2116 JOURNAL OF CLIMATE VOLUME 11
Coupled Modes of the Warm Pool Climate System. Part I: The Role of Air±Sea Interaction in Maintaining Madden±Julian Oscillation
BIN WANG AND XIAOSU XIE* Department of Meteorology, University of Hawaii, Honolulu, Hawaii (Manuscript received 7 April 1997, in ®nal form 3 September 1997)
ABSTRACT Over the warm pool of the equatorial Indian and western Paci®c Oceans, both the climatological mean state and the processes of atmosphere±ocean interaction differ fundamentally from their counterparts over the cold tongue of the equatorial eastern Paci®c. A model suitable for studying the coupled instability in both the warm pool and cold tongue regimes is advanced. The model emphasizes ocean mixed layer physics and thermody- namical coupling that are essential for the warm pool regime. Different coupled unstable modes are found under each regime. In contrast to the cold tongue basic state, which favors coupled unstable low-frequency SST mode, the warm pool regime (moderate mean surface westerlies and deep thermocline) is conducive for high-frequency (intra- seasonal timescale) coupled unstable modes. The wind±mixed layer interaction through entrainment/evaporation plays a central role in the warm pool instability. The cloud-radiation feedback enhances the instability, whereas the ocean wave dynamics have little impact. The thermodynamic coupling between the atmosphere and ocean mixed layer results in a positive SST anomaly leading convection, which provides eddy available potential energy for growing coupled mode. The relatively slow mixed layer response to atmospheric forcing favors the growth of planetary-scale coupled modes. The presence of mean westerlies suppresses the low-frequency SST mode. The characteristics of the eastward-propagating coupled mode of the warm pool system compares favorably with the large-scale features of the observed Madden±Julian Oscillation (MJO). This suggests that, in addition to atmospheric internal dynamic instability, the ocean mixed layer thermodynamic processes interacting with the atmosphere may play an active part in sustaining the MJO by (a) destabilizing atmospheric moist Kelvin waves, (b) providing a longwave selection mechanism, and (c) slowing down phase propagation and setting up the 40±50-day timescale.
1. Introduction and Schopf 1988; Battisti and Hirst 1989). However, the ocean wave dynamics are not critical to SST changes The development of El NinÄo±Southern Oscillation in the warm pool (SST is normally higher than 28ЊC) (ENSO) is generally understood as resulting from cou- of the tropical western Paci®c and Indian Oceans where pled instability of the tropical ocean±atmosphere system thermocline is deep and horizontal SST gradients are (e.g., Philander et al. 1984; Yamagata 1985; Hirst 1986; small. Neelin 1991). In the ocean models used for the coupled The interaction of atmosphere and ocean over the instability analysis, the sea surface temperature (SST) warm pool occurs on a broad range of timescales rang- variations are controlled by dynamic processes associ- ing from intraseasonal to interannual or even longer. ated with thermocline displacement and temperature ad- The interaction on the intraseasonal timescale appears vection by currents and upwelling. These dynamic pro- to be at least as vigorous as that on the ENSO timescale. cesses are indeed of central importance for the coupled Prior to the Tropical Ocean Global Atmosphere Coupled instability of the eastern Paci®c and for sustaining bas- Ocean±Atmosphere Response Experiment (TOGA inwide ENSO cycle (e.g., Cane and Zebiak 1985; Suarez COARE), Krishnamurti et al. (1988) estimated that the magnitude of SST variations on the 30±50-day time- scale was about 0.5Њ±1.0ЊC. Recent TOGA COARE ob- *Current af®liation: Jet Propulsion Laboratory, California Institute servations reveal prominent intraseasonal variability in of Technology, Pasadena, California. both the SST and mixed layer depth. The amplitude of SST variation associated with the Madden±Julian Os- cillation (MJO) (Madden and Julian 1971) exceeds Corresponding author address: Dr. Bin Wang, Department of Me- teorology, University of Hawaii, 2525 Correa Rd., Honolulu, HI 1.0ЊC (Lau and Sui 1997). This intraseasonal variability 96822. is larger than that on the annual and ENSO timescales. E-mail: [email protected] The warm pool air±sea interaction involves processes
᭧ 1998 American Meteorological Society AUGUST 1998 WANG AND XIE 2117 that differ from those over the cold tongue region. The and energy conversion of the coupled unstable modes coupling between cloud±wind and ocean mixed layer to understand why the warm pool mean state and the thermodynamic processes is of central importance. thermodynamical coupling favor development of cou- TOGA COARE analyses indicate that the variability in pled high-frequency modes. In section 5, the coupled precipitation and cloud is, to a large extent, controlled instability of the cold tongue regime is brie¯y discussed by the MJO (Johnson 1995). Thus, the MJO can regulate in order to contrast the warm pool instability. Section shortwave radiative heating in the ocean mixed layer 6 compares TOGA COARE observations with the the- via changing cloudiness. Strong turbulent mixing and ory and articulates the potential roles of the ocean mixed entrainment as well as latent heat loss associated with layer interacting with the atmosphere in sustaining MJO. westerly wind bursts during the wet phase of MJO may The last section summarizes key points and discusses also cause considerable cooling of the mixed layer. This the limitations of the warm pool instability theory. results in a spatial phase relationship between SST and atmospheric circulation: positive SST anomalies lead convective anomalies by about a quarter of period (e.g., 2. The coupled atmosphere±ocean model for the Kawamura 1988; Nakazawa 1995; Zhang 1996). On the warm pool climate system other hand, atmospheric thermodynamic variables, such a. The model equations as precipitable water vapor and convective available po- tential energy, tend to follow SST variation and lead The atmospheric model describes linear motion of the convection anomalies (Chou et al. 1995; Fasullo and lowest baroclinic mode, which is stimulated by con- Webster 1995). Signi®cant variation of SST on intra- densational heating (Gill 1980). The heating rate is as- seasonal timescales has the potential to impact on at- sumed to be proportional to anomalous SST. This ele- mospheric MJO, but the degree to which and the way mentary form was derived from different perspectives by which SST affects the MJO are not known. in the past by considering various processes such as Whereas the warm pool system involves active convective parameterization (Philander et al. 1984; Hirst ocean±atmosphere interaction on both intraseasonal and 1986), evaporation (Zebiak 1986), longwave Newtonian ENSO development timescales, the nature of the cou- relaxation (Davey and Gill 1987), or equivalent SST pled mode has not been explored with an adequate the- gradient effect (Lindzen and Nigam 1987; Neelin 1989). oretical model. The majority of coupled-instability mod- The governing equations in p coordinates for the gravest els treat the atmosphere as a ``slave'' and thus exclude baroclinic mode on an equatorial  plane are ץ Uץ (high-frequency coupled modes. Lau and Shen (1988 and Hirst and Lau (1990) realized this problem. They ϩaU Ϫ yV ϭϪ , (2.1a) t x ץ ץ took into account the transient atmospheric waves in ץ Vץ -their atmospheric models. However, the ocean±atmo ϩ V ϩ yU ϭϪ , and (2.1b) yץ t aץ sphere coupling in their models is essentially the same dynamic coupling that is suitable for the cold tongue VRgץ Uץ ץ -regime; the processes that are important for SST vari ϩ ϩ C 2(1 Ϫ I) ϩϭϪ ␣ T, (2.1c) y 2Cp aץ xץt aaץ ation in the warm pool, such as wind- and cloud-induced anomalous diabatic heating, are absent. It is, therefore, p2 necessary to investigate the coupled modes that may where U, V, and are lower-tropospheric zonal and result from the distinctive cloud±wind±mixed layer ther- meridional wind and geopotential, respectively; a and modynamic coupling processes prevailing in the warm a are, respectively, coef®cients for Rayleigh friction pool. and Newtonian cooling; Ca is a dry atmospheric Kelvin Speci®c questions to be addressed include the fol- wave speed; I denotes a heating coef®cient associated lowing. What is the nature of the coupled ocean±at- with wave-induced moisture convergence, which is a mosphere modes under warm pool basic state and pre- function of mean SST (Wang 1988a), thus the moist 1/2 vailing thermodynamic coupling processes? How does Kelvin wave speed C a* ϭ Ca(1 Ϫ I) ; ␣a is a latent the ocean±atmosphere interaction contribute to the sta- heating coef®cient; T is SST or ocean mixed layer tem- bility of the coupled warm pool system? To what degree, perature anomaly; and R, g, Cp, and p 2 denote the gas and how, does the ocean feed back to the MJO? constant, gravity, speci®c heat at constant pressure, and The main purpose of the present study is to investigate the pressure at the middle troposphere, respectively. the impacts of oceanic mixed layer processes and ocean± Equations (2.1a)±(2.1c) are essentially the same as the atmosphere thermodynamic coupling on coupled insta- atmospheric model used by Hirst and Lau (1990), except bility. In the next section, a new theoretical model is a heating term related to the zonal wind anomaly was formulated for coupled stability analysis in the warm neglected. The effect of the neglected term on the cou- pool regime. In section 3 a family of progressively re- pled instability was discussed in detail by Hirst and Lau duced physics models are used to identify roles of dif- (1990). ferent coupling processes in stimulating coupled insta- The ocean model differs from those used in previous bility. Section 4 further examines the dynamic structure coupled stability analyses. The atmosphere and ocean 2118 JOURNAL OF CLIMATE VOLUME 11
interact through both momentum and heat exchanges at Both the variations in h and h1 would further modify the ocean surface. The heat exchange is in¯uenced by mixed layer temperature. cloudiness via changing shortwave radiational heating Typical values of the model parameters used in the and by surface winds via changing turbulent entrainment present model are given in Table 1. The empirical co- and surface evaporation. Detailed derivation of the ef®cient measures the proportionality between per- ocean model is given in the appendix. turbation cloudiness and surface wind convergence. A Over the warm pool, the governing equations for the value of 2 ϫ 105 s means that an anomalous surface perturbation motion in the active upper ocean and the wind convergence of 5 ϫ 10Ϫ7 sϪ1 may result in an embedded mixed are (see appendix) increase in total cloudiness by one-tenth. The characteristic parameters for the western and hץ uץ Ϫ y ϭϪb ϩ ␣ooU Ϫ u, (2.2a) eastern Paci®c mean states are listed in Table 2. Over x the warm pool, it is assumed that mean surface frictionalץ tץ Ϫ1 h velocity u* ϭ 0.48 cm s and the entrainment layerץ ץ Ϫ yu ϭϪb ϩ ␣ooV Ϫ , (2.2b) depth is 10 m. The corresponding mean entrainment rate y Ϫ6 Ϫ1ץ tץ w e ϭ 2 ϫ 10 ms [see Eq. (A.3a)], which induces a cooling rate that balances mean net downward heat Ϫ2ץ uץ hץ ϩ H ϩϭ0, (2.2c) ¯ux Q 0 Џ 15Wm [see Eq. (A.3b)]. The estimated y mean downward solar radiation ¯ux for mean cloudinessץ xץt ץ C ϭ 0.6 is 190 W mϪ2. The typical value ␣ ϭ 12 kg h 3Uh Hh aץ 11sϪ3 KϪ1 and mean SST T 303 K yield a mean surface ϭ EUU ϩ w e ϩϪ , ϭ Ϫ2 tUHHH221 latent heat ¯ux of about 120 W m . The estimated meanץ longwave radiation ¯ux with a coef®cient ␥ ϭ 1.8 W (2.2d) mϪ2 KϪ1 (Seager et al. 1988) is Ϫ55 W mϪ2. These and estimates approximately match TOGA COARE obser- vations (Godfrey et al. 1995; Lau and Sui 1997). V 3Uh1 There are a number of coupling coef®cients whoseץ Uץ Tץ ϭ D ϩϪD Ϫ rad ent values depend on the basic state (Table 2). The wind yUH1ץ xץt ץ stress coef®cient ␣o ϭ (aCD|U|/oH) determines the UT ef®ciency of the atmospheric momentum ¯ux into the 3 Ϫ3 Ϫ DevaϩϪ oT, (2.2e) ocean, where C is drag coef®cient, ϭ 10 kg m UTϪ 293 D o the reference density in the deep ocean layer, a ϭ 1.2 where dependent variables are vertical mean upper- kg mϪ3 the air density in the atmospheric boundary layer, ocean zonal (u) and meridional () velocity, thermocline and U the mean surface zonal wind. The latent heating depth (h), mixed layer depth (h1), and mixed layer tem- coef®cient ␣a ϭ aCELCKq|U| determines the ef®ciency perature (T); b is the buoyancy (reduced gravity); o is of latent heat ¯ux into the atmosphere, where Lc and CE Rayleigh damping coef®cients in the upper ocean; H1 represent latent heat and moisture exchange coef®cients, Ϫ4 Ϫ1 and H denote the mean depths of the mixed layer and respectively; Kq ϭ 8.9 ϫ 10 K (Wang and Li 1993). thermocline, respectively; H 2 ϭ H Ϫ H1; ␣o is a wind The coef®cients Drad, Dent, and Deva [de®ned by (A.6a), stress coef®cient; E is Ekman pumping coef®cient [Eq. (A.6b), and (A.6c)] measure, respectively, the degree of
(A1eЈ)]; w e is the mean entrainment rate [Eq. (A3a)]; in¯uence on SST from cloud-dependent radiation ¯ux, o is a Newtonian cooling coef®cient in the ocean mixed wind-dependent entrainment, and evaporation. layer; and Drad, Dent, and Deva are, respectively, the mixed layer heating coef®cients associated with short- wave radiation, entrainment, and evaporation processes. c. Numerical scheme These coef®cients are de®ned by Eqs. (A.6a)±(A.6d). Normal mode solutions of Eqs. (2.1) and (2.2) are sought of the form: b. Model physics and parameters (U, V, ,u,,h,h1, T) ϭ Re[U,˜ V,ÄĘ ,uÄ, Ä, h˜,h˜ , T]ei(kxϪt), (2.3a) Figure 1 schematically summarizes the physics of the 1 present coupled model. The mixed layer temperature where ``ϳ'' denotes corresponding meridional structure, affects atmospheric heating and changes lower-tropo- k is nondimensional wavenumber, and is frequency spheric geopotential and surface winds. Changes in sur- (eigenvalue). The lateral boundary conditions are face winds alter mixed layer temperature through di- (U, V, ,u,,h,h, T) ϭ 0aty ϭϮY . (2.3b) rectly changing evaporation rate, entrainment rate, and 1 b surface convergence/cloudiness (thus the shortwave ra- With application of (2.3a,b), a ®nite-difference diation ¯ux). The surface winds also control thermocline scheme in the meridional direction transforms Eqs. (2.1) displacement h via generating oceanic waves/currents and (2.2) into a linear algebraic system, which was and in¯uence mixed layer depth h1 via Ekman pumping. solved by matrix inversion. As the meridional boundary AUGUST 1998 WANG AND XIE 2119
FIG. 1. Schematic diagram highlighting the model physics.
(Yb) was set beyond 20Њ and the resolution (⌬y) less cluded: (a) turbulent entrainment, surface evaporation, than 0.75Њ, the solutions of the principal unstable modes and Ekman pumping in the mixed layer, which are de- were not sensitive to the changes in Yb and ⌬y, indicating termined, to a large extent, by surface winds; (b) short- that Yb ϭ 20Њ and ⌬y ϭ 0.75Њ are adequate to resolve wave radiational heating, which is controlled by the the equatorially trapped coupled unstable modes. amount of cloudiness or low-level convergence; and (c) the thermocline displacement associated with ocean wave dynamics. For convenience, we will refer to the 3. Coupled unstable modes in the warm pool coupling associated with the above processes (a), (b), regime and (c) as wind-entrainment±evaporation feedback, a. A hierarchy of reduced-physics models cloud-radiation feedback, and wind±ocean wave dynam- ics feedback, respectively. In the full ocean model for warm pool system [Eq. To identify the roles of the above three coupling pro- (2.2)], three types of processes affecting SST are in- cesses on the coupled instability, two simpli®ed versions
TABLE 1. Typical values of model parameters.
Tr Reference temperature in the deep ocean 283 K
H1 Mean mixed layer depth 40 m Ϫ3 CD Drag coef®cient 1.5 ϫ 10 Ϫ3 CE Moisture transfer coef®cient 1.5 ϫ 10 A Surface albedo 0.06 Ϫ2 S0 Surface solar radiation ¯ux under clear sky 320Wm Ϫ1 Ca Dry atmospheric internal Kelvin wave speed 50ms m Turbulent mixing coef®cient due to wind stirring 1.25 Ϫ5 Ϫ1 rs Viscosity coef®cient in ocean Ekman layer 1 ϫ 10 s Ϫ6 Ϫ1 a Rayleigh friction coef®cient in the atmosphere 1 ϫ 10 s Ϫ7 Ϫ1 o Rayleigh friction coef®cient in the upper ocean 1 ϫ 10 s Ϫ6 Ϫ1 a Newtonian damping coef®cient in the atmosphere 1 ϫ 10 s Ϫ8 Ϫ1 o Newtonian damping coef®cient in the ocean mixed layer 1 ϫ 10 s Proportionality coef®cient between cloud and wind convergence 2 ϫ 105 s 2120 JOURNAL OF CLIMATE VOLUME 11
TABLE 2. The basic-state parameters and associated coupling coef®cients in the warm pool and cold tongue regions. Warm pool Cold tongue H (m) Mean thermocline depth 150 70 U (m sϪ1) Mean surface wind 3 Ϫ5 T (K) Mean SST 303 299 * Ϫ1 Ca (m s ) Moist atmospheric Kelvin wave speed 10 25
T Ϫ Te (K) Difference between T and mean entrained 1.8 5.3 (upwelled) water temperature Ϫ1 Ϫ6 Ϫ6 we (m s ) Mean entrainment (upwelling) rate 2 ϫ 10 4.4 ϫ 10 Ϫ3 Ϫ1 ␣a (kg s K ) Latent heating coef®cient 12 20 Ϫ1 Ϫ7 Ϫ7 ␣o (s ) Wind stress coef®cient 0.36 ϫ 10 1.8 ϫ 10
Drad (K) ML heating coeff. due to shortwave radiation 0.2 0.0 Ϫ1 Ϫ7 Ϫ7 Dent (K s ) ML heating coeff. due to entrainment (upwelling) 0.9 ϫ 10 1.68 ϫ 10 Ϫ1 Ϫ7 Ϫ7 Deva (K s ) ML heating coeff. due to evaporation 0.72 ϫ 10 1.2 ϫ 10 of the present model were designed with a progressively this no-wave-dynamics limit, the ocean model consists reduced level of complexity in the ocean processes. of only two mixed layer equations: Model Ia eliminates the wind-wave dynamics feed- h113UHhץ back. For this purpose, the thermocline depth is assumed ϭ EUU ϩ w Ϫ and (3.1a) to be a constant H (the mean depth)Ðthat is, the per- e tUHH21ץ turbation thermocline depth h vanishes and the ocean
V 3Uh1ץ Uץ Tץ wave dynamics decouples with mixed layer physics. In ϭ DradϩϪD ent Ϫ yUH1ץ xץt ץ UT Ϫ D ϩϪ T. (3.1b) evaUTϪ 293 o Model Ib further neglects cloud-radiation feedback
by taking Drad ϭ 0 in (3.1b). The resulting oceanic equa- tions become h113Uhץ ϭ EUU ϩ w e Ϫ and (3.2a) tUH1ץ
TUh1ץ ϭϪC ϩ Dent Ϫ dT, (3.2b) tUH1ץ where
C ϭ 3Dentϩ D eva. (3.2c)
b. Unstable modes due to wind-entrainment±evaporation feedback Let us ®rst examine the nature of the coupled modes in the simplest model Ib in which the change in SST is solely due to wind-induced mixed layer entrainment, evaporation, and Ekman pumping. A simple scaling in- dicates that the ®rst term on the rhs of Eq. (3.2b) plays a dominant role in changing mixed layer temperature. This term represents the major portion of the wind- induced evaporation and entrainment cooling. The ther- mal coupling coef®cient C measures the degree of in- ¯uence of anomalous wind-induced entrainment and surface latent heat ¯ux on mixed layer temperature. FIG. 2. Dependence of the growth rate (a) and phase speed (b) of Figure 2 shows the growth rate of coupled unstable the coupled high-frequency Kelvin (FK) and Rossby (FR) modes on wind-entrainment±evaporation coupling coef®cient C (K sϪ1) in mod- modes as function of thermodynamic coupling coef®- el Ib of the warm pool system [Eq. (3.2)]. The atmospheric Rayleigh cient C. When C exceeds a critical value, the mean state friction and Newtonian damping were neglected. becomes unstable and two coupled growing modes AUGUST 1998 WANG AND XIE 2121 emerge. One propagates eastward with an average speed of 6±7 m sϪ1. This mode originates from neutral at- mospheric moist Kelvin wave, and, for convenience, it is referred to as coupled high-frequency Kelvin mode (CHFK) in this paper. The other propagates westward with a speed about 4±5 m sϪ1. It is rooted in atmospheric moist Rossby wave and is referred to as coupled high- frequency Rossby mode (CHFR). The instability thresh- old value of C required by coupled Rossby mode is lower than that for coupled Kelvin mode. For a typical value of C (1 ϫ 10Ϫ6 KsϪ1), however, the coupled Kelvin mode grows faster than the coupled Rossby mode. The growth rates of the unstable modes have an e-folding timescale on an order of one to two weeks in the absence of dissipation. The coupling of the atmo- spheric wind with mixed layer entrainment and evap- oration reduces the eastward propagation speed of the original atmospheric moist Kelvin waves (10 m sϪ1), but it increases the westward phase speed of the original atmospheric moist Rossby waves (Ϫ3.3msϪ1). The ratio 3Dent/Deva measures the relative contribution to coupled instability of evaporation vs entrainment. Figure 3 shows the growth rate and phase speed as functions of wavenumber (wavelength) in model version Ib for the given typical value of C. Both the CHFK and CHFR modes prefer planetary scales. Wind±mixed layer coupling suppresses short waves. The suppression of short waves results from the feedback of SST to at- mospheric waves. Shorter waves have less time to affect
SST persistently when they pass over the ocean, because FIG. 3. The growth rate (a) and phase speed (b) of the coupled the perturbation wind reverses its direction more fre- high-frequency Kelvin (FK) and Rossby (FR) modes as functions of quently. This would only yield a moderate change in nondimensional wavenumber for a ®xed wind-entrainment±evapo- SST, which in turn results in a weak feedback to wave ration coupling coef®cient C ϭ 1 ϫ 10Ϫ6 KsϪ1 in model Ib. The atmospheric Rayleigh friction and Newtonian damping were neglect- growth. The unstable CHFK and CHFR modes are ed. The arrows with symbols ``30 kkm'' and ``15 kkm'' indicate the weakly dispersive as seen from the dependence of phase nondimensional wavenumbers that correspond to a wavelength of speed on wavelength. 30 000 km and 15 000 km, respectively. c. Effect of cloud-radiation feedback mospheric heating representation by introducing a time lag in the latent heat release, as ®rst suggested by Davies In model Ia, the effect of atmospheric cloudiness on (1979), the shortwave instability would be effectively SST is included. Comparison of the results obtained suppressed and longwave instability retains (®gure not from models Ia and Ib would reveal effects of cloud- shown). On the other hand, for unstable CHFR mode, radiation feedback. It turns out that the growth rate and the cloud-radiation feedback remains favorable for plan- phase speed of both the coupled unstable modes (CHFK etary scales. and CHFR) are modi®ed by cloud-radiation feedback, For a typical value of C ϭ 1 ϫ 10Ϫ6 KsϪ1, the growth but their fundamental characteristics remain qualita- rate and the most unstable wavelength do not vary with tively unchanged. D signi®cantly until D becomes unrealistically large Figure 4 displays the most unstable wavelength and rad rad (say Drad is greater than 1 K). For typical value of Drad corresponding growth rate and phase speed as functions ϭ 0.2 K, the effects of cloud-radiation feedback on the of cloud-radiation feedback coef®cient Drad and wind- most unstable wavelength, growth rate, and phase speed entrainment±evaporation feedback coef®cient C for are moderate. This implies that the effects of wind-en- CHFK mode. When C is ®xed and small (C Ͻ 3.5 ϫ trainment±evaporation feedback dominate over those of Ϫ7 Ϫ1 10 Ks ), stronger cloud-radiation feedback (larger cloud-radiation feedback. value of Drad) yields faster growth and propagation. In particular it enhances shortwave instability. In the ab- sence of evaporation and entrainment cooling processes d. Effects of wind±ocean wave dynamics feedback (C ϭ 0), the cloud-radiation feedback alone would favor One might wonder what would happen when ocean the shortest CHFK mode. However, if we modify at- dynamics become progressively more important. Com- 2122 JOURNAL OF CLIMATE VOLUME 11 AUGUST 1998 WANG AND XIE 2123 putations show that the solution obtained with the full phase tilt depends on dissipation parameters and the ocean model is nearly identical to that obtained using growth rate. model Ia. Evidently, for the warm pool basic state, the For a zonal wind perturbation with an amplitude of oceanic wave dynamics have little impact on the coupled 2msϪ1, the corresponding amplitude of SST pertur- instability. bation is about 0.3ЊC. The SST response associated with Even when one further reduces the mean thermocline the CHFK mode is thus substantially stronger than that depth H from 150 to 50 m in the full model, the insta- in Hirst and Lau's (1990) intraseasonal mode (by a fac- bility of the warm pool basic state remains insensitive tor of 3). This suggests the importance of thermody- to change of thermocline depth or to the dynamic control namic coupling of the atmosphere and ocean mixed lay- of SST. No slow unstable coupled mode was found in er. The ratio of magnitude between SST and surface the present model. This indicates that the wind±mixed wind anomalies appears to be reasonable compared to layer thermodynamic coupling in the presence of mean COARE observations. The latter show that after strong westerlies suppresses slow unstable modes that other- westerly burst episodes (amplitude is about 10 m sϪ1), wise were possibly originated from oceanic processes. SST drops about 2ЊC on an intraseasonal timescale (Lau As will be shown in section 6, in an easterly mean and Sui 1997). state (the cold tongue regime), the wind±mixed layer The phase relationship between atmospheric ®elds interaction cannot suppress low-frequency unstable and mixed layer temperature (or SST) is particularly mode if the mean thermocline depth is suf®ciently shal- interesting for the coupled mode. For the growing low (less than 100 m, for instance). The suppression of CHFK mode, positive SST anomalies lead low-level the low-frequency unstable mode by the wind±mixed convergence and rainfall by about two-tenths of a cycle layer interaction depends on the mean surface wind di- (or about 70Њ in phase) (Fig. 5b). This phase shift results rection. This is because the direction of the mean surface primarily from competitive effects of two processes. On zonal winds determines the existence of the mean up- the one hand, westerly anomalies augment mean west- welling or downwelling and thus determines the dom- erlies, thus enhancing evaporation and entrainment cool- inant process responsible for SST variation. ing in the ocean mixed layer. This process alone would tend to make westerly anomalies lead negative SST anomalies by a quarter of a wavelength. On the other 4. Mechanism of the coupled warm pool instability hand, the enhanced convection tends to reduce net a. Dynamic structure downward shortwave radiation ¯ux and to cool mixed Figure 5 shows the horizontal structure of the unstable layer water. The cloud-induced cooling region tends to CHFK mode in model Ia. Similar to neutral equatorial be located a quarter wavelength to the west of convec- Kelvin waves, all ®elds are symmetric about and trapped tion anomalies and thus roughly coincide with the sur- to the equator; the zonal winds are in geostrophic bal- face westerly anomalies. As a result of the combined ance with the meridional pressure gradient. The low- in¯uences from the above two processes, the negative level westerlies are no longer precisely in phase with SST anomalies slightly lag the anomalous westerlies. but rather slightly lead the positive pressure (or negative Meanwhile, positive SST anomalies lead enhanced con- thickness±temperature) anomalies. Note that the con- vection by about 70Њ in phase. Mixed layer deepening stant phase lines away from the equator tend to tilt west- also lags anomalous westerlies by about 70Њ in phase ward and poleward. As such, the eastward propagation (Fig. 5c), because the westerlies tend to deepen mixed of the unstable mode would yield a poleward propa- layer via Ekman divergence-induced downwelling while gation component away from the equatorial region. The the entrainment-induced mixed layer depth ¯uctuation tilt of the constant phase line with latitude is a char- is of secondary importance. acteristic of unstable equatorial trapped waves. The sign change of the Coriolis force at the equator traps Kelvin b. The energy conversion for coupled unstable modes waves to the equator. The trapping scale is the equatorial Rossby radius of deformation, which depends on the To facilitate interpretation of the coupled instability, Kelvin wave propagation speed C. When the Kelvin it is useful to examine energy conversion. Let us focus wave becomes unstable (C is complex), the trapping on the atmospheric perturbation of the coupled mode. scale depends on the phase speed (real part of C), while De®ne ensemble mean atmospheric eddy kinetic energy 2 2 the phase itself becomes a function of latitude. For a per unit mass as Ka ϭ (U ϩ V )/2, and atmospheric 2 2 growing (decaying) wave, the constant phase line tilts eddy available potential energy as Pa ϭ /2C a . Use westward (eastward) and northward. The degree of an overbar to represent horizontal average over a zonal
←
FIG. 4. The wavelength (a), growth rate (b), and phase speed (c) of the most unstable mode as functions of the cloud-radiation coupling Ϫ1 coef®cient Drad (K) and wind-entrainment±evaporation coupling coef®cient C (K s ) for the coupled high-frequency Kelvin mode in model Ia. 2124 JOURNAL OF CLIMATE VOLUME 11 AUGUST 1998 WANG AND XIE 2125
wavelength and from y ϭϪYb to y ϭ Yb, where ϪYb ation of EAPE (Fig. 6c) and the development of the and Yb denote southern and northern boundaries of the CHFK mode. It is clear that the atmosphere±ocean coupled model, respectively. The following energy mixed layer interaction is responsible for the unstable equations can be derived from Eqs. (2.1): CHFK mode in the present model. A parallel analysis for the ocean component shows K -a that the SST perturbation is ampli®ed by wind-entrain ץ ϭ͗P, K͘Ϫ2aaK , (4.1a) -t ment±evaporation feedbackÐthat is, the SST perturץ bation grows when the westerly (easterly) anomalies and Paץ ϭ͗P, K͘ϩGP12aaϩ GP Ϫ 2 P , (4.1b) negative (positive) SST anomalies have a positive co- .t variance. This is indeed the case for the CHFK modeץ where The atmospheric component of the unstable CHFR mode displays a structure similar to that of m ϭ 1 Ross- .V by wave, where m is an index for meridional mode (Figץ Uץ ͗P, K͘ϭ ϩ , (4.2a) y 7a). Strong westerly anomalies are associated with twinץ xץ cyclones residing on both sides of the equator. Rainfall V areas are located not only at the equator but also at theץ Uץ GP1 ϭ I ϩ , and (4.2b) off-equatorial region. The off-equatorial rainfall regions y lag positive SST anomalies by about 60Њ, while theץ xץ ␣ equatorial rainfall region is nearly in phase with positive GP ϭϪ a T . (4.2c) 2 C 2 SST anomalies. The generation of atmospheric EAPE a due to air±sea coupling is primarily in the off-equatorial The term ͗P, K͘ denotes conversion from atmospheric regions where the low-level low pressure is strongly eddy available potential energy (EAPE) to kinetic en- correlated with positive SST anomalies (®gure not ergy. Here, GP1 denotes generation rate of atmospheric shown). EAPE due to condensational heating associated with low-level moisture convergence, which depends on the 5. Coupled modes of the cold tongue regime covariance between low-level moisture convergence and atmospheric temperature anomaly, while GP2 is the Over the equatorial eastern Paci®c, the ocean dynam- generation rate of atmospheric EAPE due to the heating ics have an essential control on SST variation, whereas induced by SST anomalies, which depends on the co- entrainment and diabatic heating processes in the mixed variance between SST and atmospheric temperature layer are of secondary importance. The governing equa- anomalies. tions for the perturbation motion in the cold tongue The structure of the unstable CHFK mode indicates regime are (see appendix) hץ uץ .a favorable energy conversion for the unstable mode Because low-level convergence and convective rainfall Ϫ y ϭϪb ϩ ␣ooU Ϫu, (5.1a) xץ tץ -tend to overlap with positive temperature (negative geo hץ ץ potential in the lower troposphere) anomalies, a positive Ϫ yu ϭϪb ϩ ␣ V Ϫ, (5.1b) y ooץ tץ -covariance between temperature anomalies and low-lev el convergence-induced heating results in a generation ץ uץ hץ of EAPE [the GP1 term in Eq. (4.1b); Fig. 6a], which is further converted to perturbation kinetic energy. ϩ H ϩϭ0, and (5.1c) yץ xץt ץ Note that the GP1 term alone is insuf®cient for un- Vץ UץTЈ hЈץ .stable growth because it is smaller than ͗P, K͘ (Fig 6b). That means the atmosphere is dynamically stable ϭ Dupϩ D rad ϩ yץ xץtH2 ץ in the absence of a coupling to the ocean. It is the additional GP2 term that makes the CHFK mode grow. U The term GP2, which is proportional to the covariance Ϫ (Dupϩ D eva) Ϫ Ä oTЈ, (5.1d) between negative low-level pressure and positive SST U anomalies [Eq. (4.2c)], is due to the atmospheric heating where coef®cients Dup andÄ o are de®ned by Eqs. associated with SST perturbation. For growing CHFK (A.8a,b). The atmospheric equations are the same as mode, positive SST and negative low-level pressure Eqs. (2.1a)±(2.1c). anomalies are positively correlated. This yields gener- For convenience, the model equations (5.1a)±(5.1d)
←
FIG. 5. Horizontal structure of the coupled unstable high-frequency Kelvin mode in model Ia: (a) low-level geopotential (contour interval 2 Ϫ2 6m s ) and winds; (b) SST (ЊC) anomaly and anomalous precipitation centers denoted by Prϩ and PrϪ; (c) mixed layer depth (m) and current anomalies. 2126 JOURNAL OF CLIMATE VOLUME 11
FIG. 6. The rate of energy generation for the most unstable coupled high-frequency Kelvin mode in model Ia. (a) Eddy available potential energy (EAPE) generation due to condensational latent
heating induced by moisture convergence (GP1); (b) the difference between GP1 and the conversion from EAPE to kinetic energy, ͗P, K͘; and (c) EAPE generation due to heating associated with
SST anomaly (GP2). All quantities are normalized by ͗P, K͘. AUGUST 1998 WANG AND XIE 2127
FIG. 7. The same as in Fig. 5 except for the coupled high-frequency Rossby mode. 2128 JOURNAL OF CLIMATE VOLUME 11
FIG. 8. Dependence of the growth rate (a) and phase speed (b) of the coupled modes on
thermocline depth H (m) in the cold tongue regime computed from model IIa. Symbols U, Ko,
and Ro denote coupled low-frequency SST mode, Kelvin mode, and Rossby mode, respectively. are referred to as version IIa. If the diabatic heating a longer timescale. In model IIa, oceanic waves (Kelvin term is further neglected, Eq. (5.1d) reduces to and Rossby waves) remain damped in the presence of ocean±atmosphere interaction and dissipation (Fig. 8a). TЈ hЈ TЈ UЈ There is only one unstable mode that owes its origin toץ ϭ Dup ϪϪ. (5.2) tHT2rϪ TU thermodynamic processes. This result agrees well withץ For convenience, the set of Eqs. (5.2) and (5.1a)±(5.1c) the result of Hirst (1986), although the model here is is called model IIb, which is essentially a simpli®ed slightly different from his. Neelin (1991) named this linear version of Zebiak and Cane's (1987) coupled type of unstable mode slow SST mode. It is originated model. from the vertical temperature advection associated with In contrast to the warm pool regime, the cold tongue upwelling process, while the horizontal temperature ad- mean state (Table 2) supports coupled unstable modes, vection may slightly modify this mode (Hirst 1986). which are associated with oceanic processes and have The growth rate of the unstable SST mode increases AUGUST 1998 WANG AND XIE 2129
SST anomalies lead westerly anomalies by only one- eighth of the cycle. The coupled high-frequency modes, which are un- stable in the warm pool regime, are suppressed in the cold tongue regime. This is true not only with the model II, but also with the full model (®gure not shown). Sen- sitivity experiments reveal that the suppression of high- frequency coupled modes is due to the increase of at- mospheric static stability in the cold tongue and due to the reversal of the mean surface wind direction from westerly to easterly. The development of CHFK mode crucially depends on both the mean surface westerlies and the reduced atmospheric static stability, whereas the unstable CHFR mode requires only the latter.
6. Roles of air±sea interaction in maintaining Madden±Julian Oscillation TOGA COARE has provided an unprecedented op- portunity to observe the behavior of MJO and accom- panying oceanic variations. Figure 10a presents a con- cise summary of the results obtained from recent em- pirical studies. The structure of MJO in the equatorial zonal plane consists of a wet and a dry phase. The wet phase features a large-scale convective envelope whose core consists of super cloud clustersÐan ensemble of two-day waves and mesoscale convective systems (Chen et al. 1996; Sui et al. 1997). The convective region is accompanied FIG. 9. The growth rate (a) and phase speed (b) of the low-frequency by large-scale rising motion and planetary-scale, equa- unstable SST mode as functions of nondimensional wavenumber. The torial upper-level easterly and low-level westerly anom- atmospheric Rayleigh friction and Newtonian cooling coef®cients used are ϭ ϭ 2 ϫ 10Ϫ6 sϪ1. The arrow with symbol ``20 kkm'' alies (Lin and Johnson 1996). The low-level (and sur- indicates the nondimensional wavenumber that corresponds to a face) westerly anomalies, however, lag behind enhanced wavelength of 20 000 km. convection by slightly less than a quarter-wavelength (e.g., Chen et al. 1996; Chou et al. 1995). This concurs with the result of Fasullo and Webster (1995), who with decreasing mean thermocline depth in model IIa found that 850-hPa zonal winds lag rainfall and maxi- (Fig. 8a). When the thermocline is deeper than about mum cloud optical depth by a temporal phase of about 120 m, the coupled instability is very weak and the a week. The above vertical structure agrees well with unstable mode propagates westward slowly (Ͻ10 cm the composite structure of MJO previously documented sϪ1) (Fig. 8b); when the mean thermocline is shallower by Rui and Wang (1990), Salby and Hendon (1994), than about 90 m, the growth rate corresponds to an e- and Hendon and Salby (1994). folding time of about 150 days and the unstable mode A number of new features concerning the phase re- propagates eastward with a faster speed (20±30 cm sϪ1). lationship between MJO and upper-ocean variations Comparison of the results obtained from model IIa have been revealed, which are critical to understanding and IIb indicates that, without surface heat ¯ux ex- the dynamics of MJO and to validating current theories. change, the SST mode grows relatively fast (the solid First, the surface latent heat ¯ux associated with MJO curve in Fig. 9a) with a reduced most unstable wave- exhibits pronounced longitudinal asymmetries: En- length. The eastward propagating speed for long waves hanced evaporation anomalies almost coincide with (longer than 20 000 km) is much reduced. The unstable low-level westerly anomalies, whereas reduced latent waves shorter than 10 000 km move westward. The SST heat ¯ux is found to be nearly in phase with easterly mode is dispersive. anomalies (Jones and Weare 1996; Zhang 1996; Lin and The structure of the unstable SST mode obtained in Johnson 1996). This feature contradicts the premise that model IIa is almost the same as that of the ocean thermal strong latent heat ¯ux occurs in the easterly phase of mode obtained by Hirst and Lau (1990) in their model MJO due to the presence of basic-state easterlies. Thus, IV (®gure not shown). In contrast to the CHFK mode, theories based on this premise cannot explain the east- the SST anomalies associated with the slow SST mode ward propagation of MJO (Wang 1988b). Second, the are nearly in phase with rainfall, whereas the positive positive SST anomalies lead enhanced convection by 2130 JOURNAL OF CLIMATE VOLUME 11
FIG. 10. Schematic diagram illustrating the equatorial vertical structure of the Madden±Julian Oscillation observed in TOGA COARE (a) and the coupled high-frequency Kelvin mode in the present model (b). The squiggly lines denote surface latent heat ¯ux. The symbols ABL, OML, LHF, WWB, CAPE, and L represent, respectively, at- mospheric boundary layer, ocean mixed layer, latent heat ¯ux, westerly wind bust, convective available potential energy, and wavelength. The observations are based on the results obtained by Chen et al. (1991), Sui et al. (1996), Zhang (1996), Lin and Johnson (1996), Jones and Weare (1991), Fasullo and Webster (1995), and Chou et al (1995). slightly less than a quarter of a wavelength. This is in matic ®gure is based on the model results shown in Fig. good agreement with the previous results derived from 5. The overall structure of the CHFK mode resembles COADS analyses (Kawamura 1988). The present model that of the MJO and, in particular, the phase relationship analysis has explained what is responsible for this phase between atmospheric circulation and SST. However, the lead and what roles this phase lead plays in supporting phase relationship between the convection and low-level coupled instability. westerlies is not well captured by the model. This ap- The dynamic structure of the most unstable CHFK pears to be a common weakness of the current theo- mode, which results from the coupled instability of retical models of the MJO. In addition to the similarity warm pool mean state, is shown in Fig. 10b. This sche- in the dynamic structure, the preferred wavelength and AUGUST 1998 WANG AND XIE 2131 slow eastward propagation of the CHFK mode also bear equatorial Indian and western Paci®c Oceans. The lon- close resemblance to the MJO. The similarities between gitudinal scale of the warm pool may limit the zonal the CHFK mode and the observed MJO in the phase scale of the coupled mode, yielding the observed wave- relationships between atmospheric and ocean mixed lay- number 2 structure in the realistic warm pool geography. er anomalies further add con®dence to the relevance of The presence of the maritime continent, on the other the CHFK mode to the MJO. Thus, the behavior of the hand, may reduce the strength of the MJO through a CHFK mode may be instrumental for understanding number of processes: (a) reducing the coupled instability some of the observed features of the MJO. by prohibiting air±sea interaction; (b) raising the level Previous theories of the MJO have been based on of maximum latent heat release (due to land surface atmospheric internal dynamics without invoking vari- heating effect), which weakens the interaction between ations in the ocean mixed layer. Convective interaction convective heating and atmospheric Kelvin waves (e.g., with large-scale atmospheric dynamics has been rec- Miyahara 1987); (c) decreasing availability of moisture ognized as a key element of the theoretical modeling of supply due to continuing release of energy during the the MJO. Although these models reproduced some ob- enhanced diurnal rainfall; and (d) increasing surface dis- served features of the MJO, a fundamental dif®culty is sipation over the island topography. The above argu- the lack of planetary wave selection mechanism. The ments may explain why the maritime continent is a re- simulated eastward propagation in many models tends gion of destruction to the intensity of intraseasonal con- to be much faster than that observed. vective anomalies (Rui and Wang 1990). The present model results suggest a new mechanism The behavior of the coupled unstable Rossby waves for the MJO that is complimentary to the mechanism may be relevant to the observed westward propagating of atmospheric internal dynamic instability. The new intraseasonal mode. In boreal summer, there is an in- mechanism involves atmosphere±ocean mixed layer in- traseasonal mode that moves westward along 15ЊNin teraction through thermodynamic processes. It is dem- the western Paci®c (Murakami 1980; Chen and Mura- onstrated that the wind±cloud±mixed layer coupling via kami 1988; Nitta 1987). This mode appears to originate changing entrainment rate, latent heat, and shortwave from moist Rossby waves interacting with convective radiation ¯uxes renders growing SST perturbation and, heating and the underlying SST. When an equatorial in particular, makes positive SST anomalies lead low- eastward propagating convective complex reaches the level enhanced convection by about (2/5). There is a central Paci®c where SST decreases sharply, the anom- signi®cant feedback from ocean mixed layer to atmo- alous circulation tends to decouple with the convective spheric moist wave dynamics. The feedback may play heating and decays; meanwhile the Rossby waves em- three important roles in sustaining MJO: to provide an anate from the equatorial disturbance in the western energy source for the ampli®cation of atmospheric Kel- North Paci®c and move westward to the South Asian vin waves, to select planetary-scale most unstable monsoon region (Wang and Xie 1997). modes, and to considerably reduce their eastward prop- agation speed (by about one-third), which helps to match the MJO timescale (40±50 days). 7. Summary Without the coupling to ocean mixed layer, the pres- ent model atmosphere has no internal unstable modes. The basic state and air±sea interaction processes dif- It is the atmosphere±mixed layer thermodynamic cou- fer fundamentally between the warm pool of the Indian pling that destabilizes atmospheric moist Kelvin waves. and western Paci®c Oceans and the cold eastern Paci®c. The phase leading of SST to enhanced convection cre- This leads to the following hypothesis: The nature of ates a positive covariance between SST and atmospheric the coupled ocean±atmosphere instability over the warm temperature anomalies, which generates EAPE for the pool may essentially differ from that of the equatorial growing CHFK mode. Observations have shown that cold tongue. the equatorial Indian Ocean and the western Paci®c To test this hypothesis, a linear coupled ocean±at- Ocean are two major regions where the intraseasonal mosphere model is advanced that emphasizes ocean convective anomalies intensify (Wang and Rui 1990). mixed layer physics and thermodynamic coupling of the This geographic preference for MJO development may atmosphere and ocean. These processes are believed to be interpreted as a result of the contribution of atmo- be relevant to the warm pool system. For completeness, sphere±ocean interaction to the MJO. the model also includes ocean wave dynamics. Different The relatively slow response of the ocean mixed layer features of the ocean model are re¯ected in the following to wind anomalies favors ampli®cation of planetary- mixed layer processes: (i) diabatic heating due to cloud- scale waves. This provides a natural wave selection dependent solar radiation, wind-dependent evaporation, mechanism for the MJO. Note that the coupled CHFK and entrainment cooling; and (ii) the impact on SST of mode of the warm pool system involves only a coupling mixed layer depth variation, which is caused by both to shallow ocean mixed layer. The existence of maritime the surface momentum and heat ¯ux forcing and the continent is thus not a fundamental barrier for the MJO thermocline variation associated with the ocean dynamic to interact with ocean over the entire domain of the processes. 2132 JOURNAL OF CLIMATE VOLUME 11
The major ®ndings of the coupled model analysis IIa, which is essentially equivalent to a simpli®ed linear follow. model of Zebiak and Cane (1987). (a) In the presence of a full (dynamic and thermo- For analytical simplicity and clarity, we have made dynamic) coupling or in the presence of both the ocean a number of critical simpli®cations in the present model. wave dynamics and mixed layer physics, distinctive un- The conclusions derived from this model are, therefore, stable coupled modes exist, depending on the basic state. subject to limitations of the simpli®ed model physics. The warm pool regime with a deep thermocline and Some of the neglected physical processes are by no weak mean surface westerlies promotes high-frequency means always unimportant. For instance, the penetrated (intraseasonal) coupled unstable modes (the coupled solar radiation and rainfall effects in the mixed layer Kelvin and Rossby modes), whereas it prohibits low- that could be important in the mixed layer heat balance frequency unstable SST mode. In sharp contrast, the were neglected. One should also be cautious about the eastern Paci®c regime, in which the thermocline is shal- speci®c conclusions derived with the constraints of the low and surface mean winds are strong easterlies, sup- model's formulation. For instance, in the description of ports a low-frequency unstable SST mode, whereas it the atmospheric heating, the surface wind±dependence suppresses the high-frequency coupled instability. of the latent heat ¯ux, which certainly plays an active (b) The coupled high-frequency modes originate role in the coupled instability, was neglected. To focus from atmospheric moist Kelvin and Rossby waves. They on the role of atmosphere±ocean interaction and to com- are destabilized by thermodynamical coupling to ocean pare the warm pool instability with the eastern Paci®c mixed layer processes. The wind±mixed layer feedback instability, we have, in this part of the paper, deliberately via entrainment and surface evaporation plays a central excluded the instability mechanism internal to the at- role and favors planetary-scale unstable modes. The mosphere. We were, therefore, unable to examine the cloud effect on solar radiation selects long waves only coupled instability, which interacts with atmospheric in- when a time delay in the parameterized cumulus heating ternal instability. This, however, is a very important is included. Unstable atmospheric Rossby waves prefer issue. In view of its complexity and importance, we will planetary scales under both the wind-entrainment±evap- address this issue speci®cally in Part II of the present oration feedback and the cloud-radiation feedback. paper. (c) The destabilization of the moist atmospheric Since the coupled modes in both the warm pool and cold tongue regimes have preferred planetary scales, it waves by wind-entrainment±evaporation feedback de- would be more realistic to examine the nature of coupled pends on the direction of the basic-state winds. Only unstable modes in a spatially varying basic state in- westerlies render the coupled mode unstable. The ocean cluding both the warm pool and cold tongue conditions wave dynamics, on the other hand, have little effect on and both the thermodynamic and dynamic coupling be- coupled instability, even when thermocline depth is ar- tween the atmosphere and ocean. ti®cially reduced to one-third of the observed value. (d) The atmosphere±ocean mixed layer thermody- Acknowledgments. We would like to thank Dr. H. namic coupling creates a signi®cant positive covariance Hendon and two anonymous reviewers for their con- between the rising SST and atmospheric warming, structive comments and Dr. N. Nicholls for his consci- which allows unstable modes to gain available potential entious editorial work on a previous version of the energy. Second, the relatively slow response of the manuscript. This research has been supported by the ocean mixed layer to anomalous wind forcing favors Climate Dynamics Program, National Science Foun- ampli®cation of planetary-scale waves and signi®cantly dation and TOGA COARE under Grant ATM 96-13776. slows down eastward propagation of the unstable CHFK This is the SOEST (School of Ocean and Earth Science mode. and Technology) Publication 4637. (e) The characteristics of the coupled high-frequency mode of the warm pool system compares favorably with the observed MJO. It is suggested that, in addition to APPENDIX the atmospheric internal dynamic instability, the ocean Derivation of Linearized Ocean Model mixed layer thermodynamic processes interacting with the atmosphere may play a signi®cant role in mainte- a. Governing equations nance of the MJO. The air±sea thermodynamic coupling The ocean model was distilled from the two-and-one- furnishes eddy energy, provides a natural wave selection half layer ocean model of Wang et al. (1995), which mechanism, slows down atmospheric Kelvin wave prop- combines the reduced-gravity upper-ocean dynamics agation, and sets up a 40±50-day oscillation period for (Cane 1979) and the mixed layer physics. The major MJO. simpli®cations include (i) linearization of momentum While the mean state of the warm pool system is and mass continuity equations; (ii) neglect of horizontal unstable to intraseasonal perturbations, the mean state advection of temperature, horizontal variation of buoy- of the cold tongue supports low-frequency SST mode. ancy, and penetrative shortwave radiation ¯ux at the This was demonstrated using a reduced physics model base of the mixed layer. The equations governing the AUGUST 1998 WANG AND XIE 2133
vertical mean upper-ocean velocity v, the thermocline where Tr is the reference temperature in the model deep depth h, and the mixed layer depth h1 and temperature inert layer and he is the thickness of a thin entrainment T on an equatorial  plane are layer beneath the mixed layer base. The total downward heat ¯uxes at the ocean surface: v 0ץ (ϩ yk ϫ v ϭϪb١h ϩϪov, (A.1a (t oH Q00aϭ (1 Ϫ A)(1 Ϫ 0.622C)S Ϫ ␣ (T Ϫ 293ץ (h Ϫ ␥(T Ϫ 273). (A.1hץ (v ϭ 0, (A.1b´ϩ H١ tץ The ®rst rhs term in (A.1h) represents shortwave radi- h1 ation ¯ux with surface albedo A ϭ 0.06, C the frictionalץ (v ϭ w e, and (A.1c´ϩ H11١ t cloud cover, and S 0 the downward solar radiation ¯uxץ reaching sea surface under clear sky. The second rhs Twe0 Q term denotes surface latent heat ¯ux with a latent heatingץ ϭϪ ␦(wee)(T Ϫ T ) ϩ , (A.1d) th1ow1 ch coef®cient ␣a (Wang and Li 1993); the last rhs termץ where b is the buoyancy; the Rayleigh damping co- represents longwave radiation ¯ux with a coef®cient ␥ o ϭ 1.8 W mϪ2 KϪ1 (Seager et al. 1988). ef®cient in the upper ocean; H1 and H denote the mean depths of the mixed layer and thermocline, respectively; The prognostic equations (A.1a)±(A.1d) along with Ϫ1 Ϫ1 the diagnostic equations (A.1e)±(A.1h) consist of a close H 2 ϭ H Ϫ H1; o and cw ϭ 4186 J kg K the density and heat capacity of water, respectively; the wind stress set of the ocean model governing equations. at the ocean surface is estimated using the bulk aero- dynamic formulas: ϭ C |U|V; and ␦(w ) is a Heav- 0 a D e b. Linearization of thermodynamic processes iside function of entrainment velocity we. v1, as demonstrated 1) THE WARM POOL REGIME´١ ,The mixed layer divergence by the scale analysis of Wang and Fang (1996), is pri- marily due to the Ekman ¯ow, ve, which satis®es a In the governing equations (A.1), the nonlinearity is balance among the wind stress, the Coriolis force, and associated with mixed layer entrainment process. As- a Rayleigh friction in the mixed layer (Zebiak and Cane sume that the perturbation entrainment rate is smaller 1987). After neglect of the minor in¯uences of the me- than the mean entrainment rate. Linearization of the ridional surface wind stress and wind divergence on thermodynamic equation (A.1d) with respect to a ther- Ekman pumping, one can show: mal equilibrium basic state leads to
HE2 TЈ 1 Q0Јץ (v ϭϪ |U|U, (A.1e´١ v1eϭ´١ HH1 ϭϪ [w eeee(TЈϪTЈ) ϩ wЈ(T Ϫ T )] ϩ , tH1ow1 cHץ where U is atmospheric zonal wind at the surface, the (A.2) Ekman coef®cient is 2 where the overbar and the superscript Ј denote basic E ϭ aH 2CD/oHr s , (A.1eЈ) state and perturbation variables, respectively. and the eddy viscosity coef®cient rs in mixed layer is Assume that a thermal equilibrium basic state in the treated as a constant. warm pool regime can be obtained by considering a The entrainment velocity, we, is parameterized by the balance among the total mean downward heat ¯ux, Q 0, mixed layer model of Niiler and Kraus (1977) upon and the cooling associated with mean turbulent entrain- neglect of the generation of turbulent kinetic energy by ment, w , induced by mean frictional wind speed u . e * convection and the effect of shear instability at the From (A.1d), (A.1f), and (A.1g), one can show that the mixed layer base; that is, basic state w e and Q 0 are related to the mean surface 3 frictional wind speed u by 2mu* * wee␦(w )(T Ϫ T e) ϭ , (A.1f) ␣gh1 2mH w ϭ 2 u 3 , (A.3a) where u ϭ / is known as surface frictional velocity, e * * 0 o ␣gH1re(T Ϫ T )h m is the turbulent mixing factor due to wind stirring, and ␣ is the thermal expansion coef®cient of the water. 2mowc Q ϭ u 3 , and (A.3b) The temperature difference between the mixed layer and 0 * ␣gH1 the entrained water, T Ϫ Te, is assumed to be propor- tional to the vertical temperature gradient in the ther- T Ϫ Tr T Ϫ T eeϭ h . (A.3c) mocline layer: H2 T Ϫ T r It can also be shown that the linear versions of Eqs. T Ϫ Teeϭ h , (A.1g) h Ϫ h1 (A.1f), (A.1g) take the following forms: 2134 JOURNAL OF CLIMATE VOLUME 11
Ј UЈ TЈ hЈ HhЈ (A.1eЈ). The perturbation upwelling wup ϭϪE|U|UЈ. wЈϭw 3 ϪϩϪ1 and (A.4a) Replacing w by w and applying (A.3c), (A.4b), and eeUTϪ THHH e up []r221 (A.4c) in (A.2) yields the following mixed-layer tem- perature equation in the cold tongue: hTϪ T TЈϪTЈϭ erTЈϪ (hЈϪhЈ) . (A.4b) Vץ UץTЈ hЈץ e1HH 22[] ϭ D ϩ D ϩ tHup rad x y ץ ץ 2 ץ -Assuming that the perturbation cloudiness CЈ is pro portional to the perturbation surface wind convergence U with a coef®cient, , the perturbation downward surface Ϫ (Dupϩ D eva) Ϫ Ä oTЈ, (A.8) heat ¯ux can be expressed, from (A.1h), as U where the coef®cients VЈץUЈץ QЈϭ0.622(1 Ϫ A)S ϩ D ϭϪE|U|U(T Ϫ T )/H , (A.8a) y up e 1ץ xץ00 Ä ϭ D /(T Ϫ T ) ϩ D /(T Ϫ 293) ϩ ␥/ cH. (A.8b) UЈ oupr eva ow1 Ϫ ␣ (T Ϫ 293) Ϫ (␣ ϩ ␥)TЈ. (A.4c) aaU Equations (A.1a), (A.1b), and (A.8) are linearized governing equations for the active upper-ocean dynam- Use of (A.1e) and (A.4a) in (A.1c) leads to the fol- ics and mixed layer physics in the cold tongue regime. lowing mixed layer depth equation: hЈ UЈ hЈ HhЈ REFERENCESץ 11ϭ EUUЈϩw 3 ϩϪ . (A.5) e Battisti, D. S., and A. C. Hirst, 1989: Interannual variability in a tUHHH221ץ tropical atmosphere±ocean model: In¯uence of the basic state, A small term TЈ/(T Ϫ Tr) is neglected in deriving (A.5). ocean geometry and nonlinearity. J. Atmos. Sci., 46, 1687±1712. Substituting (A.4a)±(A.4c) into (A.2) yields the follow- Cane, M. A., 1979: The response of an equatorial ocean to simple ing mixed layer temperature equation: wind stress pattern. Part I: Model formulation and analytical results. J. Mar. Res., 37, 233±252. , and S. E. Zebiak, 1985: A theory for El NinÄo and Southern VЈ 3UЈ hЈץUЈץTЈץ ϭ D ϩϪD Ϫ 1 Oscillation. Science, 228, 1084±1087. yUH ent Chen, S. S., R. A. Houze Jr., and B. E. Mapes, 1996: Multiscaleץ xץt radץ 1 variability of deep convection in relation to large-scale circu- lation during TOGA COARE. J. Atmos. Sci., 53, 1380±1409. UЈ TЈ Chen, T. C., and M. Murakami, 1988: The 30±50 day variation of Ϫ D ϩϪ TЈ, (A.6) evaUTϪ 293 o convective activity over the western Paci®c Ocean with the em- phasis on the northwestern region. Mon. Wea. Rev., 116, 892± where coef®cients 906. Chou, S.-H., C.-L. Shie, R. M. Atlas, and J. Ardizzone, 1995: The December 1992 westerly wind burst and its impact on evapo- Dradϭ 0.622(1 Ϫ A)S 0/ ocH w 1, (A.6a) ration determined from SSMI data. Proc. Int. Scienti®c Conf. on the Tropical Ocean Global Atmosphere Program, Melbourne, Dentϭ w e(T Ϫ T e)/H 1, (A.6b) Australia, World Meteor. Org., 489±493. D ϭ ␣ (T Ϫ 293)/ cH, and (A.6c) Davey, M. K., and A. E. Gill, 1987: Experiments on tropical circu- eva a o w 1 lation with a simple moist model. Quart. J. Roy. Meteor. Soc., ϭ ␥/ cH. (A.6d) 113, 1237±1269. oow1 Davies, H. C., 1979: Phase-lagged wave CISK. Quart. J. Roy. Meteor. Equations (A.1a), (A.1b), (A.5), and (A.6) are lin- Soc., 105, 325±353. earized governing equations for the active upper-ocean Fasullo, J., and P. Webster, 1995: Aspects of ocean/atmosphere in- teraction during westerly wind bursts. Proc. Int. Scienti®c Conf. dynamics and mixed layer physics in the warm pool on the Tropical Ocean Global Atmosphere Program, Melbourne, regime. Australia, World Meteor. Org., 39±43. Gill, A. E., 1980: Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteor. Soc., 106, 447±462. 2) THE COLD TONGUE REGIME Godfrey, S. P., and Coauthors, 1995: Surface ¯uxes and mixed layer heat and freshwater budgets in TOGA-COARE. Proc. Int. Sci- Over the equatorial eastern Paci®c prevail mean sur- enti®c Conf. on the Tropical Ocean Global Atmosphere Pro- face easterlies and upwelling. The corresponding ther- gram, Melbourne, Australia, World Meteor. Org., 464±468. mocline is shallow. The ocean dynamics have an es- Hendon, H. H., and M. L. Salby, 1994: The life cycle of the Madden± Julian Oscillation. J. Atmos. Sci., 51, 2225±2237. sential control on SST variation, whereas the entrain- Hirst, A. C., 1986: Unstable and damped equatorial modes in simple ment is of secondary importance. If the mixed layer coupled ocean±atmosphere models. J. Atmos. Sci., 43, 606±630. depth can be treated as a constant. H1, from (A1c): , and K.-M. Lau, 1990: Intraseasonal and interannual oscillations in coupled ocean±atmosphere models. J. Climate, 3, 713±725. v1 ϭϪE|U|U, (A.7) Johnson, R. H., 1995: Variability of clouds and precipitation during´we ϭ wup ϭ H1١ TOGA COARE. Proc. Int. Scienti®c Conf. on the Tropical Ocean where wup is the upwelling velocity generated by Ekman Global Atmosphere Program, Melbourne, Australia, World Me- pumping, and E is the Ekman coef®cient de®ned by teor. Org., 552±556. AUGUST 1998 WANG AND XIE 2135
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