Interactions between Tropical Convection and Its Environment: An Energetics Analysis of a 2D Cloud Resolving Simulation

Transcription

1 1712 JOURNAL OF THE ATMOSPHERIC SCIENCES Interactins between Trical Cnvectin and Its Envirnment: An Energetics Analysis f a 2D Clud Reslving Simulatin XIAOFAN LI,* C.-H. SUI, AND K.-M. LAU NASA Gddard Sace Flight Center, Greenbelt, Maryland (Manuscrit received 1 Octber 1999, in final frm 2 Nvember 2001) ABSTRACT The hase relatin between the erturbatin kinetic energy (K) assciated with the trical cnvectin and the hrizntal-mean mist available tential energy ( P) assciated with envirnmental cnditins is investigated by an energetics analysis f a numerical exeriment. This exeriment is erfrmed using a 2D clud reslving mdel frced by the Trical Ocean Glbal Atmshere Culed Ocean Atmshere Resnse Exeriment (TOGA COARE) derived vertical velcity. The imsed uward mtin leads t a decrease f P thrugh the assciated vertical advective cling, and t an increase f K thrugh clud-related rcesses, feeding the cnvectin. The maximum K and its maximum grwth rate lags and leads, resectively, the maximum imsed large-scale uward mtin by abut 1 2 h, indicating that cnvectin is hase lcked with large-scale frcing. The dminant life cycle f the simulated cnvectin is abut 9 h, whereas the timescales f the imsed largescale frcing are lnger than the diurnal cycle. In the cnvective events, the maximum grwth f K leads the maximum decay f the erturbatin mist available tential energy (P) by abut 3 h thrugh vertical heat transrt by erturbatin circulatin, and erturbatin clud heating. The maximum decay f P leads the maximum decay f P by abut 1 h thrugh the erturbatin radiative rcesses, the hrizntal-mean clud heating, and the large-scale vertical advective cling. Therefre, maximum gain f K ccurs abut 4 5 h befre maximum decay f P. 1. Intrductin Trical cnvectin ccurs as a result f the release f unstable energy f its envirnment. The large-scale envirnment rvides favrable thermal and misture cnditins fr ccurrence and develment f cnvectin, n ne hand. On the ther hand, it is adjusted by redistributin f vertical thermal, misture, and mmentum structures induced by the cnvectin. Such interactin allws us t use envirnmental cnditins t estimate the rerties f the cnvectin such as the reciitatin. Since the envirnmental timescales (a few days and lnger) are much lnger than the cnvective timescales (a few hurs and shrter), the rate f rductin f available tential energy by the large-scale rcesses is nearly balanced by the rate f cnsumtin f the available tential energy by the cnvectin (Manabe and Strickler 1964). This quasi-equilibrium cncet is the basic remise f the cumulus arameterizatin scheme rsed by Arakawa and Schubert * Current affiliatin: NOAA/NESDIS/Office f Research and Alicatins, Cam Srings, Maryland. Crresnding authr address: Dr. Xiafan Li, NOAA/NESDIS/ ORA, 5200 Auth Rd., Rm 601, Cam Srings, MD xiafan.li@naa.gv (1974). The decrease f cnvective available tential energy (CAPE) that measures the thermal and misture cnditins f the envirnment ften cincides with the develment f cnvectin s that the CAPE and rain rate are negatively crrelated (e.g., Thmsn et al. 1979; Cheng and Yanai 1989; Wang and Randall 1994; Xu and Randall 1998). The hase relatin between the CAPE and rainfall must be related t the culing between envirnmental dynamic and thermdynamic fields (Cheng and Yanai 1989). The hases f CAPE and rainfall culd be different because time is needed fr develment f cluds. This hase difference may be included as relaxing the quasiequilibrium assumtin in cumulus arameterizatin (e.g., Betts and Miller 1986; Randall and Pan 1993). The minimum CAPE ccurs a few hurs after maximum rainfall. Such a hase lag is als demnstrated by Xu and Randall (1998) in their 2D clud reslving simulatins. Xu and Randall (1998) interreted the maximum hase lag as the adjustment time scale frm disequilibrium t equilibrium states in the resence f time-varying large-scale frcing. Lrenz (1955) first intrduced a cncet f available tential energy f dry atmshere that reresents a rtin f the tential energy that can be transferred int the kinetic energy. Lrenz (1978, 1979) extended this cncet t the mist atmshere by cnsidering the 2002 American Meterlgical Sciety

2 15 MAY 2002 LI ET AL mist-adiabatic rcesses. Randall and Wang (1992) and Wang and Randall (1994) further argued that the vertical cmnent f the mist available tential energy is a generalizatin f the CAPE. In this study, the hysical rcesses resnsible fr such a hase relatin are examined thrugh the analysis f energy cnversin rcesses between available tential energy and kinetic energy in a 2D clud reslving simulatin. We first establish the hase relatin between available tential energy and kinetic energy, and use a set f energetics equatins (sectin 2) t examine the essential hysical rcesses determining the hase relatin (sectin 3). The hase relatin is discussed in sectin 4, and the cnclusin is given in sectin Frmulatins fr mdel, energetics, and CAPE a. Mdel The clud reslving mdel was riginally develed by Sng and Ogura (1980), Sng and Ta (1980), and Ta and Simsn (1993), fr studying dee cnvective resnse t the secified large-scale frcing. A 2D versin f the mdel used by Sui et al. (1998) and mdified by Li et al. (1999) is used in this study. The gverning equatins with an anelastic arximatin can be exressed as fllws: u 1 w 0, (1) x z u (uu uuuu) t x 1 (wu wuwuwu) z c () Du D u, (2) x w (uw uwuw) t x 1 z (ww wwww ww) c () g 0.61q ql z Dw D w, (3) (u) 1 u w t x x z b 1 1 w w Qcn QR z z c c u w D, x z (4) q (uq) q 1 u wq t x x z q q w w (c e d s) z z q q u w D q, (5) x z C (uc) 1 [(w w TV)C] SC t x z D C. (6) Here, u, and w are znal, and vertical wind cmnents; and q are tential temerature and secific humidity, resectively; C (q c, q r, q i, q s, q g ), q c, q r, q i, q s, and q g are the mixing ratis f clud water, rain, clud ice, snw, and grauel, resectively; is a mean air density that is a functin f height nly; w TV is a terminal velcity that is zer fr clud water and ice; (/ ), R/c, R is the gas cnstant, c is the secific heat f dry air at cnstant ressure, and 1000 mb; c, e, d, and s dente cndensatin, evaratin, desitin, and sublimatin, resectively; Q cn L (c e) L s (d s) L f ( f m) dentes the net latent heat release thrugh hase changes amng different clud secies, where f and m are fusin and melting, resectively; L, L s, and L f are heat cefficients due t hase changes; Q R is the radiative heating rate due t cnvergence f net flux f slar and infrared radiative fluxes; S C is surce and sink f clud secies determined by micrhyical rcesses; D u, D w, D, D q, and D C are dissiatin terms; verbar ( ) dentes a znal mean; subscrit b dentes an initial value, which des nt vary with time; suerscrit dentes imsed bserved variables in the mdel. Sng and Ogura (1980) develed this clud reslving mdel based n the bserved scale searatin evidence that the timescale f the large-scale rcesses is much larger than the timescale f the life cycle f an individual cnvective clud. Similar arach was adted in Xu and Krueger (1991). The large-scale vertical velcity imsed in the mdel serves as the majr frcing. In such a mdel setu, the large-scale (hrizntal mean) thermdynamic states are adjusted nt nly by resnding t the imsed large-scale dynamic frcing but als by interacting with the cnvectin. The adjustment f the mean thermdynamic stability distributin due t the cnvectin in the semirgnstic arach fr the large-scale envirnment has been demnstrated t simulate the mean thermdynamic states mre reasnably (Li et al. 1999). Recently, Maes (1997) argued that dee cnvectin and its large-scale envirnment interact each ther s that it may be imrer t imse a vertical rfile f large-scale frcing in the clud reslving simulatin. Nevertheless, the clud reslving mdel is a useful tl t study ne-way interactin f cluds t the large-scale frcing, which

3 1714 JOURNAL OF THE ATMOSPHERIC SCIENCES may serve as the guidance t further study interactin between cnvectin and large-scale envirnment. The exeriment analyzed in this study in cnducted with the mdel frced by znally unifrm vertical velcity, znal wind, and hrizntal advectins, which are derived by Sui et al. (1997) based n the Trical Ocean Glbal Atmshere Culed Ocean Atmshere Resnse Exeriment (TOGA COARE) bservatins within the intensive flux array (IFA) regin at a time f 6 h. Hurly sea surface temerature at the Imrved Meterlgical (IMET) surface mring buy (1.75S, 156E) (Weller and Andersn 1996) is als imsed in the mdel. The mdel is integrated frm 0400 lcal time (LT) 18 December t 0400 LT 25 December The hrizntal dmain is 768 km. A grid mesh f 1.5 km and a 12-s ste are used in mdel integratins. Mre discussin f the mdel and its resnses t rescribed TOGA COARE frcing are rerted in Li et al. (1999). b. Energetics equatins Lrenz (1955) defined the available tential energy f the dry atmshere as the difference between actual ttal enthaly and the minimum ttal enthaly that culd be achieved by rearranging the mass under the adiabatic flw. The dry enthaly er unit mass is defined as the rduct f the temerature and the secific heat at cnstant ressure. In the absence f energy surces and sinks, the ttal kinetic energy and ttal enthaly are cnserved during adiabatic exansin. In the mist atmshere, latent heat energy shuld be included in the energy cnservatin. The latent heat energy er unit mass is defined as the rduct f the secific humidity and the latent heat f varizatin at 0C. In the absence f energy surces and sinks, the ttal kinetic energy and ttal enthaly and latent heat energy are cnserved during dry and subsequent saturated adiabatic exansin. Therefre, the mist available tential energy is defined as the difference between the actual mist tential energy (sum f the enthaly and latent heat energy) and the minimum mist tential energy that culd be achieved by rearranging the mass under mist-adiabatic rcesses. Znal-mean and erturbatin mist available tential energy are, resectively, defined by 2 2 P (h h b), (7a) 2C (h) P 2, (7b) c 2 where h c T L q ; T is temerature; (T) 1 ( b / (L /c )(q b /)) 1, which is a arameter related t static stability; the angle bracket imlies a vertical integratin: ( ) z B z T ( )dz. Here z B and z T are the heights f bttm and t f the mdel, resectively. (7a) is derived after sme arximatins similar t Lrenz (1955) and Peixt and Ort (1992). h b is a cnstant reference state here, and is calculated frm the initial bserved sunding. The erturbatin kinetic energy is defined by 2 2 (u) (w) K. (8) 2 An equatin fr the erturbatin kinetic energy (K) can be derived by multilying (2) by u and (3) by w and alying the znal mean and the vertical integratin n the resulting equatin: K C(K, K) C(P, K) G q (K) t G q l (K), (9) where u w z z C(K, K) uw ww, wt C(P, K) g, G (K) 0.61gwq, T b G (K) gwq. ql l Here, C( K, K) is the cnversin between K and K thrugh cvariance between erturbatin znal wind and vertical velcity under vertical shear f imsed hrizntal-mean znal wind, and between erturbatin vertical velcities under vertical shear f imsed hrizntal-mean vertical velcity. C(P, K) is the cnversin between P and K thrugh cvariance between erturbatin vertical velcity and temerature. G q (K) and G ql (K) are the generatin terms f K thrugh cvariance between erturbatin vertical velcity and secific humidity, and between erturbatin vertical velcity and clud mixing rati, resectively. T derive the equatins fr the znal-mean and erturbatin mist available tential energy, the fllwing equatin is frmed by multilying (4) by c and (5) by L and adding the resulting equatins: h (uh) h c u w cw t x x z z q L q wq Lw cw z z z q Lw L (d s f m) Q z f R h q u c w Lw. (10) x z z The equatins fr the znal-mean mist available tential energy ( P) and the erturbatin mist available tential energy (P) can be derived by multilying (10)

4 15 MAY 2002 LI ET AL by 1 1 c ( h h b ), and by c h, and alying the znal mean and the vertical integratin n the resulting equatins. Thus the znal mean equatin is P C(P, P ) G R (P ) G cn (P ) t where C(P, P ) (h h b) C h(k, P ) C (K, P ), (11) c c w L wq, z z G (P ) Q (h h ), R R b c G (P ) (h h )L (d s f m ), cn b f c q h b c x x C (K, P ) (h h )u c L, q C (K, P ) (h h )w c L b, c z z and the erturbatin equatin is where P C(P, P ) C(P, K) G R (P) t G cn(p) G(P), (12) G (P) Q h, R R c f c gl b ct c z G (P) hl (d s f m), cn G(P) wq [(h h )(hw)] b g T 1hw ct T b b g (h h b)tw ct b 2 [ (w w)(h) ] 2c z g (w w)th. ct b Here, C(P, P) is the cnversin between P and P thrugh cvariance between h h b and cnvergence f vertical flux f tential temerature and misture. Terms G R ( P) and G cn ( P) reresent the generatin f P thrugh cvariances between h h b and hrizntalmean radiative heating, and between h h b and hri- zntal-mean heating due t hase change f the clud cntents, resectively. Next, C h ( K, P) and C ( K, P) are the cnversin between K and P thrugh cvariances between h h b and imsed hrizntal temerature and misture advectins, and between h h b and the hrizntal-mean vertical temerature and misture advectins by imsed vertical velcity, resectively. Terms G R (P) and G cn (P) reresent the generatin f P thrugh cvariances between h and erturbatin radiative heating, and between h and erturbatin heating due t hase changes f the clud cntents, resectively. The term G(P) is the generatin f P. Nte that C( P, P) C(K, P) G(P) causes changes f P due t the vertical advectin rcesses. The key stes t derive (9) and (12) can be fund in the aendix. Nte that fr simlicity the dissiatin terms are excluded in abve derivatins because they d nt affect the fllwing discussins. c. CAPE calculatin The CAPE can be calculated by z c cl(z) env(z) CAPE g dz. (13) (z) LFC Here cl is the tential temerature f an air arcel lifted frm z B t z T while nt mixing with its envirnment ( env ). The air arcel is lifted dry adiabatically until it becmes saturated and then is lifted mist adiabatically thereafter. The level f free cnvectin (LFC) is the height where cl env, z c is the level where cl env. The CAPE is calculated fr a seudadiabatic rcess and a reversible mist adiabatic rcess, resectively, in this study. In the seudadiabatic rcess, an air arcel is lifted adiabatically while all cndensed water drs ut frm the arcel. In the reversible mist adiabatic rcess, an air arcel is lifted adiabatically while all cndensed water is ket in the arcel. Fllwing Xu and Emanuel (1989), The virtual temeratures (T va ) fr the seudadiabatic rcess and (T vre ) fr the reversible mist-adiabatic rcess are, resectively, exressed by env 1 q vs(t )/0.622 Tva T, and (14) 1 q w 1 q vs(t )/0.622 Tvre T, (15) 1 q vs(t ) where T is the temerature f a seudadiabatically dislaced air arcel, q vs is the saturatin secific humidity, and q w is the ttal water cntent f the air arcel. CAPE a fr the seudadiabatic rcess and CAPE re fr

5 1716 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 1. Time evlutin f (a) vertical velcity (mb h 1 ), and (b) znal wind (m s 1 ) taken frm the TOGA COARE (Sui et al. 1997) fr a 6-day erid. Dwnward mtin in (a) and westerly wind in (b) are shaded. the reversible mist-adiabatic rcess are calculated by using (14) and (15), resectively. 3. Results Figure 1 shws the time evlutin f vertical distributin f the large-scale vertical velcity and znal wind during December 1992 that are imsed in the mdel. Strng uward mtins with maxima f mb h 1 ccur n the late f 20 December, and during the early mrnings f 23 and 25 December between 400 and 500 mb. The latter tw maxima are quasi-2-day scillatins (Takayabu et al. 1996) in the cnvective hase f an intraseasnal scillatin during TOGA COARE. Tw less intense uward mtin centers aear during the nights f 19 and 21 December. The ccurrence f maximum uward mtin at each night is cnsistent with the diurnal signals bserved by Sui et al. (1997). The large-scale znal wind in the lwer trshere (belw 700 mb) are westerly that strengthens t 10 m s 1 arund 23 December. The midtrshere has an easterly westerly wind scillatin with maximum easterly wind f 10 m s 1 at 500 mb n 20 December. The uer trshere (abve 250 mb) is dminated by easterly winds. As mentined reviusly, the mdel is als frced by the bserved hrizntal temerature and misture advectins (nt shwn), which have much smaller amlitudes than the vertical advectins resectively. Figure 2a shws lag crrelatin cefficients between znal-mean CAPE and rain rate. Psitive lag hur dentes that CAPE leads rain rate. The maximum lag crrelatin cefficients between znal-mean CAPE and rain rate indicate that the CAPE reaches maximum abut 3 4 h befre the maximum rain rate. The minimum lag crrelatin cefficients indicate that the CAPE reaches minimum abut 2 h after the maximum rainfall. Bth maximum and minimum are abve 99% cnfidence level. The hase difference between maximum and minimum crrelatin cefficients is abut 5 h. Since a significant sectral eak aears at 9hbythewer sectrum analysis f the hurly rain rate (nt shwn), the hase difference is abut the half f the lifetime f the simulated cnvectin. Figure 2a als shw that the lag crrelatin cefficients fr CAPE re and CAPE a are similar. Since the mdel is frced by imsed vertical velcity, the relatinshi between energy and imsed vertical velcity is first analyzed. Figure 2b shws lag crrelatin cefficients between P and w (slid line), and between K and w (dashed line). Statistically sig- nificant lag crrelatin cefficients dislay that maximum K lags imsed uward mtin (sitive w ) by 1 2 h whereas minimum P lags uward mtin by

6 15 MAY 2002 LI ET AL FIG. 2. Lag crrelatin cefficients (a) between hrizntal-mean CAPE and rain rate, and (b) between P and w (slid) and between K and w (dashed). Slid and dashed lines in (a) dente cases fr a seudadiabatic rcess and a reversible mist adiabatic rcess, resectively. Lag crrelatin cefficient curves abve uer hrizntal light dtted line r belw lwer light dtted line exceed 99% cnfidence level. abut 6 hurs. This suggests that the K leads P by abut 4 5 h, which is abut the half f the lifetime f the simulated cnvectin. The statistically significant relatinshi between P/t and w (slid line) and between K/t and w (dashed line) can be als shwn by lag crrelatin cefficients in Fig. 3a. The imsed uward mtin leads minimum P/t (maximum decrease f hrizntal-mean mist available tential energy) by 3 h, whereas it lags maximum K/t (maximum increase f erturbatin kinetic energy) by 1 2 h. Thus, minimum P/t lags maximum K/t by 4 5 h. The ccurrence f maximum imsed uward mtin and maximum K/t and K within 3 h indicates that cnvectin is hase lcked with the imsed large-scale uward mtin. The negative lag crrelatin cefficient between P/t and w in Fig. 3 means that the imsed large-scale dwnward mtin leads maximum P/t by abut 3 h. Thus, the imsed dwnward mtin results in a buildu f P, and rvides a favrable envirnmental cnditins fr ccurrence f cnvectin. The hase relatin between P/t and K/t is als linked by lcal change f erturbatin mist available tential energy P/t. Minimum P/t lags minimum P/t by abut 1 h, and minimum P/t lags maximum K/t by abut 3 h (Fig. 3b), s that minimum P/t lags maximum K/t by abut 4 h. This is a statistically significant hase relatin cnsistent with that shwn in FIG. 3. Lag crrelatin cefficients (a) between P/t and w (slid) and between K/t and w (dashed), and (b) between d P/ t and P/t (slid) and between P/t and K/t (dashed). Lag crrelatin cefficient curves abve uer light dtted line r belw lwer light dtted line exceed 99% cnfidence level. Fig. 2b, althugh there are tw ther lag crrelatin cefficients between P/t and K/t that are abve the 99% cnfidence level. The tw minimum lag crrelatin cefficients are 9 h aart, indicative f the dminant life cycle f the mdel cnvective events. The maximum lag crrelatin cefficient aears between the tw minimum lag crrelatin cefficients, indicating that P/t reaches maximum abut 1 h befre maximum K/t. T further examine the dminant hysical rcesses determining the hase relatins, the lag crrelatin between each term f P/t [Eq. (11)] and P/t, and between P/t and each term f P/t [Eq. (12)], and the lag crrelatin between each term f P/t [Eq. (12)] and K/t, and between P/t and each term f K/t [Eq. (9)] are ltted resectively in Figs. 4 and 5. Figure 4a shws that nly the zer-hur lag crrelatin cefficient between C ( K, P) and P/t is marginally arund the 99% cnfidence level, and abve the 95% cnfidence level. The term C ( K, P) f P/t is a majr cmnent that cntributes t the maximum sitive zer-hur lag crrelatin between P/t and P/ t. The term C ( K, P) is related t the vertical temerature and misture advectins [see exansin fllwing (11)]. Further analysis shws that the lag crrelatin cefficient between vertical temerature advectin and imsed vertical velcity has the same sign as thse between the sum f the vertical temerature and misture advectins and imsed vertical velcity, whereas

7 1718 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 4. Lag crrelatin cefficients (a) between each term f P/ t [Eq. (11)] and P/t, and (b) between P/t and each term f P/ t [Eq. (12)]. Lag crrelatin cefficient curves abve uer light dtted line r belw lwer light dtted line exceed 99% cnfidence level. FIG. 5. Lag crrelatin cefficients (a) between each term f P/ t [Eq. (12)] and K/t, and (b) between P/t and each term f K/t [Eq. (9)]. Lag crrelatin cefficient curves abve uer light dtted line r belw lwer light dtted line exceed 99% cnfidence level. the lag crrelatin cefficient between vertical misture advectin and imsed vertical velcity has the site sign (nt shwn). This indicates that vertical temerature advectin determines the cnversin term C ( K, P). The imsed uward (dwnward) mtin causes the vertical advective cling (warming), which results in the lss (gain) f hrizntal-mean mist available tential energy thrugh the cnversin term C ( K, P). Figure 4a als shws that maximum zer-hur lag crrelatin cefficient between G cn ( P ) and P/t is slightly less than that between C ( K, P) and P/t. The cnversin G cn ( P) carries cnvective signals. As shwn in Fig. 4a, the maximum lag crrelatin is abut 9 h aart. Figure 4b shws that the lag crrelatin cefficients between C( P, P) and P/t, and between G R (P) and P/t, and between G(P) and P/t are abve 99% cnfidence level. The terms C( P, P) and G(P) have the same rder f magnitude (nt shwn), but the site signs (Fig. 4b) s that they cancel each ther in large art. In additin, the amlitude f the term C(K, P) is smaller than thse f the terms C( P,P) and G(P). As a result, the lag crrelatin cefficient between C( P, P) G(P) C(K, P) and P/t becmes statistically insignificant. This suggests that the vertical erturbatin advectin rcesses d nt lay imrtant rles in determining hase f P/t. Therefre, the term G R (P) causes P/t t P/t by abut 1 h. The radiative cling with sitive h and radiative warming with negative h cause the decrease f erturbatin available tential energy thrugh the cnversin term G R (P). Figure 5a shws three maximum negative lag crrelatin cefficients and ne maximum sitive lag crrelatin cefficient which are abve 99% cnfidence level. Again, the small amlitude f the cntributin f C(K, P) tp/t, and cancelatin between C( P, P) and G(P) make the vertical erturbatin advectin rcesses less imrtant in determining the hase f P/ t. The term G cn (P) lays a crucial rle in cntrlling the hase f P/t as shwn in Fig. 5a where the lag crrelatin cefficient between G cn (P) and K/t has the maximum negative value at 2 h, and is statistically significant. This suggests that the maximum lss f erturbatin mist available tential energy and the maximum gain f erturbatin kinetic energy are linked by the term G cn (P). The heating released by desitin and fusin with sitive h causes the lss f erturbatin mist available tential energy. Maximum G cn (P) ccurs abut 3 h after the maximum K/t. It is imrtant t ntice that the maximum K als ccurs abut 3 h after the maximum K/t (Figs. 2b and 3a), indicating the minimum G cn (P) cincides with the strngest cnvectin. Thus, the 3-h f hase difference between P/t and K/t is the time fr cnvectin t devel t the greatest strength. Figure 5b shws that the lag crrelatin cefficients between C(P, K) and P/t and between G qv (K) and

8 15 MAY 2002 LI ET AL FIG. 6. Schematic diagram fr the summary f hase relatinshi between the cnvectin and its envirnment in term f P/t, P/t, and K/t. P/t are similar, and their maximum negative values at 3 h are barely abve 99% cnfidence level, which cntribute t maximum negative lag crrelatin cefficient between P/t and K/t at 3 h (Fig. 3b). The terms G ql (K) and C( K, K) have the site signs with the terms C(P, K) and G qv (K) (Fig. 5b), the magnitude f C(P, K) is larger than the ther three terms (nt shwn) s that the term C(P, K) determines K/t. Cvariance between the temerature and vertical velcity erturbatins determines the lcal change f the erturbatin kinetic energy. The thermally direct circulatin f the uward mtin with the higher temerature and the dwnward mtin with the lwer temerature cnverts the erturbatin mist available tential energy t the erturbatin kinetic energy, feeding the cnvectin. 4. Discussin Figure 6 summarizes the hase relatins between the cnvectin and its envirnment. The imsed largescale dwnward mtin yields a grwth f P by the assciated vertical advective warming [C ( K, P) 0], building the favrable envirnment fr ccurrence f cnvectin. The near-simultaneus ccurrence f maximum K/t, K, and imsed large-scale uward mtin imlies that cnvectin is hase lcked with the large-scale frcing. The life cycle f the simulated cnvective events (abut 9 h) is much shrter than the timescales f imsed large-scale frcing (lnger than the diurnal cycle). In the cnvective events, maximum K/ t leads maximum P/t by abut 3 h thrugh erturbatin clud heating [G cn (P)] and the vertical heat transrt by erturbatin circulatins [C(P, K)]. Maximum K/t als leads maximum K by abut 3 h, indicating that 3 h is the time required by cnvectin t reach the maximum strength. Minimum P/t leads minimum P/t by abut 1 h thrugh erturbatin radiative rcesses [G R (P)] and the hrizntal-mean clud heating [G cn ( P)], and the large-scale vertical advective cling. Cnsequently, maximum K/t leads minimum P/t by 4 5 h (abut the half f the cnvective lifetime). The hase difference between erturbatin kinetic energy assciated with cnvectin and its envirnment assciated with hrizntal-mean mist available tential energy indicates that the generatin f envirnmental unstable energy by large-scale rcesses is nt simultaneusly balanced by its destructin by cnvectin. The minimum hrizntal-mean mist available tential energy ccurs 4 5 h after maximum erturbatin kinetic energy, and the hase lag is abut the half f the cnvective lifetime. This rvides cncrete evidence fr the adjustment frm disequilibrium t equilibrium states

9 1720 JOURNAL OF THE ATMOSPHERIC SCIENCES rsed by Xu and Randall (1998). The results shw that the cnvective lifetime is related t the clud micrhysical rcesses, cnvective radiative interactins, and dynamic thermdynamic culing inside the cnvective system. These suggest that cnvective lifetime (as well as the hase lag) may deend n characteristics f cnvectin (clud tye). When the hase lag is included in cumulus arameterizatin, the cnvective lifetime is intrduced in the general circulatin simulatins. The scale interactin may have accumulated effects n the large-scale variability ranging frm diurnal t interannual timescales. Hwever, the lifetime f cnvectin may deend n the envirnmental cnditins. Mre exeriments with different envirnmental cnditins are needed t establish the lifetime f cnvectin, and the hase relatin t the envirnment. Lis and Hemler (1986) shwed that 2D simulatin devels dee cnvectin earlier and has larger values f the vlume-mean kinetic energy than the 3D simulatin. The 3D simulatin may change the cnversin term C( K, K). The 2D simulatin here shws that the cnversin term C(P, K) has the dminant cntributin t the grwth f K. Thus, it is wrth t aly the similar energetics analysis in the 3D clud reslving simulatin and cmare with the 2D simulatin. 5. Cnclusins Energetics analysis has been carried ut with a 2D clud reslving simulatin t determine the hysical rcesses resnsible fr the hase difference between cnvectin and its envirnment. The clud reslving mdel is frced by imsed time-varying hrizntalmean vertical velcity, znal wind, and hrizntal advectins derived frm the TOGA COARE dataset fr a 6-day erid. The imsed vertical velcity serves as a majr external frcing in this articular mdel setu. Lag crrelatin analysis shws that the maximum erturbatin kinetic energy assciated with the simulated cnvective events and its maximum grwth rate lags and leads the maximum imsed large-scale uward mtin by abut 1 2 h, resectively, indicating that the cnvectin is hase lcked with the imsed large-scale frcing. The imsed large-scale vertical velcity has the timescales lnger than the diurnal cycle, whereas the simulated cnvective events have the dminant lifetime f abut 9 h. The imsed large-scale uward mtin decreases the hrizntal-mean mist available tential energy by the assciated vertical advective cling, rviding the favrable envirnment fr cnvectin develment. The maximum latent heating and vertical heat transrt by erturbatin circulatins cause maximum grwth f erturbatin kinetic energy t lead maximum lss f erturbatin available tential energy by abut 3 h. The maximum vertical advective cling, the hrizntal-mean clud related heating, and erturbatin radiative rcesses cause maximum lss f erturbatin mist available tential energy t lead maximum lss f the hrizntal-mean mist available tential energy by abut 1 h. Cnsequently, the maximum gain f erturbatin kinetic energy leads the maximum lss f hrizntal-mean mist available tential energy by abut 4 5 h (abut the half f the lifetime f the simulated cnvectin). Acknwledgments. We thank three annymus reviewers fr their cnstructive cmments and editrial assistance that imrve the manuscrit significantly. This research is surted under the TRMM rject f NASA s Missin t Planet Earth Office. APPENDIX Relatins Used in Derivatin f Energetics Equatins The fllwing relatins are derived t btain (9): [ ] 1 [ x z ] 1 u (uu uuuu) (wu wuwu wu) x z w (uw uwuw) (ww wwww ww) [(u) (w) ] [(u) (w) ] u w x 2 z 2 z z u w z z (u u) (w w) uw ww uw ww, (A1) cu () w () 0, (A2) x z

10 15 MAY 2002 LI ET AL where mass cntinuity and zer vertical velcity at t and bttm f the mdel atmshere are alied. The fllwing relatins are derived t btain (12): [ ] h q c [ x z z ] h h c [ x z z] 2 2 (h) (h) c [ x 2 z 2 z ] (uh) h c L q h u w cw wq Lw c x x z z z z h (u u) (w w) c L h (u u) (w w) c (w w) (u u) (w w) c (w w) g 2 [ (w w)(h) ] (w w)th, (A3) 2c z c T b q h c w Lw c z z [ ] [ ] b qb c ( b) L (q q b) wh c L wh c z z c z z b qb wh (h h b) c ( b) c L wh c z z c z z g T wt gl (h h )wh (h h ) wh 1 hw g b b wq c z c z c T T T c T b b b b g (h h b)wh (h h b) c w L wq (h h b)tw c z c z z c T b g T wt gl 1 hw g wq. (A4) ct T T ct b b b b REFERENCES Arakawa, A., and W. H. Schubert, 1974: Interactin f a cumulus clud ensemble with the large-scale envirnment, Part I. J. Atms. Sci., 31, Betts, A. K., and M. J. Miller, 1986: A new cnvective adjustment scheme. Part II: Single clumn tests using GATE wave, BOMEX, ATEX and arctic airmass data sets. Quart. J. Ry. Meter. Sc., 112, Cheng, M.-D., and M. Yanai, 1989: Effects f dwndrafts and messcale cnvective rganizatin n the heat and misture budgets f trical clud clusters. Part III: Effects f messcale cnvective rganizatin. J. Atms. Sci., 46, Li, X., C.-H. Sui, K.-M. Lau, and M.-D. Chu, 1999: Large-scale frcing and clud radiatin interactin in the trical dee cnvective regime. J. Atms. Sci., 56, Lis, F. B., and R. S. Hemler, 1986: Numerical simulatin f dee trical cnvectin assciated with large-scale cnvergence. J. Atms. Sci., 43, Lrenz, E. N., 1955: Available tential energy and the maintenance f the general circulatin. Tellus, 7, , 1978: Available tential energy and the maintenance f the mist circulatin. Tellus, 30, , 1979: Numerical evaluatin f mist available energy. Tellus, 31, Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium f the atmshere with a cnvective adjustment. J. Atms. Sci., 21, Maes, B. E., 1997: Equilibrium vs. activatin cntrls n large-scale variatins f trical dee cnvectin. The Physics and Parameterizatin f Mist Atmsheric Cnvectin, R. K. Smith, Ed., Kluwer Academic, Peixt, J. P., and A. H. Ort, 1992: Physics f Climate. American Institute f Physics, 520. Randall, D. A., and J. Wang, 1992: The mist available energy f a cnditinally unstable atmshere. J. Atms. Sci., 49, , and D.-M. Pan, 1993: Imlementatin f the Arakawa Schubert cumulus arameterizatin with a rgnstic clsure. The Re-

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