# A general theoretical framework for understanding essential dynamics of Madden–Julian oscillation

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## Abstract

Motivated by observed structure of Madden–Julian oscillation (MJO), a general theoretical model framework is advanced for understanding fundamental aspects of MJO dynamics. The model extends the Matsuno–Gill theory by incorporating (a) moisture feedback to precipitation, (b) a trio-interaction among equatorial waves, boundary layer (BL) dynamics, and precipitation, and (c) a simplified Betts–Miller (B–M) cumulus parameterization. The general model with B–M scheme yields a frictionally coupled dynamic moisture mode, which produces an equatorial planetary-scale, unstable system moving eastward slowly with coupled Kelvin–Rossby wave structure and BL moisture convergence leading major convection. The moisture feedback in B–M scheme reinforces the coupling between precipitation heating and Rossby waves and enhances the Rossby wave component in the MJO mode, thereby slowing down eastward propagation and resulting in a more realistic horizontal structure. It is, however, the BL frictional convergence feedback that couples equatorial Kelvin and Rossby waves with convective heating and selects a preferred eastward propagation. The eastward propagation speed in the model is inversely related to the relative intensity of the equatorial “Rossby” westerly versus “Kelvin” easterly associated with the MJO. The cumulus parameterization scheme may affect propagation speed through changing MJO horizontal structure. The SST or basic-state moist static energy has a fundamental control on MJO propagation speed and intensification/decay. Model sensitivity to BL and cumulus scheme parameters and ramifications of the model results to general circulation modeling are discussed.

## Keywords

Madden–Julian oscillation Convectively coupled waves Frictional convergence feedback Moisture mode Moisture feedback Kelvin waves Rossby waves## 1 Introduction

Notable progress has been made in development of the general circulation models (GCMs), but by far the MJO remains poorly simulated in many GCMs. For instance, the recent assessment of 24 GCMs, which participated in MJOTF (MJO Task Force)/GASS (GEWEX Atmospheric System Study) Global Model Evaluation Project on vertical structure and physical processes of the MJO, reveals that only about one-fourth of the models simulate realistic eastward propagation of the MJO (Jiang et al. 2015). The dynamical models’ prediction skill for the MJO has been rapidly improved, yet remains limited compared to its predictability estimate (Neena et al. 2014; Lee et al. 2015). These indicate the necessity to improve our understanding of the fundamental physics of the MJO.

A variety of theories have been proposed to explain MJO phenomenon over the past two decades, including the wave-CISK (Convective Instability of the Second Kind) (Lau and Peng 1987); the evaporation-wind feedback (Emanuel 1987; Neelin et al. 1987; Wang 1988a); the BL frictional feedback and coupled Kelvin–Rossby wave theory (Wang 1988b; Wang and Rui 1990a; Wang and Li 1994; Kang et al. 2013); the moisture mode theory (Raymond and Fuchs 2009; Sobel and Maloney 2012, 2013; Benedict et al. 2014; Adames and Kim 2015); the MJO skeleton driven by synoptic wave activity (Majda and Biello 2004; Majda and Stechmann 2009); the MJO modified by interactive upscale transfer of eddy momentum, heat and moisture (Wang and Liu 2011; Liu and Wang 2012a, b, 2013); the multi-cloud model of MJO (Khouider and Majda 2006, 2007); the MJO wave packet driven by the interference pattern of a narrow frequency band of mixed Rossby–gravity waves (Yang and Ingersoll 2011) or westward and eastward inertia-gravity waves (Yang and Ingersoll 2013); and the air-sea interaction theory (Flatau et al. 1997; Wang and Xie 1998; Wang and Zhang 2002; Fu and Wang 2004).

In addition to theoretical modeling, various processes and mechanisms have been identified or speculated to play significant roles in MJO dynamics based on numerical model experiences and observations, including shallow convection-BL circulation interaction (Johnson et al. 1999; Kikuchi and Takayabu 2004; Lin et al. 2004); stratiform cloud–wave interaction (Mapes 2000; Khouider and Majda 2006; Kuang 2008; Fu and Wang 2009; Seo and Wang 2010; Holloway et al. 2013); cloud-radiation interaction (Lee et al. 2001; Raymond 2001; Lin and Mapes 2004; Bony and Emanuel 2005; Andersen and Kuang 2012); moisture–convection feedback (Woolnough et al. 2001; Grabowski and Moncrieff 2004; Benedict et al. 2014), and moisture transport (Maloney 2009; Maloney et al. 2010; Andersen and Kuang 2012; Hsu and Li 2012; Kim et al. 2014; Pritchard and Bretherton 2014). More details were reviewed by Zhang (2005) and Wang (2005, 2012).

The diverging views presented above suggest that our knowledge and understanding of the essential dynamics of the MJO remain elusive. A number of critical issues regarding the MJO dynamics remain, particularly: (a) Why does the MJO possess a coupled Kelvin–Rossby wave structure and how could Kelvin and Rossby waves, which propagate in opposite directions, couple together with convection and select eastward propagation? (b) What makes the MJO move eastward slowly (about 5 m/s) over the warm pool, yielding the 30–60 day periodicity? (c) How does SST control the intensification of MJO against dissipation? These are central questions that will be addressed in the present study.

In Sect. 2, observations and model simulations are presented to provide motivation and validation for theoretical models. In Sect. 3, a general MJO framework is formulated that can accommodate different cumulus parameterization schemes. Section 4 demonstrates that the general model can simulate essential features of the MJO and exhibits how MJO-like mode grows/decays over the warm/cold ocean. Section 5 elucidates how the moisture feedback changes MJO structure and propagation speed, and addresses the mechanisms determining MJO’s propagation speed. Section 6 elaborates the critical role of frictional moisture convergence feedback and addresses how the Kelvin and Rossby waves could couple together with convection and select eastward propagation. The last section summarizes the results and discusses the model sensitivity to some critical parameters as well as the ramifications of theoretical results.

## 2 Observed and model simulated MJO

The aforementioned three observed features are well produced in the good group of GCMs (middle panels) but totally missed in the poor group of GCMs (lower panels). Obviously, the fatal defect of the poor MJO lies in the BL: The BL moisture convergence does not lead major convection and does not propagate eastward. It is also interesting to observe that in the poor group of models, Rossby wave-induced westerly is strong while the Kelvin wave-induced easterly is weak.

Based on the aforementioned observations, we propose that the essential features of the MJO that requests theoretical explanation includes (1) the coupled Kelvin–Rossby wave (horizontal) structure, (2) the slow eastward propagation over the warm oceans (about 5 m/s), (3) the BL moisture convergence and low-pressure leading major convection center, (4) amplification (decay) in the warm (cold) oceans (Madden and Julian 1972), and (5) the planetary zonal circulation scale. These essential features may be viewed as major targets for theoretical interpretation and validation metrics of MJO theory. Note that the feature (1) is indicated by the observation (Fig. 1) and is consistent with previous results in literatures (e.g. Adames and Wallace 2014). However, various theories produce variety of MJO-like modes with different horizontal structures. Validation of the horizontal structure may test whether a model includes adequate dynamics. Note that the feature (4) implies MJO is an unstable mode, but in some literature MJO is considered as a neutral mode (e.g., Majda and Stechmann 2009; Yang and Ingersoll 2013). Other important features such as the multi-scale convective structure (Nakazawa 1988), remarkable seasonal variations (Wang and Rui 1990b), coupling with ocean mixed layer (Krishnamurti et al. 1988), and irregularities (Zhang 2005) should also be explained as higher-level theoretical targets, which require models with higher-level complexity.

## 3 The general MJO model framework

### 3.1 The essential model physics

A theoretical model framework should be derived from the first principles with reasonable/justifiable assumptions. The model is designed to include only the essential physical processes that contribute, in a fundamental manner, to MJO dynamics. The coupled convection-Kelvin–Rossby wave structure and the eastward shift of the BL convergence to convective heating implies that the free tropospheric wave dynamics, BL dynamics and interactive heating must be described by the model.

### 3.2 Non-dimensional governing equations

*C*

_{0}, length scale (

*C*

_{0}/

*β*)

^{1/2}, time scale (

*βC*

_{0})

^{−1/2}, geopotential scale \( C_{0}^{2} \), moisture scale

*d*

_{0}Δ

*p*/

*g*, where

*d*

_{0}= 2

*p*

_{2}

*C*

_{ p }

*C*

_{0}

^{2}/Δ

*pRL*

_{ c }, the non-dimensional equations are:

*u*,

*v*and Φ represent the free-tropospheric low-level zonal wind, meridional wind and geopotential, respectively.

*μ*and

*υ*are Newtonian cooling and Rayleigh friction coefficients.

*q*is the column-integrated perturbation moisture and is essentially dominated by the moisture from surface up to mid-troposphere.

*P*

_{ r }and

*Ev*are precipitation and evaporation, respectively. \( \overline{Q} \) and \( \overline{Q}_{b} \) are normalized basic-state specific humidity at the lower tropospheric layer and the BL, respectively (see Table 1 for definition of the parameters), both are controlled by the underlying sea surface temperature (SST) (Wang 1988b), the relation between them and the SST are given in Appendix 1. For homogeneous SST, the zonal and meridional moisture advection in Eq. (4) will vanish.

*D*and

*D*

_{ b }are the lower-tropospheric and BL divergence, respectively.

*u*

_{ b }and

*v*

_{ b }are BL barotropic winds, and

*E*

_{ k }is the friction coefficient in the BL.

*d*is the nondimensional BL depth. The definition and standard values for the model parameters are listed in Table 1.

Parameters and their standard values used in the experiments

Parameter | Description | Typical value utilized here |
---|---|---|

Δ | Half-pressure depth of the free atmosphere | 400 hPa |

| Pressure at the surface | 1000 hPa |

| Pressure at boundary layer top | 900 hPa |

| Pressure at upper boundary | 100 hPa |

| Pressure at level 2 | 500 hPa |

| Dry gravity wave speed of the baroclinic mode | 50 m s |

| Dimensional Rayleigh friction | (16 day) |

| Dimensional Newtonian cooling coefficient | (8 day) |

| Convective adjustment time | 2.0 h |

| Moisture reference coefficient | 0.1 |

| Dimensional Ekman number in the boundary layer | 2.3 × 10 |

| Density scale height/water vapor density scale height | 3.45 |

| Sea surface temperature | 29.0 °C |

| Precipitation efficiency coefficient | 0.88 |

| Nondimensional boundary layer depth | 0.25 |

\( \overline{Q} \) | \( \overline{{q_{3} }} /(d_{0} ) \) | 0.98 |

\( \overline{{Q_{b} }} \) | \( \overline{{q_{e} }} /(d_{0} ) \) | 1.98 |

| \( d_{0} = \frac{{2p_{2} C_{p} C_{0}^{2} }}{{\Delta pRL_{c} }} \) | 0.008 |

### 3.3 Parameterization of precipitation heating

- (a)
Simplified Kuo scheme, in which the moisture tendency is neglected in (4), and precipitation is balanced by horizontal moisture convergence and surface evaporation:

where$$ P_{r} = bH(E_v -(\overline{Q} D + \overline{Q}_{b} dD_{b})) $$(7)*b*is precipitation efficiency coefficient.*H*(*x*) is a Heaviside function, which represents the positive-only precipitation (Justification is given in Appendix 2). - (b)
Simplified B–M scheme, in which the moisture tendency is retained in Eq. (4), and precipitation heating must be parameterized in order to close the governing equations. The B–M (Betts and Miller 1986; Betts 1986) scheme relaxes temperature and humidity back to a reference profile when some convective criteria are met. Following Frierson et al. (2004), the humidity is relaxed back to a reference value \( \widetilde{q} \), at which precipitation occurs (see Appendix 2):

where$$ P_{r} = \frac{1}{\tau }H(q - \widetilde{q}(T)) $$(8)*τ*is a convective adjustment time scale, \( \widetilde{q}(T) \) is a function of temperature. The \( q - \widetilde{q}(T) \) could be considered as a mimic of CAPE in this model (Frierson et al. 2004). Therefore, Eq. (8) represents relaxation of CAPE over a finite time scale*τ*, keeping CAPE in a quasi-equilibrium state. For simplicity, \( \widetilde{q}(T) \) is assumed to be proportional to free troposphere temperature, which in this model is related to low-level geopotential, hence, \( \widetilde{q}(T) = \alpha \varPhi \), where*α*is a constant coefficient. Thus, precipitation has the form:$$ P_{r} = \frac{1}{\tau }H[(q + \alpha \varPhi )] $$(9)The

*τ*measures the time over which convection releases CAPE and relaxes moisture back to its reference state. A small*τ*implies an intense cumulus activity and a rapid atmospheric adjustment toward the quasi-equilibrium reference state; a large*τ*means that thermodynamics is less tightly constrained (Neelin and Yu 1994). Betts and Miller (1986) suggested an appropriate value of*τ*being about two hours. The other parameter,*α*, is related to the intensity of CAPE. With a small*α*, CAPE is large and precipitation is intensive. When*α*is combined with convective adjustment time, the coefficient*τ*/*α*could be considered as buoyancy relaxation time (Fuchs and Raymond 2002).

## 4 Realistic simulation of the MJO characteristics

Figure 5a also shows that the BL low pressure and moisture convergence are located under and to the east of the major convection center. This structural difference between the BL and free troposphere is the 1 and ½ layer model view of the observed backward tiled vertical structure in the vertical velocity and moisture fields. The vertical tilt of moisture field is mainly contributed by the BL frictional convergence (FC) to the east of the convective heating. The vertical tilts of the moisture and vertical velocity fields in this model can only be resolved by BL dynamics as the free atmosphere is described by the lowest baroclinic mode. But there may exist other processes that contribute to lower-tropospheric moisture increase beside the BL convergence. The current model result suggests it is important to identify new mechanism that causes moisture increase in the lower troposphere. Also note that the BL convergence pattern may be a possible explanation of the swallowtail precipitation pattern as described by Zhang and Ling (2012).

There are two fundamental processes that interact with convective heating and support the MJO-like mode, one is the BL FC feedback and the other is moisture feedback. In the following two sections we will identify their different roles in MJO dynamics.

## 5 Important roles of the moisture feedback and mechanism of slow propagation

### 5.1 Impacts of moisture feedback

First, the moisture feedback in B–M simulation makes the horizontal circulation shape more close to the Gill (1980) pattern (Fig. 7). The shape of circulation can be measured by a “shape” parameter, which is defined by the ratio of the zonal extent of the equatorial “Kelvin” easterly (below −0.2 in normalized value) versus “Rossby” westerly (above 0.2 in normalized value) averaged between 5°S and 5°N. In the Gill (1980) model, the circulation pattern, in response to a given heat source, has a shape parameter of 3.0. The shape parameter is roughly 2.4 in the B–M simulation but it is about 1.4 in the Kuo simulation. This shape parameter in B–M simulation is close to observation (about 2.1).

Second, the moisture feedback in the B–M simulation enhances the relative intensity of the Rossby wave (vs. Kelvin wave) component. Since the equatorial westerlies (easterlies) are primarily contributed by equatorial Rossby (Kelvin) waves, the Rossby wave intensity can be measured by a “*westerly intensity*” index, which is defined by the ratio of the equatorial maximum westerly speed *U* _{ max } versus the maximum easterly speed *abs*(*U* _{ min }). The westerly intensity index is 0.58 in Kuo simulation while it is 1.16 in the B–M simulation.

Third, the meridional extent of the circulation is smaller in the B–M simulation. This tighter equatorial trapping is a consequence of slower eastward propagation. The meridional trapping scale is determined by the equatorial Rossby Radius of Deformation, an intrinsic length scale in equatorial dynamics, which is proportional to the wave propagation speed. Since the moisture feedback reduces the propagation speed of convectively coupled MJO mode, the “effective” Rossby Radius of Deformation also decreases.

*β*-effect tends to move westward. The theoretical result here finds support from a previous aqua-planet numerical study, which shows that when Rossby wave component becomes weak, the eastward propagation becomes faster (Kang et al. 2013). The finding here is also consistent with the GCM simulation results in Fig. 1, where the poor MJO shows a much stronger Rossby wave component than the Kelvin wave component.

How does the B–M scheme enhance the Rossby wave component? We found that the B–M scheme produces a stronger coupling of the Rossby wave and convective heating than the Kuo scheme. This is evidenced by the fact that about 50% of the precipitation area coincides with Rossby westerly in the B–M simulation, while this ratio is only about 30% in the Kuo simulation (Fig. 7). This stronger coupling is also evidenced by the fact that the precipitation center and the maximum Rossby westerly are closer in the B–M simulation than in the Kuo simulation. Thus, the moisture feedback in the B–M scheme couples more tightly the convection and Rossby waves, resulting in an enhanced Rossby wave component.

In sum, compared with the Kuo simulation, the moisture feedback in the B–M scheme can couple more tightly the Rossby waves and convection, thereby (a) enhance the Rossby wave component and substantially slow down eastward propagation of the MJO mode, (b) make the horizontal structure of the MJO mode more close to observed MJO structure, and (c) reduce the meridional extent of the MJO circulation due to the slow eastward propagation. Note that the effects of moisture feedback may depend on the cumulus parameterization scheme. The effects discussed here is associated with the simplified B–M scheme.

### 5.2 Mechanisms for slow eastward propagation of MJO

_{0}is the dry Kelvin wave or gravity wave speed, which is determined by the dry static stability;

*q*

_{ c }is a scaling constant; the quantity \( C_{0}^{2} (1 - \overline{{q_{3} }} /q_{c} ) \) represents the effective static stability, which is the reduced static stability due to the precipitation heating. Thus, \( C = C_{0} \sqrt {(1 - \overline{{q_{3} }} /q_{c} )} \) is the phase speed of convectively coupled Kelvin wave. When SST increases, the (\( \overline{{q_{3} }} \)) increase accordingly, thus moist Kelvin wave phase speed decreases. As shown in Fig. 9, given all model parameters being the same (Table 1), the dry Kelvin (gravity) wave speed is 50 m/s, whereas the convectively coupled Kelvin wave (moist-K) speed decreases with increasing SST with a value of 19 m/s for SST = 29 °C. The MJO eastward propagating speed is further slowed down by the coupling of Kelvin and Rossby waves from 19 to 14.9 m/s when SST = 29 °C with the simplified Kuo scheme. This is because the Rossby wave component induced by

*β*-effect tends to move westward. Finally, the moisture feedback can further significantly slow down the eastward propagation speed to about 5 m/s with simplified B–M scheme. The SST-dependence of propagation speed is consistent with the observed slow propagation over the warm pool and fast propagation in the cold ocean in the western hemisphere (Knutson et al. 1986).

Note that an increase in SST would not only increase water vapor, which tends to reduce convectively coupled wave speed, but also increase the atmospheric static stability that tends to increase convectively coupled wave speed. Therefore, the model static stability parameter should not be fixed, rather it should increase with SST accordingly. The increasing static stability can accelerate convectively coupled Kelvin waves in warm climate, potentially consistent with the result of Kuang (2008).

## 6 Critical roles of the FC feedback in MJO structure and eastward propagation

What process is responsible for the coupling of the Kelvin and Rossby waves? Could the moisture feedback process couple the Kelvin and Rossby waves together and propagate eastward? To address this question, we use a simplified version of the model with the B–M scheme and moisture feedback but *without BL dynamics* (by setting the BL depth *d* to zero). The model configuration is the same as that used in Fig. 3; thereby the results can be readily compared with those obtained by B–M scheme with the BL dynamics (Fig. 3b). The model without BL can isolate the role of the moisture feedback, while comparison of the simulation results with and without BL can identify the role of the BL FC feedback.

One of the central questions for MJO theory to be addressed is how could the eastward propagating Kelvin wave and westward propagating Rossby waves be coupled together with convection and select eastward propagation? The answer is in the BL dynamics. The Rossby wave-induced BL convergence exhibits not only off-equatorial maximum coinciding with Rossby wave lows but also an equatorial maximum convergence to the east of the Rossby wave lows; on the other hand, the Kelvin wave-induced BL convergence displays a trapped equatorial maximum that coincides with Kelvin wave low pressure and easterly phase (Wang and Rui 1990a). Therefore, when convective heating excites Rossby wave westerly to its west and Kelvin wave easterly to its east, the Kelvin and Rossby waves would produce a unified BL moisture convergence field that coincides and leads the major convective heating (see Fig. 5a). As a result, the frictional organization of convective heating can couple the Kelvin and Rossby waves together. The BL moisture convergence can also accumulate moist static energy and increase convective instability to the east of the major convection (Hsu and Li 2012), leading to eastward propagation of the MJO.

## 7 Conclusion and discussion

A general theoretical model for understanding essential dynamics of the MJO is advanced. The model physics include free tropospheric low-frequency equatorial wave dynamics, BL dynamics, full thermodynamics with simplified Betts–Miller (B–M) convective parameterization schemes, and a full moisture conservation equation. The model describes the moisture feedback and trio-interaction among convective heating, low-frequency Kelvin and Rossby waves and the BL friction convergence (FC) (Fig. 2). A simplified version of the current model, in which a linear heating, steady BL, and highly truncated meridional structure of the motion were assumed and a complementary eigenvalue analysis of linear instability of the ‘dynamic’ moisture mode was carried out (Liu and Wang 2016). They explored the effects of BL process and the moisture feedback on the MJO by studying an eigenvalue problem, focusing on analytical solution of the linear instability of the ‘dynamic’ moisture mode. The model here is more general, allowing for considering realistic basic state SST, transient BL, and nonlinear heating etc., simulating more realistic evolution, propagation and structure, and elaborating mechanisms by which moisture feedback slowing down the eastward propagation and the linkage between propagation and horizontal structure. This general model framework can accommodate different cumulus parameterization schemes, such as Majda and Stechmann (2009) parameterization, in which precipitation tendency is linked to the moisture anomaly. The general model framework can readily include other processes that are deemed to have important impacts on MJO’s evolution, such as eddy momentum, moisture and heat transfer as well as effects of advection of the moisture. The present model can also be extended to a multi-layer model to incorporate multi-layer cloud precipitation effects and explore stratiform cloud–wave interaction and radiation-cloud interaction.

### 7.1 Conclusion

The general model with B–M scheme yields a frictionally coupled dynamic moisture mode, which reproduces the following essential characteristics of the observed MJO (Figs. 3, 4, 5, 6): (1) a coupled Kelvin–Rossby wave structure, (2) slow eastward propagation (~5 m/s) over warm pool, (3) planetary (zonal) scale circulation, (4) a vertical structure in which BL moisture convergence leads major convection, and (5) grow/decay over warm/cold SST regions.

The FC feedback provides a mechanism that couples the Kelvin wave and Rossby wave together along with convective heating, and selects eastward propagation. Without FC feedback, the Kelvin and Rossby waves are decoupled, and there is no growing mode (Fig. 10). Therefore, the FC feedback acts like an engine that drives the wave dynamic feedback and moisture feedback to generate the unstable dynamic moisture mode.

The moisture feedback in a simplified Betts–Miller scheme can enhance the relative intensity of Rossby wave response in the MJO structure (Fig. 7). Interestingly, the eastward propagation speed decreases when the relative intensity of the Rossby wave component increases (Fig. 8). Thus, the moisture feedback can substantially reduce the eastward propagation speed to a realistic value (Fig. 9). In addition, the moisture feedback makes the convection and Rossby wave couple more tightly, resulting in a more realistic horizontal structure.

Our novel finding on the inverse relationship between eastward propagation speed and the relative intensity of the Rossby wave component seems to be consistent with the difference between the models’ simulated good and poor MJOs (Fig. 1, the westerly intensity indices along equator are about 0.8 and 2.4 for the good and poor MJO), but needs to be further verified by observations and model results.

### 7.2 The sensitivity of solutions to model parameters

The model solutions vary with the model parameters. The most sensitive parameters are the BL Ekman number and the convective adjustment time in the B–M scheme. The Ekman number *E* _{ k } (Wang 1988b), *E* _{ k } = *ρ* _{ e } *gA* _{ z }/((*p* _{ s } − *p* _{ e })*h* ln (*h*/*z* _{0})), is determined by the turbulence viscosity coefficient *A* _{ z } (with typical value 10 m^{2} s^{−1}), the surface roughness *z* _{0} (with typical value 0.01 m), the height of the surface layer *h* (with typical value 60 m), as well as the BL depth (*p* _{ s } − *p* _{ e }), or nondimensional BL depth *d* (typical value is 0.25). Thus, the typical value of *E* _{ k } used in our study is 2.3e-05 s^{−1}, which corresponds to BL damping time scale of approximately 0.5 day. This value is as the same as that used in Mori and Watanabe (2008). This value is also used to compensate the neglected effects of complex nonlinear processes.

*E*

_{ k }. As the BL friction decreases (smaller

*E*

_{ k }), the BL winds and thus the BL convergence increases. The enhanced BL convergence has two effects on the simulated MJO-like modes. First, it enhances moisture feedback, which results in enhanced Rossby wave component, as manifested by the westerly intensity index (Table 2). This slows down eastward propagation. Second, the stronger BL moisture convergence results in stronger instability of the simulated MJO-like mode.

Sensitivity of B–M simulation to Ekman number *E* _{ k } for the B–M simulation, assuming *d* = 0.25 and SST = 29.0 °C

| 180 (%) | 160 (%) | 140 (%) | 120 (%) | 100 (%) | 80 (%) | 60 (%) | 40 (%) | 20 (%) |
---|---|---|---|---|---|---|---|---|---|

Damping time (day) | 0.28 | 0.31 | 0.36 | 0.42 | 0.5 | 0.63 | 0.83 | 1.25 | 2.5 |

Propagation speed (m/s) | 7.8 | 7.4 | 6.7 | 5.9 | 4.7 | 3.1 | 1.5 | −0.37 | −2.25 |

Westerly Intensity Index | 0.80 | 0.89 | 0.96 | 1.05 | 1.16 | 1.30 | 1.45 | 1.60 | 1.67 |

Growth rate (1/day) | 0.06 | 0.10 | 0.13 | 0.17 | 0.21 | 0.27 | 0.35 | 0.51 | 0.84 |

*τ*. First, for a longer

*τ*, the relative strength of the Rossby wave component is weaker, thereby the eastward propagation is faster. The reason is that a longer

*τ*means slower atmospheric adjustment toward the quasi-equilibrium reference state. Thus, the role of moisture feedback in enhancing Rossby wave would be weaker. Second, a longer

*τ*means reduced precipitation intensity, which leads to weaker instability.

Sensitivity of B–M simulation to convective time scale *τ*, assuming *d* = 0.25 and SST = 29.0 °C

Convective time scale (h) | 1 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|

Propagation speed (m/s) | 3.0 | 4.7 | 6.9 | 8.8 | 9.1 |

Growth rate (1/day) | 0.38 | 0.21 | 0.08 | 0.02 | −0.00 |

Westerly Intensity Index | 1.32 | 1.16 | 0.92 | 0.81 | 0.78 |

Our experiment results (not shown) indicate that the planetary zonal scale of the MJO mode in this model is not sensitive to the Ekman parameter and the convective adjustment time scale. This result is different to Pritchard and Yang (2016) who showed the zonal scale change with SST. A possible reason is that the present model does not include the mean circulation. In the SPCAM model they used, the Hadley circulation is enhanced with SST, but is missing when SST is 25 °C and reverses its sign below 25 °C (downward motion over equator). The dramatic change of the mean circulation may cause the differences in the zonal scale of their MJO-like mode. However, these changes could not be represented in our model.

### 7.3 Implications

The results of this study suggest that MJO propagation is sensitive to the cumulus parameterization schemes because different schemes may produce different structures of the MJO and propagation speed is related to the horizontal structure. Even within the chosen B–M scheme, the eastward propagation speed depends on sensitive parameters such as the convective adjustment time *τ*. This may explain why a variety of MJO behaviors have been produced in GCMs and why tuning parameters or change of cumulus parameterization can effectively improve the MJO simulation.

The model here does not have shallow convection due to the simple vertical structure of the model. But the BL moisture convergence-induced heating is added to the mid-level, which is located to the east of the major convection center. In this sense, the shallow convection effect is represented in the model by the “deep’ convection to the east of the major convection. In GCMs, the BL FC always exists. However, the way by which BL convergence interacts with shallow convection in different GCM may differ and this interaction could significantly affect MJO behavior. The BL convergence can enhance shallow convective heating and upward transport of moisture that further feeds back to BL moisture convergence. This positive feedback could amplify the effects of FC feedback. This implies that MJO simulation may be sensitive to shallow cumulus parameterization schemes in GCMs. This process should be taken into account in the numerical modeling of MJO with GCMs.

## Notes

### Acknowledgements

This work has been supported by the NSF Award #AGS-1540783 and the Atmosphere–Ocean Research Center sponsored by the Nanjing University of Information Science and Technology and University of Hawaii. The authors thank Dr. Sun-Seon Lee for help with plotting Fig. 1 and anonymous reviewers for their constructive comments. This is the SEOST publication 9866, IPRC publication 1222 and ESMC publication 133.

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