1. Introduction
The Madden–Julian oscillation (MJO), named after its discoverers (Madden and Julian 1971, 1972, 1994), features an equatorially trapped, slowly eastward-propagating (about 5 m s−1), planetary-scale baroclinic circulation cell in the Eastern Hemisphere (Knutson and Weickmann 1987; Wang and Rui 1990; Hendon and Salby 1994; Maloney and Hartmann 1998; Kiladis et al. 2005; Zhang 2005). The MJO circulation comprises equatorial Kelvin waves and Rossby waves, and exhibits a quadrupole-vortex horizontal structure when the MJO convection is located over the equatorial Indian Ocean and the western Pacific (Rui and Wang 1990; Hendon and Salby 1994). A phase lag usually occurs between the leading planetary boundary layer (PBL) moisture convergence and the midtropospheric counterparts (Hendon and Salby 1994; Sperber 2003). Meanwhile, the MJO also presents nondispersity (Salby and Hendon 1994; Wheeler and Kiladis 1999; Roundy and Frank 2004).
Recent observations show that the MJO is usually coupled with a multiscale convective complex (Nakazawa 1988; Hendon and Liebmann 1994; Houze et al. 2000; Slingo et al. 2003; Straub and Kiladis 2003; Haertel and Kiladis 2004; Moncrieff 2004; Kikuchi and Wang 2010). Multicloud models (Khouider and Majda 2007; Majda et al. 2007; Khouider and Majda 2008; Khouider et al. 2011) and multiscale models (Majda and Biello 2004; Biello and Majda 2005; Biello et al. 2007; Majda and Stechmann 2009a; Wang and Liu 2011; Liu and Wang 2011, 2012; Liu et al. 2011) have been built to study the multiscale nature of the MJO.
Based on the multiscale structures of the MJO, the primary instabilities and damping are assumed to occur on synoptic scales (Khouider and Majda 2006, 2008; Kiladis et al. 2009). By the key premise that modulations of synoptic-scale wave activity are included by low-level moisture preconditioning on planetary scales, Majda and Stechmann (2009b; hereafter MS09) parameterized these scale interactions and presented a general skeleton model for the MJO, which captured three important features of the MJO skeleton: 1) slow eastward propagation at a speed of roughly 5 m s−1 (Fig. 1b); 2) peculiar dispersion relation of dω/dk ≈ 0 (Fig. 1a), where ω and k are the frequency and wavenumber, respectively; and 3) quadrupole-vortex horizontal structure (Fig. 2). However, the MS09 model only produces neutral modes and does not explain why the MJO skeleton can grow or be maintained against thermal and momentum damping, or why faster growth occurs on the preferred planetary scale.
(a) Frequency and (b) phase speed as a function of wavenumber for low-frequency modes of the skeleton model. Gray dots denote results of MS09 without the PBL effect, rb = 0. Crosses and circles denote results derived from the frictional skeleton model with rb = 0.03 and rb = 0.06, respectively. Red and blue denote unstable and damped modes, respectively. The growth rates are proportional to the diameters, with respect to the maximum growth rate of 0.11 day−1.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
Velocity vectors, wave activity (shading), with lower-tropospheric temperature (contours) of the neutral (a) westward and (b) eastward modes are plotted for wavenumber 2, where rb = 0. Positive (negative) contours are solid (dashed), and positive (negative) shading is dark (light) gray. Contour interval is one-third of the magnitude, and zero contours are not drawn. Only wave activity above one-third of the magnitude is shaded.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
Observations and general circulation model (GCMs) simulations show that the MJO major convection region is leaded by its PBL moisture convergence (Wang 1988; Hendon and Liebmann 1994; Hendon and Salby 1994; Jones and Weare 1996; Maloney and Hartmann 1998; Matthews 2000; Sperber 2003; Lin et al. 2004; Tian et al. 2006; Yang et al. 2008). This vertical tilted structure is one of the fundamental features of the MJO and has important implications for MJO dynamics. Wang (1988) demonstrated that low-frequency, moist equatorial Kelvin waves are driven by an instability arising from the PBL-induced moisture convergence. Frictional moisture convergence–induced heating was shown to be able to couple with the equatorial Kelvin and Rossby waves and select a slow eastward movement for the coupled Kelvin–Rossby wave packet (Wang and Rui 1990). However, in their works (Wang 1988; Wang and Rui 1990), the moisture pumped by the PBL was parameterized directly into the diabatic heating, without considering scale interaction. When coupled with the multicloud model of Khouider and Majda (2006), the PBL can amplify greatly the convectively coupled Kelvin waves (Waite and Khouider 2009). The frictional moisture convergence mechanism has also been confirmed further in more complex models, with different convective parameterization schemes (Salby et al. 1994; Ohuchi and Yamasaki 1997; Moskowitz and Bretherton 2000). These results encourage us to study the role of the PBL in sustaining the MJO skeleton.
To understand the role of the PBL in sustaining the MJO skeleton, we try to build a frictional skeleton model, where a slab PBL is implemented into the MS09 skeleton model (section 2). In section 3, we present a new instability source for the MJO skeleton, arising from the interaction between the boundary layer dynamics and the MJO skeleton, and discuss the preferred eastward propagation and long-wave selection for the MJO. The role of the equatorially trapped sea surface temperature (SST) on selecting the eastward propagation of the MJO is also discussed in section 4. The last section summarizes the results, and discusses limitations and future work.
2. The frictional MJO skeleton model
a. The neutral skeleton model for the MJO
The neutral skeleton model was proposed by MS09 and developed further by Majda and Stechmann (2011). In this skeleton model, the effects of synoptic wave activity (and upscale eddy moisture and heat transfer) are parameterized in the planetary-scale temperature and moisture equations, which drive the MJO skeleton. The recent works of Liu and Wang (2011, 2012) demonstrate that the out-of-phase eddy heat and moisture transfer supports this parameterized relationship between the temperature and moisture dynamics. They showed that, on planetary scales, the westward-propagating 2-day waves provide negative eddy heat transfer but positive eddy moisture transfer, and the eastward-moving moist Kelvin waves provide positive eddy heat transfer but negative moisture transfer. In this skeleton model, only the amplitude of wave activity, rather than details of synoptic-scale motion, is considered.
b. The PBL
c. Eigenvalue problem
Equations (7) and (9) compose a set of linear partial differential equations, for which the eigenvalue problem can be readily solved. For the zonal propagating plane waves, we assume that they have the structure of 
When only the lowest N meridional modes of each variable are kept for the meridional expansion of parabolic cylinder functions, we project (7) and (9) on to the σ–k space and obtain a linear matrix of (5N × 5N) for the five variables in (9). Here N = 1 represents the lowest equatorially trapped mode. The frequency and eigenvectors are calculated through the matrix inversion corresponding to each wavenumber. Only the Kelvin and Rossby waves are kept in (9) because of the long-wave approximation in the free troposphere. Following the idea that the wave activity has only the lowest meridional truncation of parabolic cylinder functions (MS09), the Rossby and Kelvin waves, on their lowest meridional modes, can be studied by using N = 3, and sensitive experiments show that a higher N does not affect the results.
To derive this skeleton model, the multiscale processes are included in the wave activity implicitly (MS09), so this frictional skeleton model is only valid for the long waves. In this paper, we only present the results with wavenumber smaller than 10.
3. The instability mechanism
When the PBL is included, the frictional model still captures the MJO-like features that the original neutral model (MS09) produced (Figs. 1 and 3). The inclusion of the PBL, however, makes the eastward-propagating long waves and westward-propagating short waves unstable, while the westward-moving long waves and eastward-moving short waves become damped (Fig. 1). For eastward modes, the strongest instability is associated with wavenumber 1; for westward modes, the growth rate increases as wavelength decreases, and the “ultraviolet catastrophe” problem exists for short westward modes. This result indicates, at least on planetary scales, that the PBL prefers the eastward-propagating mode. This result is consistent with the findings of Wang and Rui (1990), namely that the PBL provides an instability source to couple the eastward Kelvin waves and westward Rossby waves, and results in coupled eastward-moving modes when only considering the planetary modes. Coupled with the same PBL, the nonlinear heating in a two-layer model can select the most unstable planetary-scale eastward modes (Li and Zhou 2009).
As in Fig. 2, but for the frictional modes, where rb = 0.06.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
The sensitivity experiment with a weaker PBL coefficient, for example rb = 0.03 associated with a smaller PBL depth, yields the same qualitative results, except that the growth rates are reduced (Fig. 1). Decreasing sea surface temperature also plays the same role as the weak PBL coefficient did.
The role of the PBL mainly comes from E, and sensitivity experiments show that a small E still maintains the similar behavior of the MJO skeleton (Figs. 1 and 3), except that the growth rates are enhanced. The growth rate of eastward wavenumber 1 is 0.08, 0.11, 0.13, and 0.14 day−1 for E with damping scales of 4, 8, and 16 h and of 1 day, respectively.
For the growing modes, positive available potential energy (APE) should be generated, which requires positive covariance between the perturbation temperature and wave activity (diabatic heating), or 
Relationship between the wave activity and temperature perturbations for wavenumber 2. (a) Phase lag between the meridionally averaged wave activity (dashed) and temperature perturbation (solid). For comparison, the results are shifted zonally so that the zero temperature anomalies are all located at 90°E. (b) Cross correlation (
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
Normalized available potential energy (APE) for eastward (E) and westward (W) modes with wavenumbers 2 and 8, respectively. APE is normalized by the maximum APE of the eastward wavenumber-2 mode.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
How is APE generated? Figures 6 and 7 show that generation comes from the additional moisture source pumped by the PBL. Since the PBL excites wavelength-dependent Ekman pumping (Li and Wang 1994), for long waves, the right-hand side (rhs) terms in (7) containing zonal derivatives are small compared to the remaining terms, 
Velocity vectors, lower-tropospheric moisture (contours), and Ekman pumping (shading) of the (a) westward and (b) eastward modes are plotted for wavenumber 2, where rb = 0.06. Positive (negative) contours are solid (dashed), and positive (negative) shading is dark (light) gray. Contour interval is one-third of the magnitude, and zero contours are not drawn. Only Ekman pumping above one-third of the magnitude is shaded.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
As in Fig. 6, but for wavenumber 8.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
For the long eastward mode, the positive equatorially trapped Ekman pumping leads the wave activity and prepares additional moisture for the growth of the wave activity (Fig. 6b), whereas for long westward mode the negative equatorial Ekman pumping damps it by drying the lower troposphere in front of the wave activity (Fig. 6a). For short waves, the strong off-equatorial Ekman pumping, located to the west of the wave activity (Fig. 7), fulfills an opposite role, namely damping the eastward mode and destabilizing the westward mode.
4. The role of the equatorially trapped SST
In the above section, this model presents that the instability for short westward modes increases with decreasing wavelength. Even for westward wavenumber 10, the PBL produces a considerable instability compared to the eastward wavenumber 1 (Fig. 1a). But this result is obtained by calculating the model with constant SST of 28.5°C. In observation, SST has usually an equatorial maximum and decays poleward, especially in the boreal wintertime when the MJO prevails (Zhang 2005). So we further take a look at the role of the SST in this frictional skeleton model. Two additional sensitivity experiments have been run, where 
The results are presented in Fig. 8. The equatorially trapped SST presents higher-frequency oscillations for both eastward and westward modes, but they are still located in the range of the MJO speed. Compared to the constant SST, the model with equatorially trapped SST presents wavelength selection for the growing mode, the SST distribution with an equatorial maximum can enhance the growth of the long eastward modes, while suppress the growth of the short westward modes (Fig. 8a). Of course, the ultraviolet catastrophe problem still exists for very short westward modes, for which this frictional skeleton model is not valid.
As in Fig. 1, but red and blue denote constant and equatorially trapped SST, respectively. Gray dots represent the neutral mode of MS09. The maximum growth rate is 0.12 day−1.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
Associated with the equatorially trapped SST, the mean moisture, as well as the gross moisture statics stability, has also an equatorially trapped structure. For long eastward modes, the PBL Ekman pumping has an equatorially trapped structure (Fig. 6b), where the mean state moisture is also enough near the equator with a SST magnitude of 31°C, then the moisture convergence is enhanced, as well as the instability. On the other hand, for short westward modes, the maximum PBL Ekman pumping is in the off-equatorial region (Fig. 7a), where the mean state moisture is lower because the SST is cooler than that over the equator, thus the moisture convergence is relative weak compared to the constant SST.
When wavenumber is less than 10, it is interesting that the PBL prefers destabilizing the long modes (Fig. 8a), as the PBL has a wavelength-dependent Ekman pumping. When taking the lower-tropospheric moisture as a reference, Fig. 9 reveals that the relative moisture pumped by the PBL is weak for short waves. Since, for short waves, the wave activity is balanced largely by the wave convergence (Fig. 9), the perturbation pressure, which forces the Ekman pumping of (7), is relatively weak. However, for long waves, the retaining difference between the wave activity and wave convergence still produces strong perturbation pressure, which excites Ekman pumping and provides a strong instability source for the MJO.
Magnitude ratios of divergence to wave activity (Div/a) and Ekman pumping moisture to lower-tropospheric moisture (mb/q) as a function of wavenumber.
Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-12-020.1
This result means that the realistic equatorially trapped SST does help to provide a strong instability for long waves and prefers the eastward propagation of the MJO.
5. Discussion
The frictional skeleton model demonstrates the critical role of the planetary boundary layer (PBL) in sustaining the Madden–Julian oscillation (MJO) skeleton. The physical mechanism lies in the intrinsic characteristics of the equatorial waves and their associated PBL. On the planetary scale, the moisture convergence of the PBL is controlled mainly by the meridional variation of the pressure anomaly and is equatorially trapped and lies to the east of the MJO (Fig. 6b), thus the PBL Ekman pumping tends to moisten the low troposphere and precondition the atmosphere for deep convection of the MJO (Johnson et al. 1999). In future work we need to study these features in the observations. Hsu and Li (2012) also determined that the asymmetric moisture convergence in the PBL is helpful for the eastward propagation of the MJO, while the leading moisture convergence is mainly coming from the zonal wind convergence, since in their work the convection center was chosen in the off-equatorial region.
The equatorially trapped sea surface temperature prefers coupling the long waves and provides a strong instability for eastward propagation of the MJO. During the boreal summer the mean state has strong off-equatorial moisture sources, such as in the South Asian monsoon region, where the off-equatorial moisture convergence in the PBL should be important (Wang and Xie 1997; Wang et al. 2005). The proposed frictional skeleton model may be instrumental to understanding the mechanism of the Indian monsoon intraseasonal oscillation.
Keep in mind that this frictional skeleton model is only valid for large-scale waves (MS09). Although it presents unrealistic unstable short westward modes, in the large-scale wave regime, this frictional skeleton model prefers the unstable planetary scales and selects the eastward propagation of the MJO.
Here we only include the role of scale interaction implicitly, which is parameterized as planetary-scale wave activity (MS09). Actually, the scale interaction between the MJO and synoptic-scale motions also provides instability sources for the MJO (Biello and Majda 2005; Wang and Liu 2011; Liu and Wang 2011, 2012; Liu et al. 2011). So it is necessary to find out how the PBL regulates the synoptic-scale motions, and their relations with the MJO. This calls for future theoretical, model, and observational studies.
This study was supported by the Climate Dynamics Program of the National Science Foundation under award AGS-1005599 and the Global Research Laboratory (GRL) Program from the Ministry of Education, Science, and Technology (MEST), South Korea. Additional support was provided by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), by NASA through Grant NNX07AG53G, and by NOAA through Grant NA17RJ1230 through their sponsorship of research activities at the International Pacific Research Center.
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