### 2.1. Air-Sea Interaction Model

[6] The framework, based on a coupled atmosphere-ocean model derived for the warm pool climate system [*Wang and Xie*, 1998], mainly captures the strong air-sea interaction in the ISM ISO (Figure 1c): 1) For perturbations, positive SST systematically induces positive precipitation, i.e., the SST can affect the pressure through surface evaporation [*Zebiak*, 1986] and longwave radiation [*Davey and Gill*, 1987; *Lindzen and Nigam*, 1987]. 2) The SST is also affected by the precipitation (via cloudiness) through changing downward short wave radiation (SWR) [*Wang and Xie*, 1998; *Fu et al.*, 2003]. We assumed that in the warm pool region, the oceanic dynamical effect on the SST is negligibly small compared to the effects of surface heat flux exchange following the formulation of *Wang and Xie* [1998]. 3) To represent the role of the planetary boundary layer, we use the Lindzen-Nigam model [*Lindzen and Nigam*, 1987; *Neelin*, 1989; *Wang and Li*, 1993]

EV→B+yβk→×V→B=−∇ϕ+G∇T. *y* is the meridional displacement, V→Bthe boundary-layer horizontal wind,*ϕ* the geopotential anomaly, *T* the SST, *E* the boundary layer friction, *β* the meridional variation of Coriolis parameter, and *G* the forcing parameter associated with the SST gradient. Then the Ekman pumping at the top of the boundary layer is [*Xie and Wang*, 1996]

w=d1∂xx+d1∂yy+d2∂x+d3∂yϕ˜, where *x* is the zonal displacement, ϕ˜=ϕ−GT and {*d*_{1},*d*_{2},*d*_{3}} = {*E*/(*E*^{2} + *β*^{2}*y*^{2}), −βE2−β2y2/E2+β2y22, −2Eβ2y/E2+β2y22}.

[7] When winds are assumed to be adjusted to the geopotential anomaly in a time scale of *O*(*ε*^{−1}), the air-sea interaction model can be written under the p-coordinate*β* plane

εu−βyv=−ϕx,βyu=−ϕy,ϕt+μϕ=−C02D+C02ID−rw−ISαE+αLT,Tt+μoT=DradD−rw. *u* and *v*are lower-tropospheric velocities, and*D* = *u*_{x} + *v*_{y} the divergence. I=RLCΔpqL/2p2CpC02, *r* = *b*Δ*p*_{B}(*q*_{B} − *q*_{L})/(Δ*pq*_{L}), *I*_{S} = *Rg*/(2*C*_{p}*p*_{2}), and *D*_{rad} = 0.622 *L*_{C}*q*_{L}(1 − *A*)*S*_{0}*γ*/(*ρ*_{0}*C*_{w}*h*) are coefficients associated with intensity of interior wave convergence, boundary layer role compared to lower troposphere, SST forcing and SWR, respectively [*Wang and Xie*, 1998]. *C*_{0} is the gravest gravity wave speed, *ε* and *μ*the lower-tropospheric momentum and Newtonian damping, respectively,*μ*_{o}the oceanic mixed-layer Newtonian cooling.*p*_{2}, Δ*p* and Δ*p*_{B}are the middle-tropospheric pressure, lower-tropospheric and boundary layer pressure depths, respectively,*b* the available moisture ratio determining how much the boundary layer moisture can be used for the diabatic heating. *q*_{b} and *q*_{L} stand for the mean specific humidity at the boundary layer (1000 hPa through 900 hPa) and the lower troposphere (900 hPa through 500 hPa), respectively [*Wang*, 1988]. *A* is the surface albedo, *S*_{0} the sea surface solar radiation flux under clear sky, *C* the perturbation cloud cover, which is assumed to be proportional to the perturbation precipitation with a coefficient *γ*.

[8] The constants include the specific gas constant *R* = 287 J · kg^{−1} · K^{−1}, the specific heat at constant pressure *C*_{p} = 1004 J · kg^{−1} · K^{−1}, the water heat capacity *C*_{w} = 4186 J · kg^{−1}·K^{−1}, the latent heat of condensation *L*_{C} = 2.5 × 10^{6} J · kg^{−1}, the water density *ρ*_{0} = 1.0 × 10^{3} kg·m^{−3}, the gravity acceleration *g* = 9.8 m·s^{−2}, and the meridional Coriolis parameter variation *β* = 23 × 10^{−11} m^{−1}·s^{−1}. Unless otherwise mentioned, other parameters based on observations are valued in Table 1 [*Wang and Xie*, 1998].

Table 1. Parameter Values Used in the Conceptual Model*C*_{0} = 50 m · s^{−1} | Gravest gravity wave speed |

*A* = 0.06 | Sea surface albedo |

*S*_{0} = 320 W · m^{−2} | Sea surface downward solar radiation flux under clear sky |

*G* = 49 m^{2} · s^{−2} · K^{−1} | Forcing parameter associated with SST gradient |

Δ*p*, Δ*p*_{B} = 400, 500 hPa | Pressure depth of lower troposphere and boundary layer |

*p*_{2} = 500 hPa | Middle-tropospheric pressure |

*ε*, *μ* = 3 day^{−1} | Lower-tropospheric momentum and Newtonian damping |

*E* = 0.4 day^{−1} | Boundary layer friction |

*μ*_{0} = 30 day^{−1} | Mixed-layer Newtonian damping |

b = 0.3 | Available moisture ratio of the boundary layer |

*q*_{b}, *q*_{L} = 0.015, 0.008 g Kg^{−1} | Mean specific humidity of boundary layer and lower troposphere for a SST of 28.5°C |

*α*_{E} = 12 kg · s^{−3} · K^{−3} | Evaporation parameter |

*α*_{L} = 16 kg · s^{−3} · K^{−3} | Long wave radiation parameter |

*I* = 0.9 | Intensity of interior wave convergence |

*r* = 0.1 | Boundary layer role compared to lower troposphere |

*I*_{S} = 2.8 × 10^{−5} m^{2} · kg^{−1} | SST forcing parameter |

*D*_{rad} ≈ 1 K | SWR parameter |

[9] The diabatic heating *Q* comes from the moisture convergence of the lower troposphere and frictional boundary layer, respectively, i.e., *Q* − *L*_{C}*q*_{L}*D* + *b*Δ*p*_{B}*L*_{C}(*q*_{b} − *q*_{L})*w*/Δ*p*. The evaporation coefficient *α*_{E} = *ρ*_{a}*C*_{E}*L*_{C}*K*_{q}|*Ū*|, where *ρ*_{a} = 1.2 kg·m^{−3} is the surface air density, *C*_{E} = 1.5 × 10^{−3} the moisture transfer coefficient, and *K*_{q} = 8.9 × 10^{−4} K^{−1}. For simplicity, the same mean surface wind of |*Ū*| = 3 m · s^{−1} has been used for calculating the evaporation, thus the typical value *α*_{E} = 12 kg·s^{−3}·K^{−3} denotes that a mean SST of 303 K yields a mean surface latent heat flux of about 120 W·m^{−2}. The evaporation anomaly usually depends on the SST and wind anomalies. In the SST equation, the evaporation term associated with the SST anomaly was neglected because it basically acts as a Newtonian damping for the SST. Because this is a box model, the wind anomaly averaged over the box is small near the convection center, thus the evaporation term associated with the wind anomaly was also neglected.

[10] The longwave radiation coefficient is *α*_{L} = *μC*_{p}Δ*p*/*g* [*Wang and Li*, 1993]. *h*is the oceanic mixed-layer depth, and both boxes are first assumed to have the same depth of 20 m. Since the precipitation is simply*Q*Δ*p*/(*L*_{C}*g*) [*Wang*, 1988], the typical value *γ* = 45.4 kg·s·J^{−1}means that an anomalous precipitation of 1 mm day^{−1}may result in an increase in total cloudiness by one-fifth.

### 2.2. Two-Box Model

[11] Two boxes are located on the convection centers at the monsoon region (15°N, suffix “m”) and the equatorial region (Equator, suffix “e”). These two boxes are characterized by different atmospheric processes, i.e., Kelvin wave in the EIO and Rossby wave in the BoB, and by different oceanic features, i.e., a shallower mixed layer in the BoB than that in the EIO [*Bellon et al.*, 2008].

[12] As a planetary-scale system, we neglect the zonal variation near the convection center and the dynamic instability mainly comes from the frictional boundary layer convergence [*Wang*, 1988], then (3) becomes independent of *x.* Assuming that perturbations vanish at the lateral boundary, i.e., Φ_{15S} = Φ_{30N} = 0, the linear interpolation yields Φ_{y=7.5S} = Φ_{e}/2, Φ_{y=7.5N} = (Φ_{m} + Φ_{e})/2, and Φ_{y=22.5N} = Φ_{m}/2. Then the central difference method gives Φ_{my} = −Φ_{e}/(4Δ*y*), Φ_{myy} = (Φ_{e} − 2Φ_{m})/(2Δ*y*^{2}), Φ_{ey} = Φ_{m}/(4Δ*y*), and Φ_{eyy} = (Φ_{m} − 2Φ_{e})/(2Δ*y*^{2}), where the meridional step Δ*y* = 7.5°. With the momentum equations in (3) we can obtain the divergence *D* in two different boxes, thus the divergence and Ekman pumping in these two boxes are

DmDewmwe={−ε2Δyβ2ym3Φe−ε2Δy2β2ym2Φe−2Φm,0,−d1Δy2Φ˜m+d12Δy2−d34ΔyΦ˜e,−d1Δy2Φ˜e+d12Δy2+d34ΔyΦ˜m}. The air-sea interaction model on these two boxes (subscript e or m) is written as

Φm,et+μΦm,e=−C02Dm,e+C02IDm,e−rwm,e−ISαE+αLΠm,e,Πm,et+μoΠm,e=DradDm,e−rwm,e, which can be calculated analytically as a linear system.