## 1. Introduction

The Madden–Julian oscillation (MJO) is the dominant component of intraseasonal (≈30–60 days) variability in the tropics (Madden and Julian 1971, 1972, 1994). It is an equatorial wave envelope of complex multiscale convective processes, coupled with planetary-scale (≈10 000–40 000 km) circulation anomalies. Individual MJO events propagate eastward at a speed of roughly 5 m s^{−1}, and their convective signal is most prominent over the Indian and western Pacific Oceans (Zhang 2005). In addition to its significance in its own right, the MJO also significantly affects many other components of the atmosphere–ocean–Earth system, such as monsoon development, intraseasonal predictability in midlatitudes, and the development of El Niño–Southern Oscillation (ENSO) (Lau and Waliser 2012; Zhang 2005, 2013).

Besides its strong tropical signal, the MJO interacts with the global flow on the intraseasonal time scales. Teleconnection patterns between the global extratropics and the MJO have been described in early observational analyses by Weickmann (1983), Weickmann et al. (1985), and Liebmann and Hartmann (1984). Their results demonstrate coherent fluctuations between extratropical flow and eastward-propagating outgoing longwave radiation (OLR) anomalies in the tropics. In a later study, Matthews and Kiladis (1999) illustrate the interplay between high-frequency transient extratropical waves and the MJO. More recently, Weickmann and Berry (2009) demonstrate that convection in the MJO frequently evolves together with a portion of the activity in a global wind oscillation. Gloeckler and Roundy (2013) argued by using lagged composite analysis that the high-amplitude extratropical circulation pattern is associated with simultaneous occurrence of both the MJO and the equatorial Rossby wave events.

Besides observational analyses, models have also been used to study the interactions between the MJO and extratropical waves. By including tropical convection forcing data in a barotropic model, Ferranti et al. (1990) found significant improvement in the model’s predictability. Hoskins and Ambrizzi (1993) argued from their model that a zonally varying basic state is necessary for the MJO to excite extratropical waves by forcing perturbations to a barotropic model. To view the extratropical response to convective heating, Jin and Hoskins (1995) forced a primitive equation model with a fixed heat source in the tropics in the presence of a climatological background flow and obtained the Rossby wave train response as a result. To diagnose the specific response to patterns of convection like those of the observed MJO, Matthews et al. (2004) forced a primitive equation model in a climatological background flow with patterns of observed MJO. The resulting global response to that heating is similar in many respects to the observational analysis. The MJO initiation in response to extratropical waves was illustrated by Ray and Zhang (2010). They show that a dry-channel model of the tropical atmosphere developed MJO-like signals in tropical wind fields when forced by reanalysis fields at poleward boundaries. In addition, Lin et al. (2009) showed the significance of midlatitude dynamics in triggering tropical intraseasonal response by including extratropical disturbances in a tropical circulation model. Frederiksen and Frederiksen (1993) used a two-level primitive equation eigenvalue model and found large-scale basic-state flow and cumulus heating to be necessary for generating MJO modes with realistic structures. Many other interesting studies on tropical–extratropical interactions have been carried out. For example, see the review by Roundy (2011).

Among the past studies based on climate models, typically the effect of the MJO is represented by forced perturbations (Hoskins and Ambrizzi 1993; Jin and Hoskins 1995; Matthews et al. 2004), or the influences of the midlatitude variations are treated as boundary effects for the tropical circulation model (Ray and Zhang 2010; Lin et al. 2009; Frederiksen and Frederiksen 1993; Roundy 2011). Such simplifications are useful for isolating individual processes within these complex models. As a next step, it would be desirable to design a simplified model where both the MJO and extratropical waves are simultaneously interactive, rather than externally imposing one of these two components; such an approach was recently taken by Chen et al. (2015), as described next.

Chen et al. (2015) developed a simplified model that includes both the MJO and tropical–extratropical interactions. Specifically, this model combines (i) the interactions of the dry barotropic mode and first baroclinic mode, which have been studied by Majda and Biello (2003) and Khouider and Majda (2005), with (ii) the MJO skeleton model of Majda and Stechmann (2009, 2011). The MJO skeleton model includes the interactive dynamics of moisture *q* and convective activity envelope *a*. It has captured the main features of the MJO at the intraseasonal/planetary scale: (i) the slow phase speed of ≈5 m s^{−1}; (ii) the peculiar dispersion relation of

Last, it is worth noting that, for an investigation of MJO initiation and termination such as the present study, the MJO skeleton model has several important properties that make it an appropriate choice of model. First, the MJO skeleton model has been shown to reproduce the initiation and termination of wave trains of two to three MJO events in succession (Thual et al. 2014), similar to MJO events in nature (Yoneyama et al. 2013). Second, the MJO skeleton model reproduces statistics of MJO events, such as the number and duration of events, that are similar to the statistics of MJO events in nature (Stachnik et al. 2015). These aspects of MJO events are in addition to the MJO’s more basic features; in particular, the MJO skeleton model predicts the speed and structure of the MJO (Majda and Stechmann 2009, 2011; Thual and Majda 2015, 2016).

The paper is organized as follows. Section 2 describes the barotropic–first baroclinic MJO skeleton model, including SST regional variations and the resulting Walker circulation. Unbalanced moisture and cooling source terms with spatial variations are taken into account in the MJO skeleton to represent the effect of SST, in which case the Walker circulation can be found as the steady-state solution of the baroclinic system. The energy principle and asymptotic expansions are also presented. In section 3, the resonance condition is identified in the presence of an idealized Walker circulation, which mediates the interaction between the MJO and the barotropic Rossby waves. Two cases are numerically computed for the ODE system: (i) MJO initiation and (ii) MJO termination and excitation of barotropic Rossby waves. Section 4 considers more general Walker circulation cases composed of two different wavenumbers. New ODE systems are derived for the resonant condition, and numerical results are presented. Section 5 investigates the effect of a zonally uniform shear flow. Finally, section 6 is a concluding discussion.

## 2. Model description

### a. The barotropic–first baroclinic MJO skeleton model

*β*-plane equations with water vapor and convection can be written asfor the barotropic mode andfor the first baroclinic mode. These equations combine the MJO skeleton model (Majda and Stechmann 2009) and nonlinear interactions between the baroclinic and barotropic modes (Majda and Biello 2003). The details of this model are described in Chen et al. (2015). Here,

*ψ*can be used to rewrite (1a) and (1b) asThe other variables,

*θ*are baroclinic velocity and potential temperature; and

*q*is water vapor (sometimes referred to as “moisture”). The coefficients

*δ*is a small parameter that modulates the scales of the tropical convection envelope. We define

^{−1}divided by the reference heating rate scale at 10 K day

^{−1}. Likewise,

### b. Walker circulation and energy evolution

Note that the right-hand side of this equation depends on the strength of the Walker circulation, and in general it is not zero, so the energy is not conserved. The Walker circulation here behaves as an energy source/sink for the MJO mode and the barotropic Rossby wave.

### c. Asymptotic ansatz

*W*stands for Walker circulations, and the subscript

*a*stands for the leading-order anomalies from the Walker circulation.

### d. Meridional basis truncation

*L*is the meridional wavenumber. For the baroclinic variables, the meridional structures are assumed to bewhere

The asymptotic expansions in (8) are then applied to the meridional truncated system, which is described in the appendix. At the leading order, the truncated system is linear, and the baroclinic and barotropic systems are decoupled. The four major eigenmodes for the baroclinic system were described in Majda and Stechmann (2009), and they are the Kelvin, MJO, moist Rossby, and dry Rossby modes, as shown in Fig. 1.

## 3. Direct tropical–extratropical interaction mediated by Walker circulation

This section provides the reduced ODE model that includes direct tropical–extratropical interactions mediated by the Walker circulation. In particular, numerical computations for two cases will be given for this interaction mechanism: (i) MJO initiation and (ii) MJO termination and excitation of barotropic Rossby waves.

### a. The reduced model

*d*and

*h*are shown in Table 1 and are pure real values, and where the asterisk denotes complex conjugate. Three groups of interacting terms appear in this ODE system: the cubic self-interaction term

*q*–

*a*interaction, the linear self-interaction terms

*d*and

*h*are from the procedure of multiscale asymptotic analysis. In contrast to the ODE system derived by Chen et al. (2015), in which the coupling terms are quadratic, here the coupling terms

Coefficients in (18).

The values of

*E*for the anomalies iswhich is only conserved when

Here the simplified asymptotic equations in (18) are utilized to gain insight into the interactions between the MJO and the barotropic Rossby waves. For this purpose, the reduced model is integrated numerically for two sets of initial data: (i) MJO initiation:

Note that the wavenumbers

### b. MJO initiation

To simulate a case of MJO initiation, the initial conditions are set to be *β* will excite *α* through the coupled linear terms. The numerical simulation in Fig. 4 shows this behavior initially when the MJO gains energy and the barotropic Rossby wave is losing energy, and the total energy is increasing until it peaks at around 70 days. After this time, the MJO mode decays in amplitude as the barotropic Rossby wave gains energy and returns to the original state. This pattern repeats itself to be a nonlinear cycle with time period of roughly 140 days.

To illustrate the spatial variations, Fig. 5 shows the Hovmöller diagram for ^{−1}, and the wave amplitude is zero at 0 days, peaks at around 70 days, and returns to zero amplitude at 140 days. This corresponds to a wave train of roughly one or two MJO events, depending on the spatial location, similar to the organization of sequences of MJO events in nature (Yoneyama et al. 2013; Thual et al. 2014). In Fig. 6, the horizontal velocity fields in the lower troposphere are shown for the MJO, the barotropic Rossby wave, and the Walker circulation. The Walker circulation is a stationary field. For the MJO, the velocity field is zero at 0 days and achieves its maximum at 70 days. The barotropic Rossby wave is at its maximum initially and achieves its smallest magnitude at 70 days.

*αβ*) factor.] Based on linear theory, if we linearize system (18) around

### c. MJO termination and excitation of barotropic Rossby waves

To consider MJO termination and the excitation of barotropic Rossby waves, the initial condition is set to be

## 4. More general Walker circulation

### a. Single MJO interacting with two barotropic waves

*k*

_{W}= 2 and 3. These two resonant triads lead to the reduced ODE system:where coefficients

*d*and

*h*are shown in Table 2. The derivation, not shown here, is similar to Chen et al. (2015). From system (26), we can see that both barotropic waves interact with the MJO mode

*α*, but there is no direct interaction between the two barotropic Rossby waves

Coefficients in (26).

In principle, either one of the barotropic waves can potentially initiate the MJO. To consider each wave separately, two cases are computed numerically: (i)

### b. Two MJO modes interacting with two barotropic Rossby waves

*d*and

*h*are shown in Table 3. Again, the derivation, not shown here, is similar to Chen et al. (2015). In this ODE system, besides the existing coupled linear terms between the MJO–barotropic Rossby wave interactions, additional cubic interactions appear between the two MJO modes. Specifically, the terms for MJO–MJO interactions are

*q*–

*a*interaction in the MJO skeleton model, similar to the cubic self-interaction terms in Chen et al. (2015).

Coefficients in (28).

Figure 10 shows the MJO initiation with initial conditions

Note that the Hovmöller diagram in Fig. 11 displays a westward group velocity, which occurs here in the presence of the Walker circulation. This westward group velocity has also been documented in cases without extratropical wave interactions (Majda and Stechmann 2011), in the presence of a warm pool, and it is also consistent with observational analyses of the MJO, as seen in Hendon and Salby (1994) and Adames and Kim (2016). In a more idealized setting with a zonally uniform base state (Majda and Stechmann 2011), the MJO skeleton model instead displays an eastward group velocity at some wavelengths.

## 5. Effects of wind shear

*d*,

*f*, and

*h*when

Coefficients in (31).

Numerical simulations are performed for MJO initiation with the effects of barotropic shear. The resonance condition is the same as in section 3. Four different barotropic shear profiles are considered: (i)

## 6. Concluding discussion

Asymptotic models have been designed and analyzed here for the nonlinear interaction between the MJO and the barotropic Rossby waves. The models involve the combination of the barotropic and equatorial baroclinic modes together with interactive moisture and convective activity envelopes. An important feature of this framework is that the tropical and extratropical dynamics are interactive, whereas other models commonly specify one of these components as an external forcing term or boundary condition.

In the presence of the Walker circulation, the MJO and the barotropic Rossby waves can interact directly. In section 3, the reduced ODE model is derived by identifying resonant triads that include the MJO, the Walker circulation, and the barotropic Rossby wave. Two cases are presented: (i) MJO initiation and (ii) MJO termination and excitation of barotropic Rossby waves. In contrast to the results in Chen et al. (2015), in which the barotropic Rossby wave exchanges very little energy with other modes, here the barotropic Rossby wave and the MJO exchange energy directly. The time period between initiation and termination is about 140 days, depending on spatial location; this is a realistic time scale, since the MJO’s oscillation period is 30–60 days and MJO events commonly appear as wave trains of two or three successive events (Yoneyama et al. 2013; Thual et al. 2014; Stachnik et al. 2015).

To explore more realistic conditions, Walker circulations were also considered with more general zonal variations. Multiple resonant triads are identified to generate energy exchange between different modes. In particular, a four-wave MJO–MJO–barotropic Rossby–barotropic Rossby interaction is found with MJO at wavenumbers 1 and 2, in which the two MJO modes interact through the nonlinear coupling term between moisture and convective activity envelope in the MJO skeleton equation. In this case, rather than an idealized MJO with a single zonal wavenumber, a wave packet of MJO events arises with an amplitude that is zonally localized.

As a final element of additional realism considered here, horizontal and vertical shear were incorporated in the model. The barotropic and baroclinic shear, if zonally uniform, have little effect on the energy exchange between the MJO and the barotropic Rossby waves. This is in contrast to the significant effect of zonally varying wind shear as part of the Walker circulation. Further investigations are needed to better understand the role of wind shear in these different settings.

Besides the MJO skeleton model used here, other models of the MJO are also in use. For example, the MJO is described as a “moisture mode” by Sobel and Maloney (2013) and Adames and Kim (2016), and other models by Yang and Ingersoll (2013) are formulated without moisture. One could use these models to carry out a study of tropical–extratropical interactions. Here, the MJO skeleton model was used for several reasons. For instance, the MJO skeleton model predicts the speed and structure of the MJO (Majda and Stechmann 2009, 2011; Chen and Stechmann 2016) and also its vertical tilts (Thual and Majda 2015, 2016). Furthermore, it has been shown to reproduce the initiation and termination of wave trains of two to three MJO events in succession (Thual et al. 2014), similar to MJO events in nature (Yoneyama et al. 2013). In addition, the MJO skeleton model reproduces statistics of MJO events, such as the number and duration of events, that are similar to the statistics of MJO events in nature (Stachnik et al. 2015).

While the simplified asymptotic models in this paper include several realistic aspects of tropical–extratropical interactions, some other physical mechanisms are not included. For instance, the meridional structures of the variables here are set to be the leading parabolic cylinder functions. With more complicated meridional structures, the interaction mechanism will be richer and more realistic, and it would allow the model to cope with different background states, such as the boreal summer and/or winter, when the ITCZ is off the equator. Such topics are interesting avenues for future investigations.

## Acknowledgments

The research of A.J.M. is partially supported by Office of Naval Research Grant ONR MURI N00014-12-1-0912. The research of S.N.S. is partially supported by Office of Naval Research Grant ONR MURI N00014-12-1-0912, ONR Young Investigator Award ONR N00014-12-1-0744, National Science Foundation Grant NSF DMS-1209409, and a Sloan Research Fellowship. S.C. is supported as a postdoctoral research associate by the ONR grants.

## APPENDIX

### Asymptotic Expansion of the Meridional Truncated System

*δ*, the first-order system isthe second-order system isand the third-order system isHere,

**F**represents terms from the nonlinear interactions between

*q*and

*a*, and

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