## Abstract

The eastward-propagating Madden–Julian oscillation (MJO) typically exhibits complex behavior during its passage over the Maritime Continent, sometimes slowly propagating eastward and other times stalling and even terminating there with large amounts of rainfall. This is a huge challenge for present-day numerical models to simulate. One possible reason is the inadequate treatment of the diurnal cycle and its scale interaction with the MJO. Here these two components are incorporated into a simple self-consistent multiscale model that includes one model for the intraseasonal impact of the diurnal cycle and another one for the planetary/intraseasonal circulation. The latter model is forced self-consistently by eddy flux divergences of momentum and temperature from a model for the diurnal cycle with two baroclinic modes, which capture the intraseasonal impact of the diurnal cycle. The MJO is modeled as the planetary-scale circulation response to a moving heat source on the synoptic and planetary scales. The results show that the intraseasonal impact of the diurnal cycle during boreal winter tends to strengthen the westerlies (easterlies) in the lower (upper) troposphere in agreement with the observations. In addition, the temperature anomaly induced by the intraseasonal impact of the diurnal cycle can cancel that from the symmetric–asymmetric MJO with convective momentum transfer, yielding stalled or suppressed propagation of the MJO across the Maritime Continent. The simple multiscale model should be useful for the MJO in observations or more complex numerical models.

## 1. Introduction

The Maritime Continent is a region in the tropical warm pool, consisting of islands, peninsulas, and shallow seas. Because of strong insolation near the equator and low heat capacity of the land surface, tropical convection prevails over the Maritime Continent and releases a huge amount of latent heat to the atmosphere. Thus, the Maritime Continent is considered as an important energy source region for the global circulation (Ramage 1968; Neale and Slingo 2003). Tropical convection over the Maritime Continent is organized on multiple time scales, ranging from cumulus clouds on the daily time scale to intraseasonal oscillations. In particular, on the daily time scale, the diurnal cycle of tropical convection over the Maritime Continent is very significant compared with that over the Indian Ocean and the western Pacific Ocean (Hendon and Woodberry 1993; Kikuchi and Wang 2008). On the intraseasonal time scale, the Madden–Julian oscillation (MJO), the dominant component of the intraseasonal variability in the tropics, typically propagates eastward slowly across the Maritime Continent and can stall or terminate there along with large amounts of rainfall (Zhang 2005).

However, the contemporary general circulation models (GCMs) still do a poor job of resolving tropical convection over the Maritime Continent. One of the significant errors is that the GCMs cannot correctly simulate the precipitation over the Maritime Continent. For instance, obvious discrepancies of the diurnal amplitude in precipitation over the islands of the Maritime Continent during boreal winter have been noticed in present-day GCMs (Yang and Slingo 2001; Stratton and Stirling 2012). Another one of the significant errors is that the GCMs typically poorly represent the eastward-propagating MJO over the Maritime Continent (Sperber et al. 1997; Inness and Slingo 2003). One possible reason is the inadequate treatment of the diurnal cycle and its impact on the intraseasonal variability of atmospheric flow. In fact, current numerical models have difficulty in reproducing the diurnal variability of tropical precipitation (Randall and Dazlich 1991; Dai and Trenberth 2004; Tian et al. 2004), although superparameterization has enhanced fidelity (Khairoutdinov et al. 2005; Benedict and Randall 2011). To improve comprehensive numerical simulations with more realistic features, it is important to have a better understanding of the intraseasonal impact of the diurnal cycle and check whether such upscale impact from the diurnal cycle can influence the MJO.

In fact, many observational studies focus on the scale interaction between the diurnal cycle of precipitation and the MJO over the Maritime Continent (Chen and Houze 1997; Slingo et al. 2003; Rauniyar and Walsh 2011; Peatman et al. 2014). Among the previous studies, the modulation of the diurnal cycle of tropical convection by the MJO has been investigated by evaluating difference of the magnitudes and phases of the diurnal cycle between the convectively active and suppressed phases of the MJO (Sui and Lau 1992; Sui et al. 1997; Tian et al. 2006). However, the upscale impact of the diurnal cycle of tropical convection on the MJO is not well understood. In the theoretical direction, the resonant nonlinear interactions between equatorial waves in the barotropic mode and the first baroclinic mode have been studied in the presence of a diurnally varying heat source, but the effect of the second baroclinic mode is not considered there (Raupp and Silva Dias 2009, 2010). In contrast to that, multicloud models based on the first and second baroclinic modes for three types of clouds (congestus, deep, and stratiform) have been built (Khouider and Majda 2006c,a,b, 2007, 2008b,a) and reproduce several realistic features of the diurnal cycle of tropical convection (Frenkel et al. 2011a,b, 2013).

The goal of this paper is to provide a framework for modeling the passage of the MJO over the Maritime Continent where the diurnal cycle of tropical convection is significant and assess how the intraseasonal impact of the diurnal cycle of tropical convection will modify the kinematic and thermodynamic characteristics of the MJO. Indeed, a self-consistent multiscale model with two time scales (daily and intraseasonal time scales) has been built to assess the intraseasonal impact of the diurnal cycle of tropical convection (Yang and Majda 2014). This multiscale model provides two sets of equations governing planetary-scale tropical flow on the daily and intraseasonal time scales separately. It turns out that the planetary-scale circulation response on the intraseasonal time scale is forced by eddy flux divergences of zonal momentum and temperature from the daily time scale. These eddy flux divergence terms provide us with an assessment of upscale transfer of kinetic and thermal energy across multiple time scales in a transparent fashion.

According to this multiscale model (Yang and Majda 2014), the planetary-scale tropical flow on the daily time scale is governed by a set of linear equations, which can be thermally forced by a heat source. Here we prescribe a diurnally varying heat source within a standing convective envelope to mimic the latent heat release over the Maritime Continent. In detail, we utilize the vertical structure in the first and second baroclinic modes for the heat source to characterize the diurnal cycle (Frenkel et al. 2011a,b, 2013) and the organized tropical convection with the life cycle of three types of clouds (congestus, deep, and stratiform), which was first introduced in the multicloud models (Khouider and Majda 2006c,a,b, 2007, 2008b,a).

The planetary-scale tropical flow on the intraseasonal time scale is governed by another set of Gill-type equations in the longwave approximation (Matsuno 1966; Gill 1980), which can be forced by the spatially upscale transfer from the synoptic scale to the planetary scale and the temporally upscale transfer from the daily time scale to the intraseasonal time scale as well as a mean heat source. In fact, the upscale transfer from the synoptic scale to the planetary scale from wave trains of thermally driven equatorial synoptic-scale circulations in a moving convective envelope and the direct mean heating have been studied previously in a multiscale model for the MJO (Majda and Biello 2004; Biello and Majda 2005, 2006). In the similar model setup here, we consider three different scenarios of the MJO induced by synoptic-scale heating and planetary-scale heating, and all of them show some key features of the MJO, such as the horizontal quadrupole structure and upward-/westward-tilted vertical structure. Then, by considering the upscale impact of the diurnal cycle from the daily time scale to the intraseasonal time scale, we are able to obtain the planetary-scale circulation response during the passage of the MJO over the Maritime Continent, where the diurnal cycle of tropical convection is typically significant. The resulting flow field and temperature anomalies resemble some realistic features of the MJO behavior over the Maritime Continent, including stalling or termination.

The rest of this paper is organized as follows. The model for the diurnal cycle and its upscale fluxes over the Maritime Continent are summarized in section 2. The planetary-scale circulation response to the intraseasonal impact of the diurnal cycle is shown in section 3. Section 4 describes three different scenarios for the MJO induced by synoptic-scale heating and planetary-scale heating. In section 5, we discuss the intraseasonal impact of the diurnal cycle on the MJO over the Maritime Continent and compare the resulting flow fields and temperature anomalies with the observations. The paper ends with a concluding summary and discussion. Detailed descriptions for notations, dimensional units, parameters in the moving heat source for the MJO, and the synoptic-scale equatorial weak temperature gradient equations (Majda and Biello 2004; Biello and Majda 2005, 2006) can be found in the appendixes.

## 2. A model for the diurnal cycle and its upscale fluxes over the Maritime Continent

The diurnal variability of tropical convection has attracted the attention of the scientific community over a long history. Early investigations of the diurnal variability of tropical precipitation date back to the 1920s (Ray 1928). Because of the development of satellite measurements and computers, more global datasets in higher resolutions, such as the Tropical Rainfall Measuring Mission (TRMM), are available for the scientific community to study convection in the tropics. In fact, the TRMM dataset has already been utilized to study the diurnal variability of the global tropical precipitation over land and oceans (Nesbitt and Zipser 2003; Kikuchi and Wang 2008). By applying empirical orthogonal function (EOF) analysis to two complementary TRMM datasets (3B42 and 3G68) for 1998–2006, Kikuchi and Wang (2008) noted the persistence of the diurnal cycle of tropical precipitation with a strong amplitude in the continental regime and a weak amplitude in the oceanic regime. According to Fig. 2 in Kikuchi and Wang (2008), the diurnal cycle of tropical convection over the Maritime Continent is more significant than that over the Indian Ocean and the western Pacific Ocean during boreal winter.

In the theoretical direction, the significant diurnal variability of tropical precipitation is examined in some simple models for tropical convection by considering three types of clouds (congestus, deep, and stratiform) to characterize organized tropical convection (Frenkel et al. 2011a,b, 2013). Since latent heat released in tropical convection can drive the tropical flow through thermodynamics (Hartmann et al. 1984; Larson and Hartmann 2003a,b), the diurnal cycle of tropical precipitation can induce the diurnal variability of the flow field. By following this underlying physical mechanism, the multiscale model (Yang and Majda 2014) provides a set of equations governing the tropical flow associated with the diurnal cycle. In this section, we use this set of equations for the diurnal cycle and discuss the corresponding upscale fluxes on the planetary/intraseasonal time scale. The equations in dimensionless units appropriate for the daily time scale read as follows:

where all physical variables, such as velocity and potential temperature , have zero mean on the daily time scale. More details about the notations and the dimensional units can be found in appendix A and the papers by Majda (2007) and Yang and Majda (2014). Here we assume rigid-lid boundary conditions at the surface and top of the troposphere, , where 0 and *π* represent the surface and top of the troposphere, respectively.

The large-scale tropical flow can be modeled as an atmospheric circulation response to diabatic heating (Gill 1980). Here the thermal forcing on the right-hand side of Eq. (1c) is used to represent latent heat release during tropical precipitation; thus, a good cloud model can help to provide an appropriate heating profile. On the other hand, the multicloud model convective parameterizations (Khouider and Majda 2006c,a,b, 2007, 2008b,a) based on three cloud types (congestus, deep, and stratiform) have successfully reproduced some crucial features of organized convection and tropical precipitation. In the multicloud models, the three types of clouds are highlighted, and they serve to provide the bulk of tropical precipitation and the main source of latent heat in the troposphere. In detail, the cumulus congestus clouds heat the lower troposphere by latent heat release and cool the upper troposphere as a result of the detrainment and high reflectivity of the clouds’ tops. Deep convective clouds, which are responsible for the majority of tropical precipitation, produce warming throughout the entire troposphere. The stratiform clouds can heat the upper troposphere through deposition and growth of precipitation particles and cool the bottom because of the evaporation of rainfall and melting of ice precipitation. Therefore, the heating and cooling effects associated with these three clouds types exhibit the first and second baroclinic modes of vertical structure, and here we incorporate these two baroclinic modes into the heating profile in dimensionless units to mimic diurnal variability (Frenkel et al. 2011a,b, 2013) as follows:

where

Here, is a large-scale convective envelope function that only depends on the planetary-scale *X* in the zonal direction, while is the meridional profile of the heat source. At each location with specific longitude and latitude, we utilize the first baroclinic mode for deep convective heating and the second baroclinic mode for congestus and stratiform heating. Both these two baroclinic modes are harmonically oscillating to mimic the diurnal cycle. The phase shift between these two modes *β* and the relative strength of the second baroclinic mode to the first baroclinic mode *α* are key parameters here. The exact expressions for the envelope function and parameter values can be found in appendix B.

According to the main conclusion in Yang and Majda (2014), the diurnal cycle of tropical convection has significant intraseasonal impact through eddy flux divergence of potential temperature associated with Eqs. (1a)–(1e) only during the solstices (boreal summer and boreal winter). Meanwhile, the eastward-propagating MJO typically occurs during boreal winter. Therefore, we mainly focus on the case during boreal winter by setting the heating center of the envelope function south of the equator. Figure 1 shows the envelope function of the diurnal heating in a longitude–latitude diagram during boreal winter: that is, in Eq. (2). This envelope function reaches maximum value at 1200 km south of the equator with about 6600-km width in the zonal direction, which resembles such observations as those in Fig. 2c of Kikuchi and Wang (2008). This envelope profile mimics the localized effect of the Maritime Continent in the model here.

Figure 2 shows the diurnal heating in a time–height diagram for a given place with specific : that is, . The alternating heating and cooling at a given height is due to the opposite thermal effects by congestus clouds and stratiform clouds as well as the intensification and diminishment of deep convective clouds. In particular, the upward movement of the heating center can be used to describe the life cycle of three cloud types (congestus, deep, and stratiform) and mimic key features of the diurnal cycle (Frenkel et al. 2011a,b, 2013).

Based on several essential assumptions and systematic multiscale asymptotics, the multiscale model (Yang and Majda 2014) shows that the resulting flow field forced by the diurnal heating model can generate eddy flux divergences of zonal momentum and temperature on the intraseasonal time scale:

which can further drive the planetary-scale circulation response on the intraseasonal time scale.

It has been shown in the appendix of Yang and Majda (2014) that the existence of the second baroclinic mode for congestus–stratiform heating *α* and its phase shift from the first baroclinic mode *β* are essential for the intraseasonal impact of the diurnal cycle, which highlights the importance of the congestus and stratiform cloud heating during tropical convection for the large-scale tropical circulation. However, the exact eddy flux divergences of zonal momentum and temperature are less sensitive to these two parameters *α* and *β* in the sense that their magnitudes are determined by the product , while their spatial patterns are independent of *α* and *β*. Figure 3 shows the eddy flux divergences of momentum and temperature in the latitude–height diagram during boreal winter. According to Figs. 3a and 3d, the dimensionless eddy flux divergence of zonal momentum from the diurnal cycle is weak, and the eddy flux divergence of temperature provides dominating intraseasonal impact on the planetary-scale circulation off the equator at the Southern Hemisphere. There is a significant heating center in the middle troposphere of the Southern Hemisphere and cooling surrounding this heating center, as shown in Fig. 3d. In addition, the magnitude of the heating in the middle troposphere is about 2 times as large as that of the cooling in the upper and lower troposphere, which indicates that the first and third baroclinic modes are both significant in the intraseasonal impact of the diurnal cycle. Figures 3b and 3c show the meridional and vertical components of the eddy flux divergence of momentum , and both of them have small magnitudes. Figure 3e shows the meridional component of the eddy flux divergence of temperature , which consists of alternate heating and cooling at different latitudes. Figure 3f shows the dominating vertical component of the eddy flux divergence of temperature with heating in the middle troposphere of the Southern Hemisphere and cooling in the upper and lower troposphere.

## 3. The planetary-scale circulation response to the intraseasonal impact of the diurnal cycle

The planetary-scale tropical flow can be modeled by the large-scale circulation response to a heat source, such as latent heat release during tropical precipitation (Gill 1980; Sobel et al. 2001). In these studies, the longwave approximation and weak temperature gradient approximation are discussed to further simplify the models. According to the multiscale model (Majda 2007; Yang and Majda 2014), it turns out that the governing equations for the planetary-scale circulation response on the intraseasonal time scale are similar to the Gill-type model but also forced by upscale flux divergences of momentum and temperature from the daily time scale to the intraseasonal time scale. Because of the essential scaling assumptions for large-scale tropical flow, this set of equations is also in longwave approximation (Majda and Klein 2003; Majda and Biello 2004), and thus the eastward flow is in geostrophic balance with the pressure gradient. Furthermore, the zonal momentum damping and the radiative cooling have dissipation on the intraseasonal time scale (Mapes and Houze 1995; Lin et al. 2005; Romps 2014), and thus they can play a role here. The equations in dimensionless units read as follows:

Here, all physical variables represent the daily time-scale mean and depend on the intraseasonal time scale *T*. The meridional circulation is the secondary flow compared with that on the daily time scale. More details about the notations and the dimensional units can be found in appendix A herein and in the paper by Yang and Majda (2014). Here, we assume rigid-lid boundary conditions at the surface and top of the troposphere, , where 0 and *π* represent the surface and top of the troposphere, respectively. On the right-hand sides of Eqs. (5a) and (5c), and represent eddy flux divergences of zonal momentum and temperature from the daily time scale to the intraseasonal time scale, respectively:

Here, all these daily fluctuation components , , , and are from the model for the diurnal cycle in section 2.

Since the forcing terms and only involve the daily fluctuation components , the planetary-scale circulation is driven by the upscale feedback from the daily time scale to the intraseasonal time scale, as shown in the zonal momentum equation [Eq. (5a)] and the thermal equation [Eq. (5c)]. After solving Eqs. (1a)–(1e) to obtain the daily fluctuation components , we can calculate and based on the expression [Eq. (6)], and their spatial patterns are shown in Fig. 3. The resulting planetary-scale circulation response governed by Eqs. (5a)–(5e) can be inferred with and . Figure 4 shows the horizontal flow field and pressure perturbation due to the intraseasonal impact of the diurnal cycle at the upper troposphere () and lower troposphere (). The main feature is that there is a cyclone (anticyclone) at the lower (upper) troposphere along with negative (positive) pressure perturbation in the Southern Hemisphere. The minimum (maximum) pressure perturbation in the lower (upper) troposphere is located south of the equator and slightly west of the diurnal heating center (the diurnal heating center is at , shown in Fig. 1). Such a longitudinal difference between the pressure perturbation and diurnal heating can be explained by the westward-propagating Rossby waves off the equator.

In addition, thermodynamic characteristics of the planetary-scale circulation response on the intraseasonal time scale are crucial properties since they are related to cloudiness and precipitation in tropical convection. Figure 5 shows the temperature anomalies in a latitude–height diagram due to the intraseasonal impact of the diurnal cycle during boreal winter. The main feature is that, in the Southern Hemisphere, there is a positive temperature anomaly in the middle troposphere and negative temperature anomalies in the upper and lower troposphere. The comparable magnitudes of positive and negative temperature anomalies at different heights indicate that the third baroclinic mode is quite significant here. Also, such temperature anomalies even extend to the Northern Hemisphere but in much weaker magnitude. In a moist environment, negative potential temperature anomalies in the lower troposphere can increase the convective available potential energy (CAPE) and reduce the convective inhibition (CIN), which enhances the buoyancy of parcels in the free troposphere and provides a favorable condition for tropical convection. Meanwhile, the negative temperature anomaly reduces the saturation value of water vapor and promotes more convection in the lower troposphere. In contrast to that, the positive temperature anomaly in the middle troposphere can suppress deep convection in the opposite way.

## 4. The MJO models forced by a moving heat source

On the intraseasonal time scale (30–90 days), the eastward-propagating MJO is the most significant large-scale phenomenon in the tropical atmosphere, which typically initializes over equatorial Africa, intensifies over the Indian Ocean, weakens over the Maritime Continent, sometimes redevelops over the western Pacific, and dissipates near the date line (Rui and Wang 1990; Zhang 2005). The MJO is organized on multiple spatial scales and consists of coupled patterns of wind field and tropical convection.

Although individual MJO events may vary in the magnitude of convection and the spatial patterns of atmospheric circulation in reality, the majority of MJO events share several key features in the kinematic and thermodynamic characteristics, which should become an important criterion for model validation. First of all, the velocity field exhibits a horizontal quadrupole structure with flow convergence in the lower troposphere and divergence in the upper troposphere (Hendon and Salby 1994). At the lower troposphere, the easterly winds near the equator are accompanied by anticyclones to the east of the convection center. The westerly winds near the equator are accompanied by cyclones to the west of the convection center. At the upper troposphere, the horizontal quadrupole structure has opposite signs for wind directions and pressure perturbation. Second, the westerly wind burst has a distinct upward/westward tilt, meaning that the onset region of the westerly winds at the lower troposphere is located to the west of that at the surface (Lin and Johnson 1996; Yanai et al. 2000).

In the theoretical direction, several mechanisms have been proposed to improve our understanding of the MJO, and a lot of numerical modeling has been done to capture the primary observed features of the MJO (Zhang 2005). Having noticed that the planetary-scale circulation associated with the MJO also lives on the intraseasonal time scale, we can use the same equations [Eqs. (5a)–(5e)] from section 3 to model the MJO in an eastward-propagating convective envelope. In fact, besides the upscale flux divergences of zonal momentum and temperature from the daily time scale, these equations in the full multiscale model (Yang and Majda 2014) are also forced by the upscale flux divergences of zonal momentum and temperature from the synoptic scale (Majda and Biello 2004; Biello and Majda 2005). The latter has been interpreted as upscale transfer from synoptic to planetary scales of momentum and temperature and used to construct a multiscale model for the MJO (Biello and Majda 2005). Here, we build three such MJO models in different scenarios forced by the planetary-scale mean heating and the synoptic-scale heating in a moving convective envelope, which exhibit several key features of the MJO as mentioned above.

### a. The symmetric MJO with horizontal quadrupole structure induced by the planetary-scale heating

Although individual MJO events may behave differently from each other, the statistical composites of reanalysis data provide insight into the horizontal structure of the MJO envelope with key features (Hendon and Salby 1994). One of the significant features of the MJO is its horizontal quadrupole structure with cyclone–anticyclone pairs at both the lower troposphere and upper troposphere.

In a long period with multiple MJO events, the overall convection field intensifies and diminishes with changing rainfall at each specific location, which corresponds to the alternating active and suppressed phases of the MJO. Here, we prescribe the planetary-scale heating for latent heat release during tropical convection as follows:

where

The envelope function is used to mimic the eastward-moving convective envelope in Eq. (7), and below the MJO phase speed is prescribed by . Different from the standard mean heating used by Biello and Majda (2005), the envelope function used here is positive in the middle and negative on both sides, which resembles the active phase of the MJO in the middle and suppressed phases on the two sides. In fact, such an envelope function is crucial for the quadrupole structure of the resulting circulation response. The meridional profile is a Gaussian shape function, symmetric about the equator. The relative strength of the second baroclinic mode is a parameter to adjust the heating center in height. The exact expressions for the heating profile and parameter values can be found in appendix B.

As for the vertical structure of the heating in Eq. (7), the first baroclinic mode represents deep convection with maximum latent heat release in the middle troposphere. The second baroclinic mode with negative strength coefficient *α* can be interpreted as stratiform precipitation with latent heat in the upper troposphere and cooling in the lower troposphere due to rain evaporation. The combination of these two baroclinic modes leads to the top-heavy heating profile, as shown in Kiladis et al. (2005). Figure 6 shows the longitude–height diagram for the planetary-scale heating at the equator. There is top-heavy heating in the middle of the convection envelope and cooling to the east and west of the convection region. The planetary-scale heating decays as the latitude increases.

After prescribing such planetary-scale heating as shown in Fig. 6, we can obtain the planetary-scale circulation response on the intraseasonal time scale by letting this heating thermally force Eqs. (5a)–(5e). Figure 7 shows the horizontal flow fields with pressure perturbation at the lower troposphere (*z* = 5 km) and upper troposphere (*z* = 11 km). The horizontal quadrupole structure is clear at both levels. In addition, the pressure perturbation is quite weak in the sense that its magnitude is much less than 1 in dimensionless units. Meanwhile, the zonal winds at the lower and upper troposphere are out of phase, which is consistent with the low-level flow convergence and upper-level flow divergence.

### b. The symmetric MJO with westerly wind bursts induced by synoptic-scale heating and planetary-scale heating

A multiscale model for the MJO with two spatial scales (the synoptic scale and planetary scale) has been developed by Majda and Biello (2004) and Biello and Majda (2005, 2006). This model accounts for the upscale transfer from the synoptic scale to the planetary scale of momentum and temperature from wave trains of thermally driven equatorial synoptic-scale circulations in a moving convective envelope as well as direct mean heating on the planetary scale. In addition, the model prescribes the heat source with dominant low-level congestus convection to the east of the moving convective envelope and dominant upper-level supercluster activity to the west.

Here, we construct the two-scale MJO model driven by both the synoptic-scale heating and planetary-scale heating in a similar way. The planetary-scale mean heating is similar to that in Eq. (7) but with and in Eqs. (7) and (8), which is used to mimic deep convection at the alternating active and suppressed phases of MJO on the planetary scale. On the synoptic scale, there is equatorial synoptic-scale heating in an eastward-moving planetary-scale convective envelope. The synoptic-scale heating, in dimensionless units, reads as follows:

where

Here, is the moving envelope function, where corresponds to 5 m s^{−1}. The magnitude of the convective envelope is chosen to yield realistic magnitudes of wind and other quantities. The meridional profile is a Gaussian shape function, symmetric about the equator. The first and second baroclinic modes are modulated by wave trains on the synoptic scale. All the parameter values and their interpretation can be found in appendix B and Biello and Majda (2005).

In general, the synoptic-scale heating for the MJO can be on both the daily time scale and the intraseasonal time scale. For simplicity, here we only consider the case where the MJO is driven by the heating across multiple spatial scales on the intraseasonal time scale and discuss how the diurnal cycle on the daily time scale can impact the behavior of the MJO. Through the assumption that the synoptic-scale heating only depends on the intraseasonal time scale *T* instead of the daily time scale *t*, the synoptic-scale fluctuation components of all physical variables satisfy the synoptic-scale equatorial weak temperature gradient (SEWTG) equations (shown in appendix C), which has been discussed in Majda and Biello (2004) and Biello and Majda (2005, 2006). In the SEWTG equations, the momentum and thermal damping do not play a role because of their longer time scale. Figure 8 shows the contours of synoptic-scale heating on the synoptic-scale longitude–height diagram with the maximum heating and cooling in the upper troposphere, which resembles the diabatic heating observed in reality (Kiladis et al. 2005). The heating is upward/westward tilted with consistent rising and sinking motions, which are used to characterize organized convective superclusters in the convective envelope.

According to the full multiscale model (Yang and Majda 2014), the planetary-scale circulation response can also be forced by the spatially upscale transfer from the synoptic scale to the planetary scale, besides the temporally upscale transfer from the daily time scale to the intraseasonal time scale. This spatially upscale transfer from the synoptic scale of zonal momentum and temperature can be expressed as follows:

Here, , , , and are the fluctuation components with zero mean on the synoptic scale. The overbar represents spatial averaging on the synoptic scale, and its exact definition can be found in appendix A.

Then we can consider the superimposition effect of the planetary-scale heating [Eq. (7)] and the upscale transfer from the synoptic scale of zonal momentum and temperature [Eqs. (11)] and let the combined forcing drive the planetary-scale circulation response on the intraseasonal time scale [Eqs. (5a)–(5e)]. Here, for clear display, we reduce the magnitude of the planetary-scale heating to 4/5 of its original value. Figure 9 shows the horizontal flow field with pressure perturbation from this MJO model. The horizontal quadruple structure can be found clearly at the surface and in the lower and upper troposphere. In addition, the horizontal flow field indicates flow convergence at the lower troposphere with upward-/westward-tilted westerlies.

On the other hand, the potential temperature anomaly field is one of the crucial thermodynamic characteristics of the MJO. Figure 10 shows the horizontal flow field and temperature anomalies from the same MJO model above. One of the significant features is that there is a very significant third baroclinic mode with a cold temperature anomaly at height 8 km (Fig. 10c) and warm temperature anomalies at heights 4 (Fig. 10b) and 12 km (Fig. 10d) around the center of the convective envelope, which is intuitively consistent with the hydrostatic balance assumption. The magnitude of the cold temperature anomaly at the middle troposphere is larger than those of the warm temperature anomalies in both the upper and lower troposphere, which also indicates the significance of the first baroclinic mode for the deep convection.

### c. The asymmetric MJO with upward/westward tilt induced by synoptic-scale heating and planetary-scale heating

Some MJO observations indicate that seasonal variations in convective activity can also affect the planetary-scale atmospheric flow (Lin and Johnson 1996). On the other hand, the zonal winds and temperature anomalies associated with the MJO exhibit an upward-/westward-tilted vertical structure according to the observations (Lin and Johnson 1996; Kiladis et al. 2005). Therefore, it is interesting to construct a model for the MJO with a tilted vertical structure of easterlies and temperature anomalies off the equator, following Biello and Majda (2005).

Here, we consider a meridionally asymmetric MJO model forced by both the synoptic-scale heating and planetary-scale heating in a moving convective envelope off the equator. Meanwhile, the heating on both the synoptic and planetary scales is upward/westward tilted, which reflects the similarity of tropical convection across multiple scales. The synoptic-scale heating can be expressed by Eq. (9), except that the maximum heating is located at 900 km south of the equator. In contrast to the planetary-scale heating with constant relative strength of the second baroclinic mode [Eq. (7)], here we vary the relative strength of the second baroclinic mode *α* so that the heating center is located at the lower troposphere to the east and the upper troposphere to the west. Such planetary-scale heating can be used to characterize the low-level congestus heating to the east of the convection envelope and upper-troposphere supercluster heating to the west. The planetary-scale heating, in dimensionless units, reads as follows:

where

where the envelope function is used to mimic the eastward-moving convective envelope. Compared with the planetary-scale heating in Eq. (7) with constant relative strength of the second baroclinic mode, the heating in Eq. (12) has a relative strength coefficient in a linear function so that the vertical profiles of the heating are different within the convective envelope. The meridional profile is a Gaussian shape function that is asymmetric about the equator. The exact expression for the heating profile and parameter values can be found in appendix B.

Similarly, we can consider the superimposition effect of the planetary-scale heating [Eq. (12)] and the upscale transfer from the synoptic scale of zonal momentum and temperature [Eqs. (11)] with the off-equator meridional profile and let the combined forcing drive the planetary-scale circulation response on the intraseasonal time scale [Eqs. (5a)–(5e)]. Figure 11 shows the horizontal flow field and pressure perturbation from the meridionally asymmetric MJO model with the synoptic-scale and planetary-scale heating centered 900 km south. At the equator, there is flow convergence in the lower troposphere and flow divergence in the upper troposphere. The horizontal profiles of flow field and pressure perturbation exhibit strong asymmetry with only one anticyclonic–cyclonic pair of gyres south of the equator.

Again, the potential temperature anomaly field is one of the crucial thermodynamic characteristics of the MJO. Figure 12 shows the horizontal flow field and temperature anomalies in the meridionally asymmetric MJO model with the synoptic-scale and planetary-scale heating 900 km south. One of the significant features is that the temperature anomalies exhibit significantly the first and third baroclinic modes with a cold temperature anomaly in the middle troposphere and warm temperature anomalies in both the upper and lower troposphere.

## 5. The intraseasonal impact of the diurnal cycle over the Maritime Continent on the MJO

The MJO consists of a large-scale dynamic field and a tropical convection field in a coherent structure and typically propagates eastward from the Indian Ocean to the Maritime Continent to the western Pacific Ocean (Zhang 2005). Because of complex topography and tropical convection over the Maritime Continent, the MJO exhibits quite different velocity and thermodynamic characteristics there from those over other regions (Wu and Hsu 2009). For example, the convection field associated with the MJO usually gets weakened during its passage over the Maritime Continent (Rui and Wang 1990). Furthermore, when the MJO is over the Indian Ocean, its convective center sits in the region with flow convergence at the surface. After the MJO goes across the Maritime Continent, the dynamic field has faster propagation speed than the convection field so that the upper-level easterlies and low-level westerlies include the convection center (Rui and Wang 1990).

As we already know that the diurnal cycle of tropical convection is very significant over the Maritime Continent (Kikuchi and Wang 2008), one possible reason for the complex MJO behavior is its scale interaction with the diurnal cycle of precipitation based on the observational evidence (Peatman et al. 2014). In the theoretical direction, based on the multiscale model (Yang and Majda 2014), we conclude in sections 2 and 3 that the diurnal cycle has significant impact on both the atmospheric circulation and temperature anomalies during boreal winter. By adding the intraseasonal impact of the diurnal cycle during the passage of the MJO over the Maritime Continent, we can investigate how the intraseasonal impact of the diurnal cycle will modify the velocity and thermodynamic characteristics of the MJO and interpret the complicated behavior of the MJO. Since we have already built three different models with some key features of the MJO in section 4, in this section, we will discuss the intraseasonal impact of the diurnal cycle on these different MJOs separately.

### a. The symmetric MJO with horizontal quadrupole structure induced by the planetary-scale heating

The relative phase between the surface winds and convection center varies during the eastward propagation of the MJO from the Indian Ocean to the western Pacific Ocean. When the MJO intensifies in the Indian Ocean, the convective center matches the surface flow convergence. During the passage of the MJO over the Maritime Continent, however, the westerlies dominate, and thus the convection center is situated in low-level westerly winds, which is suggested by several observational studies. For example, Sui and Lau (1992) studied multiscale variability of the atmosphere during the boreal winter in 1979 and identified two intraseasonal oscillations (ISOs) within the equatorial belt. They found that persistent westerly winds are established in the region between 120°E and 180° throughout the northern winter season. Such persistent westerly winds are also observed in the monsoon intraseasonal variability of 1987/88 between 105° and 150°E in the Southern Hemisphere (Lau et al. 2012). In addition, Rui and Wang (1990) investigated the development and dynamical structure of intraseasonal low-frequency convection anomalies in the equatorial region with 200- and 850-hPa wind data and found that there are strong westerlies over the convection region when the convection anomaly reaches the Maritime Continent.

If we assume that the eastward-propagating MJO can keep the coupled structure of atmospheric circulation and convection as the one in the Indian Ocean, there are easterly winds to the east of the convection center and westerly winds to the west. However, as the observation shows, there are persistent westerly winds during the passage of the MJO over the Maritime Continent. The significant diurnal cycle over the Maritime Continent could be the essential reason. Because of the intraseasonal impact of the diurnal cycle of tropical convection, the resulting cyclone dominates in the lower troposphere of the Southern Hemisphere and generates westerlies at low latitudes of the Southern Hemisphere, as shown in Fig. 4, which can explain the persistent lower-level westerly winds over the Maritime Continent. If the strong westerlies due to the intraseasonal impact of the diurnal cycle can dominate over the Maritime Continent in the Southern Hemisphere, the resulting low-latitude westerlies can be significant during the passage of the MJO. Here, we consider both the MJO with horizontal quadrupole structure (section 4a) and the intraseasonal impact of the diurnal cycle in section 3. Figure 13 shows the horizontal velocity field under the impact of both the MJO and the diurnal cycle at 5 km. The black box denotes the region between about 15°S and 0° over the Maritime Continent, where the diurnal cycle is significant during boreal winter. One crucial feature is that, during the passage of the MJO, there are persistent westerly winds in the region denoted by the black box in Fig. 13, which matches well with the observation mentioned earlier.

As for the upper troposphere, the convection center is situated in upper-level easterlies during the passage of the MJO across the Maritime Continent (Rui and Wang 1990). If we assume that the eastward-propagating MJO can keep the coupled structure of atmospheric circulation and convection as the one in the Indian Ocean, there are westerly winds to the east of the convection center and easterly winds to the west at the upper troposphere, which does not match the observation described above. One of the reasons is the intraseasonal impact of the diurnal cycle. Because of the anticyclone in the upper troposphere of the Southern Hemisphere induced by the diurnal cycle (shown in Fig. 4), the resulting upper-level easterlies at low latitudes of the Southern Hemisphere can explain the persistent upper-level easterly winds over the Maritime Continent. Here, we consider both the MJO with horizontal quadrupole structure (section 4a) and the intraseasonal impact of the diurnal cycle in section 3. Figure 14 shows the horizontal velocity field under the impact of both the MJO and the diurnal cycle at 12 km. The white box denotes the region where the diurnal cycle is significant during boreal winter. There are strong easterly winds over the region denoted by the white box when the convection center moves to the Maritime Continent, which matches the observation well.

To explore the primary structure of the vertical motion, Rui and Wang (1990) also calculate the differential divergence , which can be considered as an estimate of the vertical motion at middle troposphere. One significant feature is that, at the period when the convection center reaches the Maritime Continent, the large differential divergence anomaly also moves into the Maritime Continent and reaches its maximum magnitude at the low latitude of the Southern Hemisphere, indicating the intensifying rising motion in the middle troposphere. On the other hand, the intraseasonal impact of the diurnal cycle induces a heating center in the middle troposphere of Southern Hemisphere and cooling in the upper and lower troposphere (shown in Fig. 3). Correspondingly, there is rising motion dominating in the middle troposphere of the Southern Hemisphere. Thus, one possible reason for the intensifying rising motion in the middle troposphere is because of the diurnal cycle. Here, we use the same model setup as above, and Fig. 15 shows the contour of vertical motion at the middle troposphere (*z* = 7.85 km) associated with both the MJO and the intraseasonal impact of the diurnal cycle. The white box denotes the region where the diurnal cycle is significant during boreal winter. One significant feature in this figure is that, when the MJO moves to the region denoted by the white box, the rising motion associated with the MJO is doubled because of the intraseasonal impact of the diurnal cycle, which resembles the observation described above.

### b. The symmetric MJO with westerly wind bursts induced by synoptic-scale heating and planetary-scale heating

Although individual MJO events vary in propagation extent and convection strength, some common features of the MJO events can be obtained by using composite MJO based on a longer time period of observational satellite data. By focusing on the composite MJO using 10 yr of outgoing longwave radiation (OLR) and 7 yr of wind data, Rui and Wang (1990) found that the eastward-propagating convective anomaly typically gets weakened over the Maritime Continent (Rui and Wang 1990). One explanation for such a weakening convection anomaly is attributed to the direct topographic effects, such as blocking and wave-making effects (Wu and Hsu 2009). Alternatively, here we try to explain the weakening MJO convection by the intraseasonal impact of the diurnal cycle of tropical convection over the Maritime Continent, which can be interpreted as the indirect topographic effect, since the significant diurnal cycle is associated with the low heat capacity of the land (Frenkel et al. 2011a,b, 2013).

Here, we consider both the symmetric MJO with westerly wind bursts in section 4b and the intraseasonal impact of the diurnal cycle in section 3. The relative strength of the diurnal cycle is adjusted to ¾ so that the magnitude of its temperature anomaly is comparable with that from the MJO. To fully discuss the intraseasonal impact of the diurnal cycle on the MJO, it is interesting to consider different phases of the MJO during its passage over the Maritime Continent. Here, we use three phases (phase I, phase II, and phase III) to denote different longitudes where the MJO convective center is located. Phase I corresponds to the case when the MJO convective center is 8.1 × 10^{3} km to the west of the diurnal cycle heating center. In phase II, the MJO convective center is 2.4 × 10^{3} km to the west, and phase III is the case with the MJO convective center 3.2 × 10^{3} km to the east. Figures 16–18 show the total planetary-scale circulation response with temperature anomalies as the MJO propagates across the Maritime Continent. The center of the diurnal cycle heating is set at . One important feature is that, at phase II, the temperature anomaly south of the equator associated with the MJO is weakened by the intraseasonal impact of the diurnal cycle. In fact, the intraseasonal impact of the diurnal cycle introduces temperature anomalies in the first and third baroclinic modes in opposite sign with those from the MJO model, which is quite clear at phase I (shown in Fig. 16) and phase III (shown in Fig. 18). Such temperature anomaly cancellation can potentially explain the fact that the MJO convection field gets weakened and even stalls during its passage over the Maritime Continent.

### c. The asymmetric MJO with upward/westward tilt induced by synoptic-scale heating and planetary-scale heating

It is also interesting to consider the asymmetric MJO with upward/westward tilt when the MJO convective center is located south of the equator. Here, we consider both the asymmetric MJO with westerly wind bursts in section 4c and the intraseasonal impact of the diurnal cycle in section 3. The relative strength of the diurnal cycle is adjusted to 0.8 so that the magnitude of its temperature anomaly is comparable with that from the MJO. Again, we consider the three phases (phase I, phase II, and phase III) as in section 5b. Figures 19–21 show the temperature anomaly under the intraseasonal impact of the diurnal cycle during the passage of the asymmetric MJO at phases I, II, and III, respectively. The intraseasonal impact of the diurnal cycle introduces temperature anomalies in the first and third baroclinic mode in opposite sign with those from the MJO model, which is quite clear at phase I (shown in Fig. 19) and phase III (shown in Fig. 21). During phase II, the temperature anomaly in the active phase of the MJO is cancelled by that from the intraseasonal impact of the diurnal cycle. Such a weakening temperature anomaly can potentially explain the fact that some MJOs gets weakened and even stall during their passage over the Maritime Continent.

## 6. Concluding summary and discussion

Tropical convection over the Maritime Continent is organized on multiple spatiotemporal scales ranging from cumulus clouds on the daily time scale over a few kilometers to intraseasonal oscillations over planetary scales. The diurnal cycle, the significant process on the daily time scale, has stronger magnitude over the Maritime Continent than over the Indian Ocean and the western Pacific Ocean. On the other hand, the MJO, the significant component of the intraseasonal variability of tropical convection, typically propagates eastward across the Maritime Continent during boreal winter. To improve the present-day comprehensive numerical simulations for tropical convection over the Maritime Continent, a better understanding of the scale interaction between the diurnal cycle and the MJO is necessarily required. In this article, we focus on the intraseasonal impact of the diurnal cycle over the Maritime Continent on the MJO during boreal winter.

In the theoretical direction, the multiscale analytic model with two time scales (daily and intraseasonal) provides assessment of the intraseasonal impact of planetary-scale inertial oscillations in the diurnal cycle (Yang and Majda 2014). In detail, this multiscale model provides two sets of equations governing planetary-scale tropical flow on the daily and intraseasonal time scales separately. Here, we use the set of equations on the daily time scale to model the diurnal cycle and that on the intraseasonal time scale for the planetary-scale circulation response on the intraseasonal time scale. The latter is forced by eddy flux divergences of zonal momentum and temperature from the daily time scale. Furthermore, the full multiscale model considers two spatial scales (synoptic and planetary) and two time scales (daily and intraseasonal), and thus the planetary-scale circulation response is also forced by eddy flux divergences of zonal momentum and temperature from the synoptic scale to the planetary scale. In fact, the upscale transfer from the synoptic scale to the planetary scale of momentum and temperature has been applied to successfully model the MJO based on its multiscale features (Majda and Biello 2004; Biello and Majda 2005, 2006).

In the model for the diurnal cycle, diurnal heating in the first and second baroclinic modes is prescribed to mimic latent heat release associated with the life cycles of three cloud types (congestus, deep, and stratiform; Frenkel et al. 2011a,b, 2013). Such organized tropical flow in the diurnal cycle can generate eddy flux divergences of momentum and temperature, which further drives the planetary-scale circulation response on the intraseasonal time scale (Yang and Majda 2014). In particular, here we consider the diurnal heating during boreal winter with the heating center sitting to the south of the equator. The resulting upscale flux divergence of temperature has the dominating impact on the circulation response and exhibits a heating center in the middle troposphere of the Southern Hemisphere and cooling at both the upper and lower troposphere surrounding the heating center. The corresponding planetary-scale circulation response on the intraseasonal time scale shows that such intraseasonal impact of the diurnal cycle can induce a cyclone (anticyclone) in the lower (upper) troposphere as well as significant temperature anomalies in the tropics. In a moist environment, particularly, the negative potential temperature anomaly in the lower troposphere can increase the convective available potential energy (CAPE) and reduce the convective inhibition (CIN), which enhances the buoyancy of parcels in the free troposphere and provides a favorable condition for tropical convection. Meanwhile, the negative temperature anomaly reduces the saturation value of water vapor and promotes more convection in the lower troposphere. A positive temperature anomaly in the middle troposphere has the opposite effect and can suppress deep convection.

By using the planetary-scale equations on the intraseasonal time scale, we model the original MJO by the circulation response in a moving heat source without the impact of the diurnal cycle. Since the real individual MJO events may differ from one another in convection magnitude and circulation pattern, we consider MJO models forced by three different types of the synoptic–planetary heating in a moving heat source. Each MJO model can capture several key features of the MJO, such as the horizontal quadrupole structure and upward/westward tilt. Then, by considering the diurnal cycle during the passage of the MJO over the Maritime Continent, we try to answer the questions of how the intraseasonal impact of the diurnal cycle will modify the behavior of the original MJO and whether the resulting velocity and thermodynamic characteristics match the observations.

The results are as follows. For the MJO with the horizontal quadrupole structure induced by the planetary-scale heating, the intraseasonal impact of the diurnal cycle tends to strengthen westerly winds in the lower troposphere and easterly winds in the upper troposphere during the passage of the MJO over the Maritime Continent, which explains the fact that the MJO convection center typically sits in the westerlies in the lower troposphere and easterlies in the upper troposphere there. In addition, the intraseasonal impact of the diurnal cycle can also strengthen the vertical motion in the middle troposphere. As for the symmetric MJO with westerly wind bursts induced by the synoptic-scale and planetary-scale heating, the temperature anomaly associated with the MJO tends to get cancelled by that from the intraseasonal impact of the diurnal cycle, which can explain the fact that MJO events typically get weakened across the Maritime Continent. In fact, such temperature anomaly cancellation is also significant in the asymmetric MJO with upward/westward tilt induced by the synoptic-scale and planetary-scale heating. Tung et al. (2014) found that, during the passage of the MJO over the Maritime Continent, the symmetric MJO signals, such as the heating and drying signals, diminish entirely, and the corresponding off-equatorial signals propagate with weakening strength. In contrast, the off-equatorial convection in the asymmetric MJO convection passes the Maritime Continent without inhibition. One possible factor developed here to support the asymmetric MJO propagating off the equator is the negative temperature anomaly induced by the intraseasonal impact of the diurnal cycle, which provides a favorable condition for tropical convection off the equator.

This study has several important implications for physical interpretation and model prediction. First, the diurnal cycle of tropical convection has significant upscale transfer of temperature from the daily time scale to the intraseasonal time scale through eddy flux divergence of temperature, which leads to another mechanism for the upscale impact of tropical convection from small spatiotemporal scales besides convective momentum transport (Majda and Biello 2004; Biello and Majda 2005). Second, the intraseasonal impact of the diurnal cycle can significantly modify the MJO during its passage over the Maritime Continent, which helps to explain the complex behavior of the MJO over the Maritime Continent and its scale interaction with the diurnal cycle. Third, it emphasizes the significance of the representation of the diurnal variability of tropical precipitation for comprehensive numerical simulations. The present model can also be elaborated in several ways. For example, the diurnal heating prescribed here is assumed to have zero mean on the daily time scale. The diurnal heating with nonzero daily mean can generalize the framework and may be more realistic for the tropical convection over the Maritime Continent. In addition, we only consider the diurnal cycle of tropical convection on the planetary scale here. The diurnal cycle on the synoptic scale or even smaller scales can be interesting for modeling individual tropical convection events, such as cumulus clouds.

## Acknowledgments

This research of A.J.M is partially supported by the office of NAVAL Research ONR MURI N00014-12-1-0912, and Q.Y. is supported as a graduate research assistant on this grant.

### APPENDIX A

#### The Dimensional Units and Notations in the Multiscale Model

The full multiscale model for the intraseasonal impact of the diurnal cycle of tropical convection (Yang and Majda 2014) is derived from the hydrostatic, anelastic Euler equations on an equatorial *β* plane, which are the appropriate equations for large-scale phenomena in the tropical troposphere. This derivation follows using multiple-scale techniques developed in Majda and Klein (2003) and Majda (2007). These equations have been nondimensionalized first so that time is measured in units of the equatorial time scale , the horizontal length scale is in units of the equatorial deformation radius , and the vertical length scale is in units of the troposphere height divided by *π*, . Here, *c* is defined as the dry Kelvin wave speed and, *β* denotes the Rossby parameter in the *β*-plane approximation. The free troposphere occupies the domain , , and . The dimensional units for all physical variables and some constant parameters are summarized in Table A1.

To consider the large-scale quantities after averaging about the small scales, two averaging operators on the synoptic scale and daily time scale have been defined as follows:

For all physical variables *f*, we can have its planetary-scale mean and synoptic-scale fluctuation decomposition , and satisfies . Similarly, we can also have the intraseasonal time mean and daily fluctuation decomposition , and satisfies .

By using the averaging operator on the daily time scale, we can define the daily time mean for all physical variables as follows: , , , , and . Here, and are at the second order, and *u*, *p*, and *θ* are at the first order in the asymptotic expansion.

### APPENDIX B

#### The Expressions and Parameters in the Heating Profile for the Diurnal Cycle and MJO

##### a. The heating profile for the diurnal cycle

In the heating profile for the diurnal cycle in Eq. (2), the envelope function and the meridional profile are chosen as follows:

Here, is chosen to be a half-cosine function to mimic the Maritime Continent in about 6600-km longitude width (), and its magnitude is . The symbol for half-cosine function used in the following context has the same meaning. The meridional profile is chosen to be a Gaussian shape function for simplicity. The value is chosen to mimic the case for boreal winter so that the latitude with maximum magnitude is at 10.8°S, , and . The dimensionless parameters and are chosen to be physically consistent with the life cycles of the three types of clouds (congestus, deep, and stratiform). The dimensionless *k* is chosen to be wavenumber 1, and *ω* corresponds to the 1-day frequency for the diurnal cycle.

##### b. The heating profile in the MJO model in section 4a

In the MJO model in Eq. (7), the envelope function of the heating and the meridional profile are chosen as follows:

Here, we choose the parameters in the envelope function , , and so that the zonal average of around the equator is zero. Such an envelope function can mimic the planetary-scale convection with the active phase in the middle and suppressed phases on the two sides. Also, the circulation response to the planetary-scale heating is not sensitive to the damping coefficients due to the zero zonal mean. As for the meridional profile, we choose and to mimic the MJO when it propagates along the equator and the convection is trapped around equatorial regions. To mimic the deep convection and stratiform cloud heating, we choose .

##### c. The heating profile in the MJO model in section 4b

In the MJO model in Eq. (9), the envelope function of the heating and the meridional profile are chosen as follows:

Here, represents the 5000-km half-width of the envelope, . The value of can be adjusted for different seasons, . It has been shown that the upscale flux divergence is insensitive to many details of the wave train (Biello and Majda 2005). Thus, we pick the cosine function for the wavelike structure for the synoptic-scale fluctuations. The typical length scale of the wave packet , and is for the time-varying phases of the convective supercluster; *α* is the ratio of stratiform to deep convective heating, and is phase difference between the stratiform and deep convective heating.

##### d. The heating profile in the MJO model in section 4c

In the MJO model in Eq. (12), the envelope function of the heating , the meridional profile , and the relative strength of the second baroclinic mode are chosen as follows:

Here, is the magnitude of the convective envelope; *L* = ⅓ represents the 5000-km half-width of the envelope; corresponds to 5 m s^{−1}; and . The maximum value for the meridional profile is chosen so that the heating reaches its maximum value south of the equator to mimic the boreal winter case. The envelope function is nonzero only in the domain ; thus, the relative strength coefficient *α* varies in the range .

### APPENDIX C

#### The Synoptic-Scale Equatorial Weak Temperature Gradient Equations

The synoptic-scale equatorial weak temperature gradient equations were first established based on the systematic derivation of the intraseasonal planetary equatorial synoptic dynamics model from the primitive equations (Majda and Klein 2003). Then they are utilized for wave trains of thermally driven equatorial synoptic-scale circulations in a multiscale model for the MJO (Majda and Biello 2004; Biello and Majda 2005, 2006). The equations, in dimensionless units, read as follows:

Here, all physical variables, including the synoptic heating, have zero mean on the synoptic scale.

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