flux F R0 : F OLR[T cu(0)] F R0. (51) d. Radiatively driven circulation If an air parcel in the subsidence region is in local therodynaical balance, the parcel should subside with a velocity w(x, z) such that the copressional

16 2212 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 heating rate excess over the lapse rate (x, z) of the environental atosphere is equal to the radiative cooling rate T/t R (x, z) of the parcel. As in Eq. (37), this relationship can be written as 1 g T c t p R w. (52) Multiplying (52) by the air density (z) yields the subsidence ass flux 1 g T c t p R w. (53) The continuity equation (1) for the atosphere is (u) (w). (54) x z The horizontal velocity u vanishes at the center of the subsidence region x x S fro the syetric boundary condition (43). Therefore, the horizontal ass flux u at an arbitrary point (x, z) in the subsidence region is obtained by integrating (54) horizontally fro x S to x. Substituting (53) into (54), we derive x 1 g xs z [ c p ] u(x, z) (z) (x, z) T (x, z) dx. (55) t R The ass fluxes w and u, respectively, given by (53) and (55), define an atospheric circulation around the cuulus. The circulation is generated within an atosphere balanced in the radiative budget [by (51)] by keeping dynaical and therodynaical equilibria in an air parcel within the subsidence region [by (49) and (52), respectively]. Therefore, the atosphere is driven not by direct convective forcing, but by radiative cooling. The circulation is a radiatively driven circulation, and the odel a kineatic circulation odel. e. Water vapor transport The only variable that develops in tie is the water vapor ixing ratio q (x, z, t) in the subsidence region. The equation for its teporal change is q q q u w q t x z c [ ] 2 4 q 1 q q 2 2 x4 4 K K, (56) x z z x where the constants K s 1 and K x s 1 are the second-order isotropic diffusion coefficient and the fourth-order horizontal diffusion coefficient, respectively. These values are still saller than FIG. 13. Horizontal ean profiles of (a) ixing ratio and (b) relative huidity converging to the equilibrated ones fro the initial conditions clear sky (solid lines) and all cloud (broken lines) in the KCM for the STD scenario. those used for the DCM. The diffusive ters are introduced only to prevent nuerical instability. When the ixing ratio exceeds its saturation value, vapor is condensed to liquid water following (16) as in the DCM. Condensed liquid water is reoved iediately. Latent heat liberated by the condensation in the subsidence region is ignored: the teperature field is assued to be instantaneously deterined by the convective adjustent (49), and the heating effect generated by the condensation in the subsidence region is negligible because of the sall aounts of water vapor condensation. f. Model paraeters, initial conditions, and convergence to equilibriu Optical paraeter values used were the sae as in the DCM for the STD scenario. The vertical height z upper 25 k is divided into 100 layers of thickness z 250. The nuber N and interval x of the horizontal grids are 32 and 2 k, respectively. The periodic horizontal length X (2N 1)x is 126 k. One horizontal grid is designated the cuulus region, providing a cuulus width 2x cu x 2 k and therefore a cuulus flank at x x cu 1 k. The relative huidity h cu in the cuulus doain is set to the value of 92.5%. This is the relative huidity vertically averaged over the free troposphere at the cuulus center obtained in the DCM for the STD scenario. The resulting relative huidity in the subsidence region is proportional to h cu. Starting with initial conditions ranging fro clear sky (no water vapor in the subsidence region) to all cloud (saturated troposphere in the whole subsidence region), the water vapor fields converge to the sae equilibriu solution, as shown in Fig. 13. The axiu error between the values of water vapor ixing ratio fro the different initial conditions are less than 3% over the whole troposphere after tie integrations exceeding 200

17 15 JULY 2002 IWASA ET AL days. The solution does not depend on either teporal or spatial resolution. 5. Results of the KCM In this section, we copare the structure of the troposphere obtained by the KCM with that obtained by the DCM STD scenario. Direct coparisons of soe atospheric features, such as precipitable water, teperature, and tropopause height, are not valid between the odels and are postponed to a pending report. This is because the KCM lacks the echanis to for a CBL with abundant water vapor and a dry-adiabatic lapse rate. However, experients of the KCM with CBL-like structure (not shown) indicate that the CBL has little effect on the circulation and the huidity distribution in the free troposphere. a. Huidity distributions Two-diensional distributions of ixing ratio q derived fro the DCM and KCM for the STD scenario are shown in Fig. 14. They are consistent in the free troposphere, showing triangular contours around the cuulus. The results near the surface are less reliable in the KCM. One reason for this is the obtained huidity distributions are distorted near the surface due to the absence of the teperature gap at the surface as entioned in section 4b. A second reason is the CBL should be fored near the surface in the realistic atosphere as obtained by the DCM. The contour profiles of relative huidity in Fig. 15 show agreeent in the free troposphere between the two odels. These suggest that the radiatively driven circulation in the KCM, without direct dynaical transport of water vapor in the subsidence region, is able to reproduce alost the sae huidity fields as the DCM. Both the DCM and KCM lack processes that ake the huidity distribution horizontally unifor in the real atosphere: horizontal advection of water vapor by the general wind, horizontal changes in cuulus location, and horizontal advection of condensed water that stays in the atosphere. If we included such huidifying processes in the odels, the iniu huidity would increase. Therefore, the iniu huidity generated in the odels can be interpreted as the iniu huidity that a ore realistic atosphere including the above processes would have. How dry can the equilibrated free troposphere be and how are the iniu values of huidity deterined at the subsidence center of the driest situation? Vertical profiles of the ixing ratio in the troposphere at the subsidence center fro the DCM and KCM, respectively, for the STD scenario are shown in Fig. 16a. The figure also shows the profile (labeled *CCM) produced by a chineylike echanis. The profile shows the ixing ratio of an air parcel in the KCM originating at the tropopause and subsiding in the troposphere. Note that it is not the profile of an exact CCM, because feedback occurring in the exact CCM is not considered. In contrast to the vertically constant profile of the *CCM, those in the DCM and KCM show that as air subsides, it gains water vapor. This suggests that air does not just subside fro the tropopause as assued in the CCM, but is also transported horizontally over the vertical range of the troposphere in the DCM and KCM. The KCM shows greater ixing ratios in the free troposphere than the DCM, as shown in Fig. 16a. It is related to a stronger greenhouse effect in the KCM: the KCM has higher teperature and hence larger saturation ixing ratios in the troposphere than the DCM. Huidity agreeent between the two odels is ore obvious in relative huidity than the absolute huidity. Figure 16b shows the vertical profiles of relative huidity corresponding to Fig. 16a. Since the KCM has a higher tropopause than the DCM, the profiles diverge in the upper troposphere. They agree in the iddle to lower troposphere, showing ostly constant profiles vertically. The DCM has the iniu value 12.0% just above the CBL (z k). The KCM has a siilar value (14.4%) at the sae level. In contrast, the chineylike echanis predicts a very dry troposphere, with a value of 5.5% at that level. The DCM has a relative huidity slightly less than the KCM because of drying due to the chineylike circulation of the interittently developing cuulus convection in the DCM; this convection is issing fro the KCM. If we take horizontal eans, the differences between the huidities in the DCM/KCM (increasing the relative huidity up to about 40%) and those of the *CCM (staying near the iniu values shown in the figure) becoe ever ore distinct. b. Subsidence flow field Radiative cooling distributed as in Fig. 17 directly controls the flow field in the KCM by Eq. (53). Vertical profiles of radiative cooling depend on the radiative schee. For exaple, radiative cooling in a gray atosphere tends to be focused on the level of unit optical depth, as shown in Figs. 9 and 17. More realistic nongray radiation schees would ake the profile ore unifor in the vertical; however, the flow fields are not greatly affected by the radiation schees (appendix B). This is because subsidence ass flux (53) also depends on the atospheric density as entioned in section 3d. The atospheric density increases downward so steeply that the subsidence ass flux becoes divergent no atter how the radiative cooling profile varies in noral situations. The divergent subsidence ass flux drives the basic property of the radiatively driven circulation; naely, the detrainent occurs over a wide vertical range of troposphere. To visualize the two-diensional flow field, we introduce a streafunction (x, z) satisfying the relationships (5). Since the vertical axis at the subsidence center

18 2214 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 14. Two-diensional distributions of ixing ratio obtained by (a) the DCM and (b) KCM for the STD scenario. The KCM has a periodic horizontal length of 126 k slightly saller than 128 k for the DCM. The contours fro the KCM are horizontally rearranged so that the horizontal location of the cuulus x 38 k is the sae as in the DCM. x x S is one of the strealines and the integration constant added to is arbitrary, we put a boundary condition (x x S ) 0. We can obtain at any point (x, z) in the subsidence region (x cu x x S ) by integrating the second equation of (5) horizontally fro x S to x. Figure 18 shows the distribution of for the STD scenario. It clearly shows a ass circulation with detrainent over the whole vertical range of the troposphere. The circulation pattern is different fro that of the CCM. The vertical profiles of ass and water vapor detrainent across the cuulus flank are in sharp contrast to

19 15 JULY 2002 IWASA ET AL those predicted by the CCM in Fig. 19. The ass detrainent u (Fig. 19a) is a axiu just below the tropopause height and decreases downward, but it reains positive down to the surface. The ixing ratio q increases downward, so that the level of the axiu water vapor detrainent q u (Fig. 19b) descends to the iddle troposphere at k. Water vapor detrainent at such a low level in the troposphere helps to huidify the subsidence region. 6. Analytic solution a. Basic equations and assuptions To obtain an analytic solution for the equilibriu atosphere sustained by the radiatively driven circulation echanis, we ake a few additional assuptions to those of the nuerical odels. We assue a polytrope atosphere (appendix E). A cuulus of infinitesial width and the center of the subsidence region are located at the fixed points x 0 and x x S, respectively. The cuulus is assued to be saturated with a water vapor ole fraction of *. A constant radiative cooling rate Q in the subsiding-troposphere is assued. Following Eq. (52) fro the nuerical KCM, local therodynaical equilibriu in the subsidence region eans that the subsidence velocity w generated by the radiative cooling is 1 g w Q, (57) c p where is the constant lapse rate given by (E8). The conservation laws of ass and water vapor content are, respectively, written as (v) 0 (58) (v) 0, (59) where (z) and are the density and the water vapor ole fraction of the atosphere, respectively. The atospheric velocity is v (v H, w) with coponents v H in the horizontal direction and w in the vertical. Since Eq. (57) has a constant subsidence velocity w, the conservation laws (58) and (59) becoe where H is horizontal. H vh w 0 z (60) v w H H 0, z (61) b. Analytic solution in two-diensional geoetry If otion is confined to the x z plane, then H / x and v H u(x, z). Hence, (60) and (61) becoe u ln w 0 (62) x z 2 2 u w 0, (63) x z where 2 (x, z) denotes the water vapor ole fraction of the two-diensional atosphere. By applying the syetric lateral boundary condition u 0 at the subsidence center x x S, we obtain the horizontal velocity u(x, z) by integrating (62) horizontally fro x S to x as ln u w (xs x). (64) z Substituting (64) into (63), we find 2 2 (xs x) 0. (65) x ln Introducing a variable x S x, (65) is rewritten as (66) ln ln Here we assue a separation of variables solution 2 A()B(), in which the factors A and B are functions only of (x) and (z), respectively. Substitution of this solution into (66) gives lna lnb. (67) ln ln Each side of (67) should equal to a constant, say,. Therefore, application of the polytrope relationship (E3) yields /(1) 2 AB T (xs x), (68) where is the polytrope exponent (appendix E). The boundary condition of saturated cuulus 2 (x 0) * applied to (68) gives [(H T/H p)1] * T x, S (69) where is substituted by (E14). On the other hand, fro (E21), the saturating ole fraction at x 0is (L /RT 0)(H T/H p) T * *, (70) 0 T 0 where T 0 is a constant teperature, and * 0 is the saturating ole fraction at T T 0. Latent heat of condensation of water and gas constant per ole are L J ol 1 and R J K 1 ol 1, respectively. Identifying T dependencies in (69) and (70), is obtained as L H T RT0 Hp. (71) H T 1 Then 2 becoes H p

20 2216 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 15. As in Fig. 14 except for the relative huidity. L 2 0 exp RT T T 0 0 (L /RT 0)(H T/H p) [(L /RT 0)(H T/H p)]/[(h T/H p)1] xs x, (72) x S where 0 is a constant. Attending to the variable-separated for, and to the one-to-one correspondence between T and z by (E7), the solution (72) can be rewritten as (x, z) 2 [(L /RT 0)(H T/H p)]/[(h T/H p)1] xs x *(z). (73) x S Equation (73) eans that the water vapor ole fraction

21 15 JULY 2002 IWASA ET AL FIG. 16. Vertical profiles of (a) ixing ratio and (b) relative huidity within the troposphere at the subsidence ceter, expected fro the chineylike echanis (*CCM, dotted line), and obtained fro the DCM (broken line) and KCM (solid line), for the STD scenario. in the subsidence doain is proportional to that of the cuulus doain at an arbitrary height, and hence the relative huidity h 2 2 / * is independent of height. Equation (73) also yields horizontally siilar distributions for different values of the horizontal interval x S between the cuulus and the subsidence center. The horizontally averaged relative huidity becoes H T x S Hp h2 dx, (74) * xs * L 0 1 R T 0 which is independent of x S. When T K, then L /R T 0 20, H T , and H p 8000, and the resulting horizontal ean relative huidity becoes h 2 20%. These features in the huidity field of the analytic solution are siilar to those in the nuerical odels. The independence of the huidity distribution fro the horizontal scale predicted here is confired using the nuerical KCM in appendix F. FIG. 18. As in Fig. 17 except for the streafunction. 7. Coparison to the CCM How does the circulation in the CCM copare with those in the DCM and KCM? Convection and radiation are both necessary to balance ass and heat in the CCM and the DCM/KCM. However, the priary processes driving the circulations of ass and water vapor differ between the odels. The CCM (illustrated in Fig. 20a) assues that cuulus convection drives the net circulation. The downward ass flux (essentially a downdraft ) in the subsidence region copensates for the upward ass flux in the updraft. The convectively forced circulation and the net circulation are identical in this odel. In contrast, the net circulation in the subsidence region of the DCM/KCM (illustrated in Fig. FIG. 17. Two-diensional contours of radiative heating rate obtained by the KCM for the STD scenario. FIG. 19. (a) Vertical profile of the noralized ass detrainent obtained by the KCM for the STD scenario (solid line). Detrainent expected fro the CCM (dotted line) and diagnostic entrainent assued to be vertically unifor within the CBL (thin line) are shown for coparison. (b) Vertical profile of the water vapor detrainent obtained by the KCM for the STD scenario (solid line, not noralized). That expected fro the CCM (dotted line), when the total ass detrainent is the sae as that of the KCM, is also shown. Entrainent within the CBL is not shown, because water vapor loss within the cuulus due to condensation and precipitation is not taken into account.

22 2218 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. 20. (a) Scheatic of the circulations in the cuulus chiney odel and (b) that extracted fro this study. White arrows indicate the net ass and water vapor circulation, while black arrows indicate the convectively forced circulation. The convective circulation in (b) is drawn as a single circulation for convenience, but it ay be coplex and is not investigated copletely in this study. 20b) is a downward ass flux that is regulated/driven by radiative cooling. The convectively forced part of the circulation is liited both teporally and in horizontal extent and has little effect on the net circulation in the subsidence region. The role of the cuulus convection in this odel is to aintain localized convective circulation in the cuulus tower, which acts as the source of water vapor for the subsidence circulation. The radiatively regulated/driven circulation in the DCM/KCM produces a uch oister atosphere than the convectively driven chiney circulation. The DCM/ KCM allows water vapor in the cuulus updraft to ove through the cuulus flank directly to the subsidence region at lower, wetter levels in the atosphere. Coparison of the odels reveals two unrealistic assuptions in the CCM. One is the vertically one-diensional geoetry that prohibits exchange of ass and water vapor between the cuulus updraft and subsidence regions anywhere but near the tropopause. This leads to a sall constant value for ixing ratio deterined by conditions near the tropopause. This sall value becoes representative of the entire subsidence region. The other ore probleatic assuption is the steady convective otion. The CCM has a vertical teperature profile balanced therodynaically between radiative cooling and copressional heating in the subsidence region. The teperature profile generally differs fro the oist adiabatic profile in the cuulus updraft region. This is not consistent with the quasi-dynaical equilibriu with a horizontally unifor virtual teperature that results fro occasional convective events in the DCM. 8. Conclusions A newly constructed DCM with explicit treatent of longwave radiation and oist convection produced an equilibriu atosphere in which radiative and convective effects balance. At odel equilibriu, the convectively forced circulation was localized in both tie and horizontal extent and did not greatly influence the subsidence region. Subsidence was consistent with radiative cooling. To aintain local therodynaical balance in the subsidence doain in the free troposphere, the subsidence velocity should cause a copressional heating rate balanced by radiative cooling. In the free troposphere in the absence of convective forcing, the radiative cooling rate and hence the subsidence velocity do not vary in the vertical. Since the atospheric density increases downward, the subsidence ass flux, the product of the subsidence velocity and the atospheric density, is in general divergent. Such a divergent ass flux cannot be aintained without inducing convergent horizontal ass flux directed fro the cuulus updraft region into the subsidence region, thus preserving ass continuity. These vertical and horizontal flow fields define a circulation with detrainent at various levels in the free troposphere. This is in contrast to the cuulus chiney circulation in the CCM. The CCM includes artificially steady updrafts and downdrafts. The CCM gives rise to a virtual teperature difference between the updraft and downdraft regions, which disagrees with the quasi-dynaical equilibriu in the DCM. Therefore, it is not appropriate to apply the CCM to an atospheric evolution over a tie period uch longer than the lifetie of a single cuulus convective event. A KCM resulted in nearly the sae free atospheric state as produced with the DCM. The KCM includes a radiatively driven circulation echanis, but lacks any direct convective ass or water vapor transports in the subsidence region. Miniu huidities were copared aong the KCM, DCM, and CCM. The DCM and KCM agree with each other in ters of relative huidity, giving a uch oister troposphere than the very dry CCM. An analytic solution was derived for the equilibriu state sustained by the radiatively driven circulation in an atosphere with two-diensional geoetry under the additional assuptions of constant radiative cooling in a polytropic troposphere. The relative huidity was found to be unifor in the vertical with a constant horizontal ean value irrespective of the horizontal scale. These analytic results agree with the DCM and KCM. The circulation in the DCM/KCM effectively transports water vapor. Air within the cuulus updraft region can enter the subsidence region directly through the cuulus flank at various levels in the free troposphere without losing large content of water vapor at high levels. The circulation echanis provides a satisfying explanation for the huidity distribution. Detailed knowledge of the convective transport is not needed. The radiatively driven circulation echanis adds hu-

23 15 JULY 2002 IWASA ET AL idity to the atosphere and avoids the excess drying present in the CCM. Acknowledgents. This work was supported by the Center for Cliate Syste Research (CCSR) of the University of Tokyo. Nuerical calculations of the DCM were perfored on a VP100 at the Coputer Center of Nagoya University. The contour figures were produced using xfarbe written by Dr. Albrecht Preusser. The Colun Radiation Model (CRM) developed at the National Center for Atospheric Research (NCAR) is a nongray radiation odel copared to the gray radiation odel in appendix B. The authors thank Dr. Kensuke Nakajia, Dr. Masaki Satoh, and Prof. Yoshi-Yuki Hayashi for valuable discussion of oist convection and radiative convective equilibriu. Dr. Tooe Nasuno and Dr. Masaki Ogawa helped the authors by providing a suary of various cuulus paraeterizations and by inforing the of the FCT ethod, respectively. Mr. Taichu Tanaka helped the authors with his coputer network skills. The authors are grateful to Prof. Teruyuki Nakajia for his suggestions and encourageents. This work would not have been possible without the indispensable assistance of Prof. Taro Matsuno, Prof. Akiasa Sui, and especially of Prof. Syukuro Manabe, who gave the authors invaluable suggestions and encourageents, and provided one of the authors (YI) with both the opportunity and a congenial environent in which to do research. APPENDIX A Description of the CAM Consider an atosphere in radiative equilibriu. The vertical profile of pressure p rad (z) in a dry, gray atosphere in radiative equilibriu is [ ] 1/4 dlnprad g FR0 3 kq n n 1 prad. (A1) dz R 4 g 2 When the paraeter values of k n q n and F R0 are given, Eq. (A1) can be solved for p rad with a boundary value p rad (0). Then, the vertical profiles of the optical depth rad (z), teperature T rad (z), and density rad (z) of the atosphere are obtained sequentially as kq n n rad p rad, (A2) g [ ] FR0 3 1 Trad rad +, and (A3) 4 2 p rad. (A4) rad RT rad In the DCM and KCM, p rad and rad are used to approxiate the pressure p(z) and density (z), respec- tively. Figure A1 shows the vertical profiles of the density (z) used in the calculations. 1/4 FIG. A1. Vertical profiles of the atospheric density (z) used for the nuerical odels. We used a CAM siilar to Huel and Kuhn (1981) to generate the initial states. The CAM has the sae vertical doain and grid configuration (appendix C) as the DCM and KCM, although the CAM lacks dynaical variables. Convective adjustent occurring as a oistadiabatic lapse rate with fixed relative huidity is applied within the troposphere. Following Abe and Matsui (1988), the lapse rate for a given relative huidity h is (1 ) lnh T R T d g, z S L L (1 )cp lnh R T T (A5) where kg ol 1 d is the ean o- lecular weight of air and c 7R p /2 is specific heat per ole at constant pressure. The subscript S on the lefthand side denotes partial differentiation done at fixed entropy. In Eq. (A5), is the ole fraction of water vapor given as d L, (A6) 1 d hq* in which d d / / is the olecular weight ratio of air to water vapor, and the saturated ixing ratio q* is obtained using Tetens for-

24 2220 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 ula (17). If the teperature is specified at the botto boundary, Eq. (A5) can be nuerically integrated to obtain the vertical profile of the tropospheric teperature. The botto teperature (and hence the tropopause height) is adjusted so that the OLR heat flux at the upper boundary of the calculation doain is F R0. The calculation of the longwave radiative transfer is the sae as that used in the KCM (no teperature gap at the surface). APPENDIX B FIG. B1. The sensitivity of the horizontal ean OLR heat flux to the increase of horizontal ean teperature at the botto, copared aong the radiation schees of gray and nongray atospheres. The GRAY case shows a gray atosphere with the optical paraeters used in this study. The GVLn and GVGn cases show gray atospheres with water vapor of lower and greater optical efficiencies than the GRAY case, respectively. CRMD and CRMA are nongray atospheres. See Table B1. Validation of the Gray Atosphere Assuption This appendix deterines the realis of the gray radiation schee used in the DCM/KCM. The gray odel cannot reproduce a radiation field identical to the real atosphere, but it ust provide reasonable values for variables of interest (botto teperature, tropopause height, and total optical depth). The CAM with a tropospheric huidity of 75% yields acceptable values as shown in Table 1. Sensitivity to waring in the atosphere was one of the factors used in deterining the optical paraeter values (Sipson 1927, 1928). Two-diensional distributions of teperature and water vapor obtained by the KCM for the STD and XTR scenarios and a scenario of k n q n kg 1 were used. For the three different atospheres, we perfored calculations with gray radiation, fixing the values of the optical paraeter pair k and k n q n. The values of the optical paraeter pair can be chosen in various ways under the restriction that the OLR heat flux is F R0 for the STD scenario. The results for soe typical pairs listed in Table B1 are plotted in Fig. B1. The gradient of each graph shows the sensitivity FOLR/T bot of the horizontal ean OLR heat flux to the increase in horizontal ean botto teperature. The sensitivity increases when k decreases (or k n q n increases) and vice versa. To investigate results fro a coparable nongray radiation schee, we used the NCAR Colun Radiation Model (CRM; Kiehl et al. 1998). The CRMD case with the paraeter values provided with the CRM distribution causes an OLR heat flux larger than F R0 for the STD scenario. In the CRMA case, therefore, we dropped all the optical constituents other than water vapor and CO 2, the ixing ratio of which is adjusted so as to ake the OLR heat flux F R0 for the STD scenario. The CRMD and CRMA cases have nearly the sae sensitivities. The GRAY case used in the ain part shows a waring sensitivity very siilar to that of the CRMA case. To show the siilarity in the circulation patterns arising in gray and nongray atospheres, we present vertical profiles of ass detrainent consistent with the vertical profiles of the radiative cooling horizontally averaged over the subsidence region obtained fro the KCM for the STD scenario (Fig. B2a). Gray and nongray schees both provide positive ass detrainent over the whole free troposphere as shown in Fig. B2b. They arkedly differ fro the profile expected fro the CCM (shown in the sae figure) in which detrainent is confined to near the tropopause. TABLE B1. Pairs of optical paraeter values used to copare the radiation odels shown in Fig. B1. *In the gray odels, they are the values of the absorption coefficient k for water vapor. In the CRMs, the paraeter values for water vapor absorption are provided by the CRM distribution. **They are the values of the optical paraeter k n q n for the NGG in the gray odels and the values for CO 2 volue ixing ratio in the CRMs. Except for the CRMD, these values are adjusted to provide an OLR heat flux of F R 0 at the top of the atosphere in the KCM for the STD case. Model GVL10 GVL5 GVL2 GRAY GVG2 GVG5 CRMD CRMA Paraeter value for water vapor* Provided with the CRM distribution Provided with the CRM distribution Paraeter value for NGG or CO 2 ** Provided with the CRM distribution

25 15 JULY 2002 IWASA ET AL APPENDIX C Nuerical Schees In the DCM, all variables are defined at coon grid points in the horizontal. In the vertical direction, dynaical and radiative variables (including vorticity, streafunction, optical depth, and Planck function, all of which are also defined at the upper and surface boundaries) are offset fro the advected variables (potential teperature and the ixing ratio). A pseudospectral ethod (Gazdag 1973; Tanaka 1975) is applied to copute derivatives in the horizontal direction. The flux-corrected transfer (FCT) ethod (Zalesak 1979) was adopted to calculate the vertical advection of teperature and the ixing ratio in Eqs. (14) and (15), respectively. A sipler finite-difference ethod was used to calculate other vertical derivatives. Tie integration is achieved using a partially corrected second-order Adas Bashforth schee (Gazdag 1976) that is copatible with the pseudospectral ethod. An extension equivalent to Gresho et al. (1980) was adopted for variable tie steps. When a prognostic equation G(, t) (C1) t for a physical variable is given, the forward tie step is split into the following pair of steps; predictor step: [ ] n n 1 t t n1 n n n1 n n1 n1 2 G G t (C2) 2 t t corrector step: 1 n1 n n n1 n (G G )t. (C3) 2 Here, each superscript indicates the tie step nuber t n1 t n t n, and the contracted notation G n G( n ) is used. We applied a factor of 1/8 to the Courant condition for atospheric flow, and the tie step can be shortened to satisfy the condition. Thus, the tie step varies as [ ] 1 x z t(t) in in,, t ax, (C4) 8 u ax(t) w ax(t) where u ax (t) and w ax (t) are the axiu absolute values of the horizontal and vertical velocities, respectively, within the calculation doain at tie t. The axiu tie step is t ax 10 s. The dynaical equations (7), (8), (14), and (15) are solved every t given by Eq. (C4). FIG. B2. (a) Vertical profiles of the radiative heating rate horizontally averaged over the subsidence region, calculated with the GRAY and the CRMA radiation schees. (b) Vertical profiles of the noralized ass detrainent obtained by the KCM, consistent with the radiative cooling profiles in (a). Profiles of the detrainent expected fro the cuulus chiney odel (dotted line) and diagnostic entrainent within the CBL (thin line) are shown for coparison as in Fig. 19a.

26 2222 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 To iniize coputations, the equations for the radiative transfer (21) and the surface boundary condition (27) are solved using average values over tie intervals that exceed t. The tie ean value of each ter is approxiated by the value of the ter in which the factors for the physical variables are replaced by their corresponding tie ean values. The tie-averaged value i of a physical variable over tie interval t i in each equation is t 1 i(t) (t) dt, (C5) t i tti where i is R or surf for the radiative transfer or the surface boundary condition, respectively. The tie interval t R 10 in for the radiative transfer calculation (21) is greater than t surf 1 in for the surface boundary condition (27). Thus, we used the latest tieaveraged radiative quantities of the atosphere in the calculation of the surface boundary condition. Siilarly, the values fro the latest radiative calculation and the latest surface boundary condition were used to calculate the advective equations (14) and (15). The pseudospectral ethod and the one-diensional FCT ethod for advection in the DCM conserve the extrees in the vertical profiles of the advected values. Therefore, soothing ethods were used when preparing soe figures. In Fig. 10b, the iniu-square ethod using a cubic curve between 3 and 12 k (the vertical range of the free troposphere in the XTR scenario) is used in the vertical at every horizontal grid. A four-point running ean ethod is used in the vertical direction at every horizontal grid in Figs. 6, 10a, and 11. In Fig. 11, a centered finite difference k 1 k2 k1 k1 k2 ( 8 8 ) (C6) z 12z is used to copute the vertical derivative /z of variable at the kth vertical grid point. In the tracer experient presented in section 3c, we used two-diensional cubic spline functions to interpolate the velocities of the tracers at their locations off the grid points. All the variables of the KCM are taken at the sae grid points in the horizontal and vertical directions, except that the horizontal velocity u is at staggered points in the horizontal, and the optical depth and radiative heat flux F R are at staggered grid points in the vertical. The vertical velocities w are obtained at the coon vertical grid points, and then vertically interpolated between the coon vertical grid points for the advective calculation. We adopted a MACHO schee (Leonard et al. 1996) to calculate the advective ters in the water vapor equation (56). The tie-forwarding schee is the sae as that used in the DCM. Equations for the convective adjustent (including radiative calculations) used in the KCM are solved at tie interval t adj to save coputational expense. A tie-averaged value adj of a physical variable over the tie interval t adj is expressed by (C5) with i adj. Since the tie interval t adj 10 in for the convective adjustent (51) is longer than t ax 1 in for the calculation of the water vapor advection (56), we used the latest convectively adjusted values in the advection calculation. APPENDIX D Diagnostic Cuulus Fluxes Net fluxes in the cuulus region in the KCM can be obtained diagnostically fro the fluxes in the subsidence region. We assue that the vertical ass flux F cu within the cuulus is horizontally unifor and vanishes at the tropopause. The flux is required to sustain the ass detrainent across the cuulus flanks (both the left and right sides of the cuulus doain). The net cuulus ass flow F 2x cu cu at an arbitrary height z is obtained by integrating the doubled ass detrainent u across the right cuulus flank x x cu downward fro the tropopause height z trop to z. Therefore, the net cuulus ass flux is written as z 1 trop cu F (z) 2(z)u(x, z) dz. (D1) cu cu 2x z Then the vertical water vapor flux F cu (z) within the cuulus doain is obtained using the ixing ratio (47) and the ass flux (D1) in the cuulus region as F (z) q (z)f (z). (D2) cu cu cu Figure D1 shows the vertical profiles of cuulus fluxes for ass (Fig. D1a) and for water vapor (Fig. D1b) calculated by (D1) and (D2), respectively. The net updraft fluxes decrease with height, consistent with detrainent over the troposphere. Note that absolute values of the detrainents and the net cuulus fluxes are proportional to the width of the subsidence region (appendix F). FIG. D1. Vertical profiles of net vertical fluxes of (a) ass and (b) water vapor in the cuulus doain diagnostically obtained by the KCM.

27 15 JULY 2002 IWASA ET AL APPENDIX E Polytrope Atosphere a. Polytrope relationships We assue a polytrope atosphere. The polytrope relationship is 1 (1)/ p T, (E1) p T where (z), p(z), and T(z) are atospheric density, pressure, and teperature, respectively; 0, p 0, and T 0 are reference values; and is the polytrope exponent. The equation can be rewritten as p T /(1) or (E2) p T / p T 1/(1). (E3) p T b. Hydrostatic equilibriu For the case of constant gravitational acceleration g, the hydrostatic equilibriu with the z axis in the upward direction becoes dp g. (E4) dz Substituting Eqs. (E1)(E3) yields [ ] [ 1 g p 0 ] [ 1 g p 0 ] 1/(1) 1 0 g 0 1 (z z 0 ), (E5) p 0 /(1) p p (z z 0 ), or (E6) T T (z z 0 ). (E7) The teperature (E7) has a constant lapse rate dt 1 0g 1 g 0 T0. (E8) dz p R Here we used the equation of state R 0 p0 0 T 0 (E9) for an ideal gas with olecular weight. We also can obtain fro the lapse rate 0 as g T0 g. (E10) 1 R R T Furtherore, denoting the scale heights at z z 0 for teperature and pressure as H T and H p, respectively, we can show that dz T H 0 T and (E11) d lnt 0 dz R T H 0 p. (E12) d lnp g Therefore we obtain the following relations: H T or (E13) 1 H p HT. (E14) HT Hp These relationships are substituted into (E5)(E7) to give 0 (H T/H p)1 z z 1 0, (E15) 0 H T H T/Hp z z p p 1 0, or (E16) 0 H T z z T T 1 0. (E17) c. Polytrope approxiation for the water vapor saturation pressure curve The saturation pressure curve for water vapor p * (T) is approxiated by the following for: p*(t) p* 0 exp, (E18) RT where p* 0 is a constant. We approxiate this curve with a polytrope, naely, we write T p*(t) p* (E19) 0 with constants p* 0 and. Applying the agreeent con- ditions of p* and its derivative dp*/dt at T T0 be- tween (E18) and (E19), p* 0 and are deterined and (E19) becoes H T T 0 L L /RT 0 L T p*(t) p* 0 exp. (E20) RT T 0 0 The approxiation (E20) becoes ore accurate in general when the value of L /R T 0 is larger. With L / R T for water vapor at T K, Eq. (E20) gives a precision better than 10% in the teperature range of 250 K T 300 K. The ole fraction for the saturation water vapor * (T) is approxiated using (E2) and (E13) as

28 2224 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 59 FIG. F1. (a) Two-diensional distributions of relative huidity obtained for different horizontal grid nubers N 8, 16, 50, 128. (b) Horizontal profiles of the horizontal ass flux vertically integrated over the entire doain for the cases of N 8, 16, 50, 90, 128. p*(t) p* 0 L T *(T) exp p(t) p R T T where (L /RT 0)(H T/H p) (L /RT 0)[ /(1)] T *, (E21) T 0 p* 0 L * exp 0. (E22) p R T 0 0 By cobining Eqs. (E20) and (E21) with the polytrope relationships (E1)(E3), we can write p* and * as functions of and p as L p* p* 0 exp RT 0 0 (L /RT 0)[H p/(hth p)] 0 0 p 0 (L /RT 0)(H p/h T) L p p* 0 exp and (E23) RT p * * 0 0 [(L /RT 0)(H p/h T)1]/[1(H p/h T)] (L /RT 0)(H p/h T)1 *, (E24) respectively. p 0 APPENDIX F Horizontal-Scale Independence of the Equilibrated State Two paraeters affect the horizontal scales in the KCM: the cuulus interval X and the cuulus width