1. Introduction
Tropical intraseasonal variability (ISV) is often characterized by the slow eastward propagation of large-scale convective systems near the equator with a broad spectrum of 30- to 90-day periods. Since the discovery of the Madden–Julian oscillation (MJO) a half-century ago (Madden and Julian 1971, 1972), long-term observations have revealed that the spatiotemporal features of the tropical ISV vary with background climate conditions. The large-scale mean-state structure fundamentally shapes the ISV locality so that intraseasonal convective activity dominates only over the Indo-Pacific warm-pool region, where the lower troposphere is warm and humid and the easterly trade wind diminishes (e.g., Zhang 2005). Indeed, the background seasonality may strongly control the pattern and location of the ISV (Zhang and Dong 2004). While the MJO is the most energetic during the boreal winter over the warm-pool region and tends to propagate to the central Pacific, the summer ISV travels from the Indian Ocean to East Asia. On an interannual time scale, the tropical background largely fluctuates due to El Niño–Southern Oscillation (ENSO), affecting the MJO in terms of the location of convective activity (Bellenger and Duvel 2012; Suematsu and Miura 2018) and periodicity (Pohl and Matthews 2007; Wei and Ren 2019; Wang et al. 2019). A changing climate may also modulate the lifetime of MJO convection as the warm-pool area expands (Roxy et al. 2019). Since these phenomena are all associated with the warm pool and the Walker and monsoon circulation systems, a geographically proper background is a necessary condition for simulating ISV diversity.
Many theories have been suggested to explain the essence of the tropical ISV in simple frameworks (cf. Jiang et al. 2020). For instance, a large-scale eastward-propagating convective system can selectively emerge as a dynamic mode of convectively coupled Kelvin–Rossby waves (Wang and Rui 1990), where strong friction in the planetary boundary layer (PBL) leads to low-level convergence of background moisture so that the equatorial Rossby and Kelvin waves can be coupled. The resulting propagation speed under warm-pool conditions ranges from 10 to 15 m s−1, much higher than the observed speed of approximately 3–7 m s−1 (Zhang and Ling 2017). Alternatively, tropospheric moisture variability has been noted as a major factor in interpreting the ISV. Sobel et al. (2001) proposed a quasi-balanced moist vortex propagating eastward due to the meridional advection of background moisture, while Raymond (2001) found that cloud–radiation interactions may destabilize such a moisture mode. The moisture mode theory has been further validated by considering various processes. The selective growth of large-scale modes may be realized by cloud-radiative effects (Adames and Kim 2016), horizontal diffusive processes (Sobel and Maloney 2013), moist convective damping (Emanuel et al. 1994), moistening due to low-level perturbation easterlies (Ahmed 2021), a combination of zonal moisture advection and surface flux feedback (Wang and Sobel 2022b), and so on. Meanwhile, propagation relies on a balance among surface evaporation, horizontal advection of background moisture, modulation of synoptic-eddy drying, and PBL frictional convergence (Sobel and Maloney 2012, 2013; Adames and Kim 2016). Moreover, explicit treatment of moisture in the dynamic model of Wang and Rui (1990) with PBL friction yields a large-scale dynamic moisture mode with a phase speed of approximately 5–12 m s−1, depending on the convective adjustment time scale (Wang and Chen 2017; Chen and Wang 2019). Other theoretical studies have also suggested that slow ISV propagation could be a result of the interference of westward and eastward inertia–gravity waves (Yang and Ingersoll 2013) or of the cyclonic vorticity generated by the vertical stretching of the planetary vorticity over the convective region (Hayashi and Itoh 2017). Previous studies have suggested a unified view of the tropical ISV based on observational analysis (Jiang et al. 2018) and theoretical models such as a moist shallow water model (Ahmed 2021; Wang and Sobel 2022a,b). Whereas idealized theories could be fruitful in interpreting general aspects of the ISV, such as the spatial scale and frequency, most theoretical models are highly simplified, and they often ignore actual climatological mean-state patterns, such as the meridional gradient asymmetry of background moisture, as well as prognostic dynamical processes to reduce the eigen solutions, leaving the connection to the observed ISV uncertain.
ISV dynamics can be further understood in realistic situations by comparing the outputs of general circulation models (GCMs). With the use of 37 atmosphere–ocean coupled GCMs, Ahn et al. (2017) showed that a close relationship between precipitation and lower-tropospheric relative humidity as well as the lower value of the mean-state gross moist stability over the warm-pool region can be critical for better simulating the MJO (see also Ahn et al. 2020). Coupling with ocean models tends to improve the MJO simulation performance, while ocean feedbacks may affect tropical mean moisture levels to modulate MJO propagation (DeMott et al. 2019). Ling et al. (2017) tracked individual MJO events in 27 GCMs (Jiang et al. 2015), showing that all the models can simulate the MJO in essence, but the background state, especially the zonal wind at 850 hPa over the warm-pool region, can be a major conducive factor of MJO occurrences regardless of the similarity of the simulation background to the observed background. This suggests that a proper approach to investigating ISV diversity must first consider realistic background conditions. However, the interaction among multiscale fields makes it difficult to effectively separate the ISV from the background when GCM outputs and observational datasets are analyzed. Therefore, developing an intermediate-complexity model that can simulate the ISV given a specific basic state is important to reveal the role of the background in determining the locality and diversity of the tropical ISV.
In this study, a linearized GCM is used to explore the diversity of the tropical ISV under various background states. A moist linear baroclinic model, developed by Watanabe and Jin (2003), is improved with a simple yet suitable convective parameterization scheme and cloud-radiative feedback to atmospheric heating such that it can better capture the slow moisture mode dynamics for the ISV as a linear mode under the prescribed global basic state (section 2a and the appendix). The advantage of this model comes from the basic state derived from reanalysis datasets, and thus, the spatial inhomogeneity, such as zonal and meridional asymmetry of the background conditions and topography, can be easily considered. This model can be utilized to examine the separate roles of the dynamic and thermodynamic background fields in terms of their impacts on the ISV. We conduct a series of experiments from the idealized to realistic basic states (section 2b) to clarify the role of the realistic background in modulating the ISV. Section 2c describes the method used to analyze the model outputs. We show the basic properties of the simulated ISV under a boreal winter basic state in section 3. In section 4, we systematically investigate the winter ISV under zonally uniform and nonuniform basic states and determine its sensitivity to model parameters and dynamics. The impacts of the ENSO and background seasonality on the ISV are examined in sections 5 and 6, respectively. Section 7 provides a summary and discussion.
2. Methods
a. Model
We use a moist linear baroclinic model (mLBM), which consists of the primitive equations linearized from an atmospheric GCM (Watanabe and Kimoto 2000) as well as a prognostic moisture equation to incorporate interactive convection (Watanabe and Jin 2003). Equations for the vorticity ζ, divergence D, potential temperature T, specific humidity q, and logarithm of the surface pressure ln ps ≡ π are linearized under a basic state in vertical sigma coordinates σ ≡ p/ps. Hereafter, the basic state is denoted as overbars, and the perturbation is denoted as primes (e.g.,
In Watanabe and Jin (2003), a linearized Betts–Miller convective adjustment scheme (Betts and Miller 1986; Neelin and Yu 1994; Yu and Neelin 1997) was implemented together with bulk schemes for the sensible and latent heat fluxes at the surface to obtain a steady response to the prescribed anomalous SST forcing. Although Watanabe and Jin (2003) intended to examine the role of moisture dynamics, the improper implementation using strong tropospheric restoration of T′ and q′ outside of the forced region with a convective relaxation time scale of 2 h effectively kills the slow internal variability associated with moisture dynamics. The forced response is thus drastically damped (e.g., Fig. A2). Proper implementation of the convective parameterization should focus on the slow mode, as noted by Neelin and Yu (1994) (see also Yu and Neelin 1997). Additionally, observations do not support a strict dependence between surface evaporation and precipitation rates in models incorporating the Betts–Miller scheme (Raymond 2001) or a short convective adjustment time scale of a few hours in the MJO (Bretherton et al. 2004; Adames et al. 2019). Therefore, the convective parameterization implementation in this model must be adapted to allow internal moisture variability.
We formulate a linear scheme for the convective process based on the separate contributions of the PBL and the free troposphere, as suggested by Zhang (2002), whose scheme resulted in better MJO simulation in a GCM (Zhang and Mu 2005). First, the convective heat source and moisture sink are related to the free-tropospheric moisture anomaly with a convective adjustment time scale τc of 12–18 h (Bretherton et al. 2004; Adames and Kim 2016; Rushley et al. 2018; Ahmed et al. 2020). Vertical profiles of the convective heat source and moisture sink terms are determined by combinations of both the vertical gradients of the dry static energy and moisture of the basic state, respectively, and the standardized vertical motion analytically derived from the slow moist mode solution reported by Neelin and Yu (1994) that approximates the first baroclinic vertical structure in the tropics. Here, the cloud bottom level is fixed to the PBL top at σb = 0.95, while the cloud top σt is determined as the basic-state tropopause (Reichler et al. 2003), assuming deep tropical convection. These convective terms ensure the conservation law of column moist static energy by multiplying the ratio of the column-integrated convective moisture sink to the heat source profiles. Second, the moisture anomaly in the PBL is allowed to be transported into the free troposphere so that convective heating is produced, similar to Zhang and McFarlane (1995). The heating profile due to PBL moisture is assumed to match that due to free-tropospheric moisture, while only q′ in the PBL is consumed. The moisture transport time scale τb is a few hours based on the typical lifetime of shallow mesoscale convective systems in the tropics (Houze 2004; Betts and Miller 1993). This time scale needs additional theoretical support to improve further (e.g., Ahmed et al. 2020). Note that this part is not essential for simulating the internal variability (section 4c) but is more important for calculating the steady response to SST forcing. The total convective precipitation rate is reduced where the column relative humidity of the basic state is lower than 50% (Bretherton et al. 2004).
Cloud-radiation feedback over moist regions is considered in this model following the radiative heating parameterization of Adames and Kim (2016). The horizontal structure R is determined by a space function weighted with nearby precipitation anomalies, where Lrx = 1000 km and Lry = 300 km are the zonal and meridional length scales of the weighting calculation, respectively. A modestly bottom-heavy vertical profile is prescribed from the surface to the cloud top, mimicking the net cloud-radiative heating anomalies associated with the MJO in observations (Ma and Kuang 2011). The amplitude of the radiative heating is proportional to the greenhouse enhancement factor r, which is set to realize approximately 10%–20% heating of the column-integrated convective heat source.
The other detailed settings are summarized as follows. We implement a large-scale condensation (LSC) process that adjusts q′ toward the anomalous saturated moisture due to T′ in each grid with a time scale τl of 2 h only where the basic state is sufficiently saturated. The sensible and latent heat fluxes at the surface are also implemented in the model so that the wind-induced surface heat exchange (WISHE; Emanuel 1987), for instance, can be considered (e.g., Sobel and Maloney 2013; Wang and Sobel 2022a). The linearized bulk formulation of surface fluxes is the same as that in the former version (Watanabe and Jin 2003), and thus, the bulk coefficients are calculated with basic-state quantities (Louis 1979), but the roughness length over the ocean is linearized via the formula of Miller et al. (1992). The linear damping terms in the momentum (ζ′, D′) and temperature (T′) equations assume Rayleigh damping and Newtonian cooling with time scales of 7 and 14 days in the free troposphere and 2 and 4 days in the PBL, respectively. In the PBL, q′ also exhibits weak linear damping with a time scale of 10 days, assuming potential moisture dissipations near the surface (this makes the time integration stable for obtaining a steady response to SST forcing). To avoid reflection from the upper boundary, these time scales are set to 0.5 days and 1 day at the two uppermost levels and third level, respectively. To focus on the tropical variability, free-tropospheric linear damping is strengthened by 10 times outside the tropics (35°S/N with a buffer range of 10° to the equator) so that any extratropical eddies can be damped. This strong damping at midlatitudes may need to be replaced with a scale-dependent synoptic-eddy feedback scheme, which will be developed to allow further examination of the teleconnection from the tropics. The time scale of the biharmonic horizontal diffusion terms τhdiff is 20 min for the smallest wavenumber at a T42 resolution. Although no coupling with ocean and land model components has yet been implemented, realistic land distribution and topography are considered. Over land areas, all the linear damping time scales below the PBL top are changed to 0.5 days, and the precipitation processes are halved (assuming land–ocean difference in surface moisture source), which is admittedly inaccurate but should not critically affect the results reported herein.
This new linear model thus fully considers the observed dynamic and thermodynamic components of the basic state and incorporates the most well-known atmospheric processes important for generating the ISV. Overall, these basic settings allow the model to produce not only tropical ISV modes but also a steady response to El Niño SST forcing (refer to Figs. A1 and A2, respectively, in the appendix), as the model can be used for ENSO research as well.
The main parameters in the present model are basically configured as follows: the free-tropospheric convective adjustment and moisture transport time scales (τc and τb) are set to 13 and 3 h, respectively, in the convective parameterization and the greenhouse enhancement factor r is set to 0.12 in the radiative heating parameterization. The dependence of the results on the model configurations is investigated in section 4c. For example, we test the sensitivity of our main results to the parameter settings by changing τc from 10 to 240 h, τb from 3 to 240 h, and r from 0.12 to 0.0. We also test to what extent the results are affected by removing each of the convective heating terms related to τc and τb. It will be revealed in section 4c that τc is the critical model parameter for controlling the propagating property of the ISV.
b. Experimental designs
With the use of a hierarchy of basic states from the idealized to realistic states, we conducted a series of time integration experiments for at least 200 days initialized with equatorial moisture perturbations in the lower troposphere. The three-dimensional and surface atmospheric basic states were derived from the ERA5 monthly dataset (Hersbach et al. 2023a,b) for 1981–2010 and interpolated onto the model grid. The basic state comprises the zonal and meridional winds (transformed into the vorticity and divergence, respectively, in the calculation process), potential temperature, specific humidity, and surface pressure as well as the surface temperature that merges the air–skin and sea surface temperatures.
The basic-state vertical motion is derived internally from the basic-state inputs. This model can examine the sensitivity of the results to each basic-state component under the prescribed basic state. Since the ISV may be affected by, for instance, the background zonal winds and moisture distributions, we conducted additional experiments by changing only the dynamical (D) or thermodynamic (T) field from a specific basic state. Here, the D field includes the zonal and meridional winds, while the T field consists of the specific humidity, potential temperature, surface temperature, and surface pressure.
1) CTL experiments
In the control (CTL) experiments, the basic states are equivalent to the long-term mean of atmospheric reanalysis (Figs. 1a–c). The internal mode under the boreal winter basic state [December–February (DJF)] is considered in section 3, as the variance in the tropical ISV is dominated by the MJO during this season in the observations. The influence of the background seasonality is also briefly examined in section 6 using the spring [March–May (MAM)], summer [June–August (JJA)], and autumn [September–November (SON)] basic states as well as the annual (ANN) mean.
Basic states of (left) the specific humidity at 850 hPa and surface temperature and (right) the zonal and horizontal winds at 850 hPa. The zonally nonuniform basic states are provided for the (a) DJF mean, (b) JJA mean, and (c) annual mean. (d) EN anomalies of each field in DJF. The zonally uniform basic states are provided for the (e) ZM, (f) EH mean, and (g) WH mean in DJF.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
To show the impact of the ENSO state on the winter ISV, composited El Niño anomalies (Fig. 1d) are added to the winter basic state in section 5. To composite the DJF anomalies of ERA5, we selected 9 El Niño winters (DJF in 1982/83, 1986/87, 1987/88, 1991/92, 1994/95, 1997/98, 2002/03, 2009/10, and 2015/16) based on the winter average of the oceanic Niño index (ONI) greater than 0.8 K. Here, the ONI is derived from Extended Reconstructed SST, version 5 (ERSST.v5) dataset (Huang et al. 2017) as a 3-month running mean of the SST anomalies in the Niño-3.4 region (5°S/N, 120°–170°W). The El Niño composite is represented as EN and multiplied by a factor d = {−1.0, −0.5, 0.5, 1.0} in the CTL+dEN experiments (hereafter, the curly brackets represent the ranges of experimental parameters). The basic state is close to the El Niño peak for positive d values and the La Niña peak for negative d values. For simplicity, the spatial complexity of the ENSO (cf. Timmermann et al. 2018) was not considered.
The initial condition is a lower-tropospheric moisture perturbation over the Indian Ocean (Figs. A4a,b in the appendix), which is positive and localized between 0.9 and 0.5 σ levels in a 15°-radius horizontal circle centered at 80°E on the equator. It has an e-folding horizontal scale of 10° and exponentially decays from the bottom to the top levels. Regardless of the initial condition details, a leading mode eventually dominates under the spatially complex basic state.
2) ZM experiments
A series of idealized experiments are conducted to better understand the essence of the internal modes in this model. The simplest basic state used in this study is the zonal mean (ZM) derived from the ERA5 basic states (Fig. 1e). The land distribution is not considered, so all the grids are ocean configurations. Since the variance associated with the ISV is concentrated mostly in the Eastern Hemisphere (EH) between 0° and 180°, we add the deviation of the EH-averaged basic states from the ZM onto the ZM, symbolized as ZM+bEH. Here, factor b ranges from −1 to 1 with intervals of 0.2. The basic state is indicated as ZM+EH for b = 1 equal to the EH average (Fig. 1f) and ZM−EH for b = −1 equal to the zonal mean over the Western Hemisphere (WH; Fig. 1g). The leading mode in each zonal-mean experiment depends on the initial value of the zonal wavenumber (k). Therefore, initial equatorial moisture anomalies with sinusoidal zonal structures of k = {1, 2, 3, 4, 5}, shown in Fig. A4 of the appendix, are used for each basic state to examine the scale dependence of the growth rate and frequency, i.e., the dispersion relationship is examined.
We investigate how the global structure of basic states affects the ISV modes based on two types of experiments. The locality of the ISV over the Indo-Pacific warm-pool region is assessed by combining the EH and WH conditions, but the zonal extent of the EH is changed. The EH condition extends from longitude 0° to LEH × 360°, where LEH ranges from 0.1 to 0.8 with intervals of 0.1, and it is linearly replaced with WH in a 30° buffer area. In addition to these idealized cases, we conduct a series of intermediate experiments between ZM and CTL. The deviation of the CTL basic state (Fig. 1a) from the ZM basic state (Fig. 1e), represented as GLB, is added to the ZM by multiplying by a factor a = {0.2, 0.4, 0.6, 0.8, 1.0} in the experiments ZM+aGLB. The case of a = 1 (ZM+GLB) is referred to as the reference (Ref) case for a series of sensitivity tests of the model parameters. The initial value is the same as the CTL experimental value since these basic states include various wavenumbers. For easy comparison, the land distribution and topography remain the same as those in the ZM experiments. Note that the difference between CTL and Ref (ZM+GLB) appears only in the land and topography treatment, which will be examined in section 4c.
c. Data analysis
The growth rate and frequency of the most unstable (least damped) mode in each experiment are estimated from the model output, and the growing or decaying oscillation is rescaled using the estimated growth rate for further analysis. We use the model output after day 51 to avoid the influence of the initial perturbation. The data period ranges from days 51 to 200, but different lengths are used when necessary as the growth rate approaches zero. The initial cutoff period of 50 days is enough to eliminate the initial condition impacts on the anomalous pattern of the leading mode as the time scales of the convective adjustment and atmospheric linear damping, described in section 2a, are much shorter than 50 days.
The growth rate of the leading mode, B/2, is approximated by the least squares fitting slope B of the logarithm of the column-integrated eddy kinetic energy (EKE) averaged over the 15°S–15°N zonal band, i.e.,
To obtain rescaled periodic oscillations with growth or decay removed, the model output is divided by the exponential growth factor exp(Bt/2) and then multiplied by a factor that normalizes the amplitude so that the zonal mean of the standard deviation of the 20°S–20°N precipitation anomaly equals 1 mm day−1. The frequency F is estimated by the inverse of the time reaching the maximum lead autocorrelation of the rescaled precipitation averaged over 90°–120°E and 15°S–15°N. The propagation direction is determined by the zonal shift of the rescaled precipitation peak during a one-eighth cycle of the period (F−1) based on the lead–lag regression of the rescaled precipitation between 15°S and 15°N onto the area-averaged one. To further confirm the cyclicity, it is examined if the lead autoregression coefficients of the rescaled anomalies with the period are approximately the unity. Thus, the regression patterns of the anomalies onto the area-averaged precipitation remain consistent over time. Wavenumber–frequency spectra are calculated from the rescaled anomalies using NCL subroutines (https://www.ncl.ucar.edu/Applications/mjoclivar.shtml).
3. Winter ISV in the CTL experiments
In the CTL experiments under the boreal winter (DJF) basic state, an eastward-propagating damped mode emerges as the leading mode with a 51-day period. The estimated growth rate of this damped oscillation is B/2 = −0.030 day−1. The results below are rescaled by dividing the model output by an exponential decay factor to focus on the periodic cycle without growth or decay.
The overall features of this propagating mode are summarized in Fig. 2. The dominant variability in the precipitation and low-level zonal wind is confined over the Indo-Pacific warm-pool region near the equator from 60°E to the date line. Its action center is shifted to the south of the equator, especially over the western Pacific. The 51-day periodic cycle is characterized by eastward propagation over the equatorial warm pool. The propagation speed of the precipitation and zonal wind anomalies are close to 3–5 m s−1. In the power spectrum within the frequency and zonal wavenumber (k) domain, the precipitation is dominated by k = {1, 2, 3}, while the zonal wind is dominated by k = {1, 2}. The simulated power spectra are consistent with the observations, where the dominant wavenumber ranges broader for the precipitation than the zonal wind (contours in the bottom panels of Fig. 2). Such differences among the variables result from the spatial complexity of the basic state.
(top) Statistical features of the precipitation and zonal wind in CTL with the DJF basic state. (middle) Variance map and time–longitude section of the 51-day cycle of the equatorial anomalies of the precipitation and zonal wind at 850 hPa from 20°S to 10°N. The solid and dashed lines (middle) indicate eastward propagation speeds of 5 and 3 m s−1, respectively. (bottom) The wavenumber–frequency power spectrum of the equatorial anomalies of the precipitation and surface zonal wind in the CTL (shading) and observational datasets (contours). The GPCP and ERA5 datasets are used for the winter spectrum.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
The horizontal and vertical structures of this ISV mode resemble those of the MJO in the boreal winter. In Fig. 3, a half cycle of the 51-day periodic propagation is presented by the rescaled anomalies of the precipitation, horizontal winds, pressure velocity, and specific humidity that are regressed to the rescaled and normalized precipitation anomalies averaged over 90°–120°E and 10°S–10°N with the following time lags and leads: 6-day lag, 0-day lag, 6-day lead, 12-day lead, and 19-day lead. The anomalous precipitation region propagates to the east from the Indian Ocean to the date line and is accompanied by wind variability. The shape of the precipitation area changes over time, but it expands by approximately 60° in longitude over the warm-pool region. The location of the ascending motion coincides with that of positive precipitation and moisture anomalies. The moisture anomaly peaks from 800 to 600 hPa over the Maritime Continent, while the vertical motion anomaly is the largest at approximately 400 hPa. The results indicate that a large-scale but local eastward-propagating leading mode is simulated over the Indo-Pacific warm pool and its overall structure is similar to the observed winter MJO. Please refer to Zhang (2005), Waliser et al. (2009), Jiang et al. (2011, 2015), and many other previous studies to compare the present results to observations.
Half cycles of the boreal winter ISV mode with the 51-day period in CTL. Lead and lag regressions of (left) the precipitation and wind anomalies and (right) the equatorial specific humidity and pressure velocity anomalies onto the normalized precipitation anomaly over 90°–120°E and 10°S–10°N. (from top to bottom) The anomalous fields at 6-day lag, 0-day lag, 6-day lead, 12-day lead, and 19-day lead are shown.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
4. Winter ISV under a hierarchy of the basic states
An MJO-like large-scale eastward-propagating mode emerged in the winter CTL experiment mainly over the EH but not over the WH. Why is the simulated ISV localized in the EH? To answer this question and to clarify the dominant factors that control the properties of this MJO-like mode, we start with a series of idealized experiments under the zonal-mean basic states from the WH to EH conditions. These zonal-mean experiments allow us to reveal the dispersion relationship of the modeled ISV mode by using several initial conditions with different zonal wavenumbers k. Here, we consider k = {1, 2, 3, 4, 5}. Then, we systematically adjust the basic state from the zonal mean to more realistic conditions to connect the idealized experiments with the CTL experiments.
a. Zonally uniform basic states
There is a clear dependence of the growth rate and frequency on both the basic state and zonal wavenumber (Fig. 4). In the ZM+bEH experiments, factor b controls the amplitude of the EH component added onto the ZM basic state (Fig. 1e) so that the zonal-mean basic-state ranges from the EH to WH zonal averages by varying b from 1 (Fig. 1f) to −1 (Fig. 1g). The growth rate increases with increasing b in general and tends to reach its maximum at k = {2, 3}. The frequency equals the eastward-propagating period of approximately 60 days for the EH basic state with b = 1, and this quantity is sensitive to both b and k. When b is negatively large and k is high, the leading mode is replaced with westward-propagating modes (stippled pattern in Fig. 4a), potentially related to a westward-propagating Rossby-like moisture mode over the WH (Mayta et al. 2022). This propagation direction transition seems consistent with that from a canonical equatorial Rossby wave to the eastward-propagating counterpart in a moist shallow water model (Ahmed 2021), but the mLBM experiments confirmed that the transition occurs even under the zonally uniform and meridionally nonsymmetric background states. Because the westward-propagating modes are highly damped in this model (B/2 < −0.06 day−1), we only focus on the eastward-propagating modes hereafter.
Dependence of the (left) growth rate and (right) frequency at each zonal wavenumber (k = 1, 2, 3, 4, and 5) on the basic states from the WH (b = −1) to EH (b = 1) zonal averages in the boreal winter (DJF). The parameter b = 0 corresponds to the ZM experiment. The frequency and corresponding period, shaded in (right), are positive for the eastward-propagating modes and negative for the westward-propagating modes.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
Snapshots of the k = 2 anomalous fields on day 111 in the ZM+EH experiment are shown in Fig. 5 as examples of the eastward-propagating modes obtained in the ZM+bEH experiments. Here, the model output on day 111 is rescaled so that the maximum of the precipitation averaged between 20°S and 10°N is 5 mm day−1, and only a half hemisphere is shown. In the time integration process, this structure remains unchanged during eastward propagation with a phase speed of 3.9 m s−1 (60-day period), but its amplitude exponentially grows with the growth rate B/2 = 0.020 day−1. As shown in the winter CTL simulation experiments, the precipitation and upper-level divergence anomalies deviate from the equator to the south, and the rotational wind variability is higher in the Southern Hemisphere (SH). The moisture anomaly between 20°S and 10°N is large below 600 hPa and tilted to the upper west. The location of the anomalous ascending motion coincides with that of the positive precipitation anomaly as well as the convective heat source and moisture sink. The convective moisture sink is largely balanced with vertical moisture advection, which is mostly due to the anomalous vertical motion as the basic state ω is very weak in the zonal-mean basic state. The radiative heating is in phase with the convective heating along the zonal direction but exhibits a small amplitude (approximately 10% of the convective heating) peaking in the lower troposphere, as prescribed in this model. The LSC is much smaller than the convective heating but effectively reduces the upper-level moisture anomaly.
Snapshots of the half-zonal wavelength of the k = 2 ISV mode in the ZM+EH experiment. (top) The horizontal structures of the anomalies of the precipitation (shading), horizontal winds at 850 hPa (vectors), velocity potential at 200 hPa (shading), and streamfunction at 850 hPa (contours). (middle),(bottom) The vertical structures of the anomalies of the specific humidity (shading), pressure velocity (contours), convective moisture sink (shading), vertical moisture advection (contours), convective heat source (shading), radiative heat source (contours), and LSC heat source (shading) and moisture sink (contours) averaged between 20°S and 10°N. Day 111 of time integration is selected.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
Moisture budget analysis for the k = 2 ISV mode in the ZM+EH experiment averaged over the free atmosphere between 20°S and 10°N. Day 111 is selected as in Fig. 5. The half-zonal wavelength is depicted. The gray shading indicates the moisture anomaly. The black dashed line represents the actual moisture tendency, and the black solid line is the total moisture tendency, including the zonal (green), meridional (red), and vertical (cyan) advective terms and the convective (magenta) and LSC (yellow) moisture sinks. The blue line indicates the sum of the moisture sinks and vertical advective term.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
The sum of these tendency terms explains the actual moisture tendency between 20°S and 10°N well. Moistening attains a positive peak to the east of the moist area, which is due to the meridional moisture advection associated with the anomalous wind mainly but partly canceled by the other terms. The zonal advective term moistens to the west of the moist area, generating a westward-propagating tendency by the easterly trade winds over the equatorial band at lower levels where the moisture anomaly is large. Because there is no zonal gradient in the basic state, the zonal advective term is attributed only to the zonal-mean flow. The vertical advective term tends to moisten the moist area but is largely compensated for by the convective moisture sink. The residual contributes to damping the moisture anomaly as well as reducing the eastward-propagating tendency. This residual tendency is partly canceled by the LSC, which plays a smaller role than the other terms in this model. In summary, the eastward propagation is caused by meridional moisture advection, which is prevented mainly by the low-level basic easterly state and partly by the residual of the vertical advection and moisture sinks. The growing tendency in the moist area is due to all the advective terms but canceled by the convective moisture sink.
The impact of the dynamic and thermodynamic (D and T, respectively) basic-state fields on the ISV is examined (Fig. 7). Both the growth rate and frequency in ZM+EH are greater than those in ZM, as shown in Fig. 4. When the thermodynamic field of the EH component is added to the ZM basic state (ZM+EHT), the growth rate is enhanced at higher k, and the frequency increases regardless of k. The frequency increase may result from the atmospheric static stability change. In contrast, the dynamic field in ZM+EHD is more efficient for enhancing the growth rate at lower k, especially at k = 2, and increases the frequency at higher k. Since the basic-state zonal wind of the EH component is westerly below approximately 500 hPa but easterly at the upper levels, the dynamic field of the EH is further separated along the vertical direction into two parts by multiplying step-function-like smooth functions of the σ coordinate: {tanh[10(σ − 0.5)] + 1}/2 for the lower levels and –{tanh[10(σ − 0.5)] − 1}/2 for the upper levels. In Fig. 7, the corresponding experiments are referred to as ZM+EHD(low) and ZM+EHD(up), respectively. The results indicate that only the low-level basic-state winds contribute to the change from ZM+EH to ZM+EHD. Therefore, the reduced basic-state easterly wind at lower levels in the EH, characterized by the western branch of the local Walker circulation, plays a role in generating large-scale modes even though the EH thermodynamic condition prefers higher k.
Growth rate and frequency of the leading modes for each zonal wavenumber k. The black and gray lines indicate the results of the ZM+EH and ZM experiments, respectively, while the green and red dashed lines indicate the results of the ZM+EHT and ZM+EHD experiments. The orange and blue triangles denote the ZM+EHD(low) and ZM+EHD(up) experiments.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
Positive feedback due to the cloud-radiative effect is assumed by
Dependence of the growth rate and frequency of the leading modes at each k on radiative heating in the ZM+EH experiment. (top) The dotted, dashed, and solid lines correspond to the radiative heating parameters r = 0.00, 0.06, and 0.12, respectively. (bottom) The black lines are the same as in (top), while the orange lines denote the horizontal diffusion time scale of 1 h. The asterisk symbols indicate the results with the sinusoidal half-wavelength vertical profile of the radiative heating with v1 = v2 = 0.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
b. Zonally nonuniform basic states
The basic state is zonally nonuniform in actuality, as characterized by the local Walker circulation and monsoonal circulation over the Indo-Pacific warm-pool region. While the above results indicate that the EH condition is favorable for exciting eastward-propagating modes, it remains unclear how such large-scale modes are shaped in a zonally localized EH basic state as in the zonally uniform case. Thus, we examined the dependence of the growth rate and frequency on the zonal extent of the EH condition by systematically varying LEH from 0.1 to 0.8 (Fig. 9a). Here, LEH is the relative zonal length of the EH to Earth’s circumference along the equator, and the EH basic state (Fig. 1f) is linearly replaced with WH conditions (Fig. 1g) at the edges. In these idealized experiments, the method for estimating the frequency (section 2c) uses the rescaled 15°S–15°N precipitation averaged over a 30° zonal range centering the middle of the EH domain so that the EH condition is focused on.
Growth rate and frequency of the leading modes in the experiments under the combined EH and WH basic states. (a) The zonal structure of the combined basic state controlled by the relative EH length LEH from 0.1 to 0.8. The ZM+EH experiment corresponds to LEH = 1.0 (dashed line). (b),(c) The standard deviations of the rescaled anomalies of the 20°S–10°N averaged precipitation and 850-hPa zonal wind in each experiment (solid lines) and in the ZM+EH experiment with k = 3 (dashed lines). In (d),(e), the black dots and dashed lines denote the growth rate and frequency, respectively, at each LEH and LEH = 1.0 with the radiative heating parameter r = 0.12, while the gray triangles and dashed lines denote r = 0.00. The red solid and green dash–dotted lines are the same as the black dots and lines, respectively, except that one of the EHD and EHT conditions is restricted by LEH.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
The variability is localized over the EH condition and damped over the WH condition, as shown in the zonal structures of the standard deviations of the equatorial precipitation and 850-hPa zonal wind anomalies (Figs. 9b,c). When the EH extent is greater than the half-circumference (LEH ≥ 0.5), LEH scarcely influences both the growth rate and frequency (Figs. 9d,e). The growth rate is close to but slightly lower than that in the zonally uniform EH case (LEH = 1.0; dashed line) since the propagating anomaly is strongly damped outside of the EH condition (Figs. 9b,c). As the frequency closely approaches that for LEH = 1.0, the propagation property remains unaffected under the EH condition. However, the growth rate rapidly starts decreasing when LEH is less than 0.4, which corresponds to 144° longitude. This is further confirmed in two additional experiments, EHT and EHD, that consider the zonally uniform EH basic state of the thermodynamic or dynamic field, but the extent of the other field is restricted by LEH (the red and green lines in Figs. 9d,e). This rapid decrease in the growth rate occurs even without radiative heating (gray triangles) when LEH is less than 0.3, reflecting the notable impact on zonal scale selection. In summary, both the dynamic and thermodynamic fields of the EH, such as the reduced low-level easterly wind and sufficient equatorial moisture, must extend further than the wavelength of the leading mode to avoid adversely affecting the modal properties of the leading ISV mode. In other words, the expansion of the warm-pool condition may enhance the instability of the ISV mode and expand its domain of activity, and vice versa.
The impact of the global pattern under the basic state is further examined by changing the basic state from the zonal-mean stepwise in the ZM+aGLB experiments. The departure of the winter basic state from its zonal mean is added to ZM with factor a ranging from 0.2 to 1.0. To enable comparison to the ZM and ZM+EH experiments, the land and topography remain the same as those in ZM. The experiment for a = 1.0 is the same as CTL with dynamic (D) and thermodynamic (T) fields and is referred to as ZM+GLB. As a result (the black line in Figs. 10a,b), both the growth rate and frequency linearly increase with respect to factor a from the levels of the leading mode (i.e., k = 2) in the ZM experiment. This is partially due to the dynamic field of the winter basic state associated with the western branch of the local Walker circulation, as observed in the ZM+aGLBD experiments (the red line in Figs. 10a,b), but the thermodynamic fields are twice as efficient and further increase these levels in ZM+aGLB. Note that unstable stationary modes emerge at some locations in the ZM+aGLBT experiments, so the propagating mode cannot be properly analyzed by only focusing on the leading mode (not shown). The growth rate and frequency in ZM+GLB are greater than those in ZM+EH, indicating that the zonal inhomogeneity in the warm-pool basic state can locally enhance the ISV activity. The locality of the variability in the equatorial precipitation and 850-hPa zonal wind anomalies is sharply enhanced over the Indo-Pacific warm-pool region when the inhomogeneity parameter a exceeds 0.6 (Figs. 10c,d), and then, the leading mode becomes destabilized (Fig. 10a). The local favoring and trapping of the ISV mode by the dynamic and thermodynamic structures of the warm-pool basic state suggest that accurate simulation of the Indo-Pacific basic-state structures in climate models is an important condition for simulating the associated ISV variability. Our result is consistent with the notion from a recent analysis regarding the importance of the mean-state flow in the climate’s ability to simulate the ISV (Nakano and Kikuchi 2019).
(a) Growth rate and (b) frequency of the leading modes in the ZM+aGLB (black) and ZM+aGLBD (red) experiments. The results of the ZM+aGLBT experiments are not shown because the stationary modes are dominant. The numbered blue and red circles indicate the ZM and ZM+EH experiments, respectively. The equatorial (20°S–10°N) averages of the standard deviation of (c) the rescaled precipitation and (d) 850-hPa zonal wind anomalies in the ZM+aGLB experiments and the ZM experiment with k = 2.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
c. Sensitivity tests
We further clarify the essence of the propagating mode in this model based on sensitivity tests of the model parameters by adopting the ZM+GLB experiment as the Ref [refer to section 2b(2)]. Figure 11 shows a summary of the results of the sensitivity tests of the parameters that control several damping and diffusive terms (gray), surface fluxes (green), tropospheric heat sources and moisture sinks (orange), and radiative heating (purple). Overall, the growth rate is affected by these parameters, while the frequency is less sensitive to the parameters except for the moisture adjustment time scale.
Dependence of the growth rate and frequency on the model parameters. (from left to right) The red bars indicate Ref. The gray bars indicate the experiment without PBL damping (noBLdamp) and those with halved land precipitation (landPr0.5), land damping (landdamp), both landPr0.5 and landdamp (land), the CTL configuration (CTL, see section 3), free-atmospheric moisture damping (qdamp), and horizontal diffusion time scales of 1 and 2 h (τhdiff = 1.0 h and τhdiff = 2.0 h). The light green bars indicate noSFLX. The orange bars indicate noCONV; noLSC; the experiment with free-atmospheric convective moisture adjustment time scales of 10, 18, and 240 h (τc = 10 h, τc = 18 h, and τc = 240 h); the experiment with PBL convective moisture adjustment time scales of 6, 13, and 240 h (τb = 6 h, τb = 13 h, and τb = 240 h); the experiment without the LSC processes but with τc = 240 h (noLSCτc=240h); the experiment without the LSC processes but with τb = 240 h (noLSCτb=240h); and noCeff. The purple bars indicate the experiment with half and zero radiative heating amplitude (r = 0.06 and r = 0.00), SinRad, and the experiment with the smaller zonal scale in R (Lrx = 300 km).
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
In the experiment without increased PBL damping (noBLdamp), the growth rate increases, but the frequency remains unchanged, suggesting that the propagation mechanism is not associated with the dynamic coupling of Rossby and Kelvin waves through PBL friction. In the CTL experiment (section 3), the growth rate is reduced mainly due to the increased PBL damping over land, as in the landdamp experiment, while it is slightly affected by the reduced land precipitation, as in the landPr0.5 experiment. The difference between the CTL and land experiments resulting from the topography is minor. Applying nonzero linear damping to the free-atmospheric moisture by setting αq = αN (qdamp) could reduce the growth rate, but the propagation feature remains. Changing the horizontal diffusion is effective in altering the growth rate and modulating scale selection in the ZM+EH experiment (τhdiff = 1.0 h and τhdiff = 2.0 h). Removing the surface fluxes results in a higher frequency (35-day period) but an unchanged growth rate (noSFLX).
The frequency is the most affected by the convective scheme. While removing the LSC does not affect the results (noLSC), the propagation period is greatly changed from 44 to 194 days when convective heating is removed (noCONV). This slowdown is primarily related to the moisture adjustment time scale of the free atmosphere (τc = 10 h, τc = 18 h, and τc = 240 h; see also noLSCτc=240h). The period ranges from 39 to 107 days for the time scale τc ranging from 10 to 240 h. In contrast, the period is still 68 days even when the time scale τb = 240 h, indicating that the impact of the convective heating associated with PBL moisture on the propagation mechanism is secondary (τb = 6 h, τb = 13 h, and τb = 240 h; see also noLSCτb=240h). Tuning these adjustment time scales based on theory (e.g., Ahmed et al. 2020) is interesting to simulate the propagating mode more realistically. Note that the result is insensitive to the convective efficiency (noCeff), which reduces precipitation under a dry basic state.
When radiative heating is removed or reduced (r = 0.06 and r = 0.00), only the growth rate is affected. Additionally, the result is hardly influenced by its vertical profile (SinRad), possibly because the radiative heating is much lower than the total diabatic heating. The growth rate increases when the zonal scale of the spreading effect R is reduced (Lrx = 300 km) because the relative amplitude of the radiative heating to the convective heat source is increased. It is important that the frequency is not affected by any parameters regarding the radiative heating parameterization.
In summary, the moisture adjustment process in the free atmosphere is the dominant factor controlling the propagation features of the ISV in this model.
5. Impact of El Niño on the winter ISV
This linear model can be used to study the one-way contribution of the ENSO to the ISV. The El Niño (EN) composite anomalies in DJF are added onto the winter basic state in the CTL+dEN experiments, where factor d controls the amplitude of the EN anomalies. Furthermore, we examine the influences of the dynamic and thermodynamic EN anomalies on the winter ISV by adding only the dynamic (D) or thermodynamic (T) field in CTL+dEND and CTL+dENT, respectively.
The ENSO basic states affected the oscillation in the winter ISV. Both the growth rate and frequency rapidly increased for d > 0 but moderately decreased for d < 0 (Fig. 12). For instance, the winter ISV becomes unstable with a period of 37 days when d = 1. The growth rate and frequency also increased monotonically with d even in CTL+dEND and CTL+dENT, but the rapid increase for d > 0 in CTL+dEN was only achieved when the thermodynamic EN (ENT) anomaly was considered. This suggests that the increased equatorial moisture near the date line during El Niño (Fig. 1) efficiently supports the eastward propagation of the winter ISV over the central Pacific, as suggested in observational studies (Wei and Ren 2019). The wind change associated with the ENSO fulfills a secondary role in modulating the oscillation property.
Dependence of the growth rate and frequency on the background ENSO state during the boreal winter. The gray bars indicate the results of the CTL+dEN experiments, while the triangles and circles indicate those of the CTL+dEND and CTL+dENT experiments, respectively.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
The favorable location of the winter ISV is also modulated by the ENSO state. Figure 13 shows that the standard deviations of the precipitation and zonal wind at 850-hPa shift to the southeast near the date line for d > 0 and are localized near the equator to the west of the date line for d < 0. This indicates that the basic state during El Niño winters enhances the MJO amplitude near the date line on the southern side of the equator, whereas the La Niña basic state suppresses the MJO activity in the central Pacific and modulates the structure, namely, the symmetry increases toward the equator over the Maritime Continent. A more appropriate study should also consider El Niño and La Niña asymmetry aspects. Interestingly, both the dynamic (D) and thermodynamic (T) fields of the ENSO state separately contribute to the pattern modulation effect. For d > 0 in CTL+dEND, the low-level westerly anomaly of El Niño near the date line may maintain eastward propagation through zonal advection of the mean wind. In contrast, the equatorial Indian Ocean and Maritime Continent are more favorable at d < 0 because the basic-state westerly wind is enhanced locally. The zonal shift in ISV activity in CTL+dENT can be attributed to the equatorial moisture anomaly of El Niño over the central Pacific, which enhances the meridional and vertical moisture advection processes for d > 0. Therefore, the low-level wind and moisture anomalies associated with the ENSO may jointly modulate the ISV location and time scale and potential intensity.
Dependence of the standard deviation of the precipitation and zonal wind at 850 hPa on the background ENSO state in the CTL+dEN experiments. (top) The horizontal structures of the precipitation (shade) and zonal wind (contour) as in (top) panel of Fig. 2. (bottom) The equatorial (20°S–10°N) average of the standard deviation for the rescaled anomalies of (left) the precipitation and (right) 850-hPa zonal wind in the CTL+dEN, CTL+dEND, and CTL+dENT experiments.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
6. Seasonality of the ISV
The seasonal cycle is the dominant controlling factor of tropical ISV in observations (e.g., Zhang 2005). To investigate the seasonal dependence of the oscillation modes simulated by the mLBM, the time integration experiments for the winter (DJF) basic state described in sections 3 and 4 are repeated under four basic states: the boreal spring (MAM), summer (JJA), autumn (SON), and ANN mean. For example, the summer basic state is characterized by strong monsoonal west-northerly flow in the northern Indian Ocean that converges over East Asia, and the moist warm area expands to the north (Fig. 1b). In contrast, the annual-mean basic state is more symmetric toward the equator than that in winter and summer (Fig. 1c). Since the modeled wintertime ISV mode was sensitive to the low-level wind and thermodynamic fields of the basic state, as described in section 4, the background seasonality could likely modulate the internal mode as well.
The seasonal differences in the growth rate and frequency in the ZM+EH, ZM+GLB, and CTL experiments are shown in Fig. 14. In ZM+EH, the propagating mode attains the highest growth rate at k = {2, 3} regardless of the season, and the winter is the most favorable for ISV growth. The frequency also depends on the season. The frequency is the highest in DJF and tends to be lower during the other seasons. The growth rate in CTL is lower than that in ZM+GLB overall mainly due to land friction (section 4c), but the frequencies are similar during any season. The seasonal dependence in CTL and ZM+GLB is overall similar to that in ZM+EH. An exception is that the realistic basic states of winter and summer can excite oscillating modes at a higher frequency than that in ZM+EH (see also Fig. 10), indicating that the local basic-state pattern is critical for ISV modulation.
Dependence of the growth rate and frequency of the leading modes on the background seasonality. The basic state is changed to the annual, MAM, JJA, SON, and DJF mean states in the CTL (open triangles), ZM+GLB (closed triangles), and ZM+EH (numbered gray circles) experiments. In the ZM+EH experiments, the results for k = {2, 3, 4} are shown.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
The horizontal pattern of the leading modes depends on the basic-state season. As shown in Fig. 2, the variability of the precipitation and zonal wind appears over the Indo-Pacific warm-pool region but mainly on the southern side of the equator in DJF. When the annual-mean basic state is applied (Fig. 15), the variability is suppressed between 120°E and 180°, and the wind amplitude is more symmetric toward the equator. Such an equatorial symmetric structure is also clearly observed in the propagating wind fields (Fig. 16). The result is similar to that of the annual-mean experiment during the spring and autumn seasons. In contrast, the summer (JJA) basic state generates an oscillating mode on the northern side of the equator over the Maritime Continent and northwestern Pacific (Fig. 15). The precipitation anomaly propagates toward the northern east over the western North Pacific and is accompanied by wind anomalies (Fig. 16). There is also a clear signal near the eastern edge of the ITCZ. The higher variability in the northern tropics in JJA can be attributed to the reduced easterly wind in the basic state.
As in (top) of Fig. 2, but for the basic-state season. (from top to bottom) The MAM, JJA, SON, DJF, and annual mean states are used.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
As in (left) of Fig. 3, but for the JJA and annual mean basic states.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
This leading mode of the JJA basic state resembles the boreal summer intraseasonal oscillation (BSISO) to some extent compared to the observations (e.g., Kikuchi 2021). However, its oscillation frequency is yet too slow, and the northward propagation over the northern Indian Ocean is not smooth. These biases may result in a poor representation of the northwest–southeast tilted precipitation anomalies, which consist of mesoscale convective systems and might be a by-product of the eastward-propagating branch (Kikuchi 2021). Thus, the mLBM is yet premature to reproduce the observed BSISO-like pattern, as in many climate models (Sperber et al. 2013). Summertime ISV simulation can be improved potentially by implementing a land–ocean model to consider the air–sea–land interaction and by improving the convective heating and cloud-radiative heating schemes, which are beyond the scope of this study. The contrast between the results for the winter and summer basic states is basically consistent with a recent study that also obtained both MJO- and BSISO-like linear modes in a moist shallow water system on an equatorial beta plane by varying the meridional gradient of background moisture that is meridionally symmetric (Wang and Sobel 2022a). Because the basic-state structure is critical for simulating the tropical ISV, it will be important to consider the meridionally asymmetric basic states, especially in the boreal winter and summer, even in a theoretical framework.
7. Conclusions and discussion
In this study, we examined the explicit role of the background state of the Indo-Pacific warm pool in determining the tropical ISV mode within a linearized atmospheric GCM framework, i.e., mLBM. The model originally developed by Watanabe and Jin (2003) was improved to simulate the internal moisture variability in the tropics. We used a convective adjustment scheme that separately relates precipitation to free-tropospheric and boundary layer moisture anomalies by following the basic idea of Zhang (2002) and replaced an overly damped restoring-type scheme resulting from the improper use of the linearized Betts–Miller convective parameterization in the original model of Watanabe and Jin (2003). A simple cloud-radiative heating scheme was adopted from the study of Adames and Kim (2016), and a linearized bulk formula for the sensible and latent heat fluxes at the surface was implemented, as adopted in Watanabe and Jin (2003). The control experiment under the boreal winter basic state derived from the ERA5 dataset resulted in a local ISV mode propagating eastward with a phase speed of 3–5 m s−1 over the Indo-Pacific warm-pool region. This mode resembles the observed MJO well in terms of the spatial pattern and zonal wavenumber–frequency spectrum. The modeled ISV was significantly modulated by replacing the winter basic state with the El Niño winter and different seasonal basic states, capturing the fundamental features of the observed ISV diversity. The leading mode of the summer basic state propagates eastward along the equator and northward to the western North Pacific, which captures aspects of the BSISO, yet its frequency and structure are biased. Nevertheless, the series of experiments under various basic states derived from a reanalysis dataset indicate that realistic interannual and seasonal backgrounds can turn a moisture low-frequency wave mode into a versatile ISV mode.
The characteristics of the MJO-like winter ISV were further explored by conducting a series of experiments from the idealized to realistic situations, which is summarized in Fig. 17. In the zonally uniform experiments, the eastward-propagating ISV at the zonal wavenumber k = {2, 3} can dominate under the EH condition due to not only the more humid tropics but also the reduced lower-tropospheric easterly winds. Large-scale selection appears to result in part from horizontal diffusion, but the cloud-radiative feedback and the reduced easterly basic state are favorable for enhancing the growth rate at k = {1, 2, 3}. Additional experiments involving adjusting the zonal extent of the EH winter condition revealed that both EH basic states of the winds and thermodynamic fields extending over at least 150° longitude are necessary to maintain the ISV mode with a zonal wavelength of k = 3. The expansion/contraction of this warm-pool condition may enhance/reduce its instability and expand/reduce its domain of activity. We further demonstrate that the reason MJO is locally contained in the Indo-Pacific warm region is due to the local thermal/moisture condition and the local Walker circulation that greatly enhance its instability, but outside this region, this mode is heavily damped.
Schematic figure of the leading tropical modes under various winter basic states. The x and y axes represent the dominant zonal wavenumber (k) and frequency (cycle per day). The solid red and dashed blue arrows indicate the enhanced and declined growth rate, respectively, due to the background changes (see the legend) and the implementation of cloud-radiative feedback and horizontal diffusion. The light green circle represents the leading modes that are locally determined by the nonuniform background of the Indo-Pacific warm pool. The dark green dashed circle shows the leading modes over the central Pacific Ocean during EN.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
During El Niño, the ISV becomes dominant in the central Pacific as the warm and humid conditions and the reduced low-level easterly winds expand to the east of the western Pacific warm-pool edge. Under geographically realistic background conditions, the location and propagation speed of the ISV are locally determined and thus sensitive to the ENSO phase and seasonality and possibly to the changing climate and GCM biases, the impacts of which will be examined in the future.
The sensitivity of the results to the model parameters under the identical basic state indicates the essential processes for ISV simulation. The growth rate can be affected by many factors, but the frequency is less sensitive to the parameters. For instance, the frequency is not affected by boundary layer damping, LSC, or cloud-radiative heating. The surface fluxes and land friction tend to slightly reduce the eastward propagation speed, but this is a minor effect. In contrast, the convective adjustment time scale is effective in modulating the frequency, which becomes lower than 0.01 cpd (100-day period) when the relationship between convection and free-tropospheric moisture is diminished. Therefore, the convective process associated with the free-tropospheric moisture plays a key role in eastward propagation mainly due to meridional advection of the mean-state moisture, as evidenced by the moisture budget analysis results. This supports the results of a recent observational study (Kang et al. 2021) that showed the role of the background meridional moisture gradient across the Maritime Continent region in MJO propagation.
Previous studies have suggested that other processes not considered in the mLBM may modulate the tropical ISV. Coupling with the ocean tends to improve the MJO simulation performance of GCMs (Klingaman and Woolnough 2014), but a recent study indicated that this occurs because the background moisture is enhanced near the equator, resulting in a sharper meridional moisture gradient to drive MJO propagation (DeMott et al. 2019). A strong diurnal cycle over Maritime Continent islands is a factor that interrupts MJO propagation (Zhang and Ling 2017). This may occur because the diurnal cycle shapes the lower-tropospheric mean moisture pattern (Jiang et al. 2019). Therefore, these processes are potentially included in part through the background condition and thus can be assessed within the presented framework. The upscale feedback from the embedded synoptic eddies in the ISV onto the ISV wind anomalies is not considered in the mLBM but is often included in theoretical models to enhance the propagation speed (Majda and Stechmann 2009; Sobel and Maloney 2013; Adames and Kim 2016). Although in observations, low-level MJO westerlies favor stronger synoptic-scale eddies, while the MJO easterlies suppress them (Kiranmayi and Maloney 2011), it remains unclear how notable the upscale feedback from the high-frequency eddies embedded in the ISV can be. A recent observational study used a sophisticated filtering method, indicating that the horizontal eddy moisture transports can facilitate the eastward propagation of the MJO, particularly from the Maritime Continent to the western Pacific (Tulich 2023). This may be incorporated into our model and evaluated when a properly validated parameterization for this kind of multiscale dynamic interaction becomes available.
The model developed in this study is applicable to research on other climate variabilities, but several improvements are desirable. A linearized multiscale cloud parameterization including shallow clouds may improve our model’s depiction of the life cycle of the MJO and the steady response to El Niño SST forcing. It may also be critical to implement a more sophisticated cloud–radiation scheme (e.g., Chou and Neelin 1996) and to couple our model with a simple but consistent interactive ocean–land model to better study the ISV mode. Diffusive damping on synoptic eddies with weak upscale feedback from storm tracks on a planetary scale as formulated in Jin et al. (2006) may replace the artificial strong free-tropospheric damping at midlatitudes. This may allow the model to simulate the teleconnection from the tropics. Once this type of intermediate complexity model is improved considering these aspects, it will be a useful tool to better understand not only the ISV but also the tropical ocean–atmosphere variability.
Nevertheless, the inadequacy of our current version of the mLBM is unlikely to represent a major hindrance to altering the main conclusions inferred from the consistency within the large body of work completed considering various aspects. The lower-tropospheric winds and thermodynamic fields under the background state over the Indo-Pacific warm pool, together with the convective–radiative moisture mode dynamics, favor a versatile tropical ISV mode, which is subject to notable modulation by changes in the basic state at seasonal and interannual time scales. This may serve as a useful framework for understanding the tropical ISV variability.
Acknowledgments.
The authors thank Prof. M. Watanabe for providing the original LBM package. Advice given by Prof. G. J. Zhang was a great help in formulating the convective scheme. We are grateful for the review provided by Dr. F. Ahmed and two anonymous reviewers. M. Hayashi was supported by JSPS KAKENHI Grant 21K13993 and the Advanced Studies of Climate Change Projection (SENTAN, JPMXD0722680395) of the Ministry of Education, Culture, Sports, Science and Technology of Japan. F.-F. Jin was supported by U.S. National Science Foundation Grant AGS-2219257 and NOAA’s Climate Program Office’s Modeling, Analysis, Predictions, and Projections (MAPP) Program Grant NA23OAR2007440. A part of this study was conducted when M. Hayashi was supported by JSPS Overseas Research Fellowships (201860671).
Data availability statement.
The original LBM package is available upon reasonable request to M. Hayashi from the LBM website: https://ccsr.aori.u-tokyo.ac.jp/∼lbm/sub/lbm.html. The ERA5 dataset is available on the Copernicus Climate Change Service (C3S) Climate Data Store: https://cds.climate.copernicus.eu/. The ONI data are available at https://origin.cpc.ncep.noaa.gov/products/analysis_monitoring/ensostuff/ONI_v5.php.
APPENDIX
Model Description
a. Parameterizations
Variables and definitions of the model parameters.
1) Convective heat source and moisture sink terms:
2) Large-scale condensation:
3) Radiative heating:
An example of the vertical profiles at 120°E along the equator. For the boreal winter basic state, (a) F, (b) A, and (c) B. (d) The vertical profile of the radiative heating Ar with the default parameters (v1, v2) = (−2.5, 1.0).
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
4) Surface sensible and latent fluxes:
b. Example of a steady response
With the use of the mLBM, we calculated the steady response to an El Niño SST anomaly to reveal the differences from the original model of Watanabe and Jin (2003). The model configuration is the same as that in the CTL experiments, but a steady SST forcing is imposed instead of the initial moisture perturbation. With the use of ERSST.v5 (Huang et al. 2017), we derived a positive-only SST anomaly in the eastern equatorial Pacific by averaging nine El Niño winters from 1979 to 2016 (section 2).
Figures A2 and A3 show the steady responses to the SST forcing under the boreal winter basic state in the original mLBM of Watanabe and Jin (2003) and the new mLBM version in the present study, respectively. The results are averaged over days 51–200 in the time integration process. In the original mLBM (Fig. A2), the precipitation and moisture responses are confined within the SST forcing region (red dotted line). Since atmospheric diabatic heating is proportional to the anomalous moist static energy in the PBL, the model generates anomalous ascending motion and associated circulation anomalies near the SST forcing area only, and remote responses to the forcing are damped overall. Importantly, the descending motion anomaly in the western equatorial Pacific is not produced, as there is no local negative SST forcing (Watanabe and Jin 2003). In the new mLBM (Fig. A3), the positive-only SST forcing induces not only positive precipitation and moisture anomalies over the forcing region but also negative precipitation and moisture anomalies over the western equatorial Pacific outside of the forcing region. This negative moisture anomaly results in diabatic cooling and thus the descending anomaly, accompanied by the northwestern subtropical anticyclonic circulation anomaly. The strong cyclonic anomaly on the southern side of the equator near the date line, which is related to the southward shift in westerly anomalies favorable for terminating El Niño events, is also prominent in the new mLBM. These nonlocal responses are closely related to the seasonal background and thus critical for reproducing the interaction between the ENSO and the annual cycle, referred to as the combination mode (Stuecker et al. 2013).
Steady response to EN SST forcing under the winter basic state in the former model version of Watanabe and Jin (2003). (top) The surface winds (vectors), precipitation (shading), velocity potential (shading), and surface streamfunction (contours, 0.5 × 106 m2 s−1). The red dotted line indicates the SST forcing region, and the red contours indicate the SST forcing (0.5, 1.0, and 1.5 K). (bottom) The vertical structures of the specific humidity (shade), pressure velocity (contours, Pa s−1), total moisture sinks (shade), and heat sources (contours, 0.5 K day−1) averaged between 20°S and 10°N.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
As in Fig. A2, but for the new mLBM developed in this study. Note that the shading ranges are different from those of Fig. A2, and the contour interval of the streamfunction is 1 × 106 (m2 s−1).
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
A deficiency in the new mLBM emerges in the eastern Pacific, where the moisture anomaly is less prominent. This may be attributed to the single-mode assumption of the vertical motion profile in the heating parameterization that produces less heating at lower levels, resulting in weak vertical moisture advection. Implementing shallow convection may improve the response over cold SST regions such as the eastern Pacific cold tongue.
c. Initial conditions
In the mLBM experiments, the time integration is conducted with the initial moisture perturbation. Figures A4a and A4b show the initial moisture perturbation for the zonally nonuniform basic states, including the CTL experiment. The moisture anomalies are localized over the equatorial Indian Ocean and have higher amplitude in the lower troposphere. Because of the complicated horizontal pattern of the basic state, the initial perturbation immediately spreads to various wavenumbers, and thus, the emergence of the leading mode is not affected by the initial condition (not shown). In the zonally uniform basic states, the zonal wavenumber of the leading mode is determined by the initial condition. Figure A4 also shows the zonal wavenumber 1, 2, 3, 4, and 5 cases, for instance. The vertical profiles are the same as in the zonally nonuniform basic states, but the zonal patterns are provided as the sinusoidal function with each of the wavenumbers.
Examples of the initial moisture perturbations used for the time integration experiments of the mLBM. (a),(b) The horizontal and vertical structures of the localized perturbation over the Indian Ocean used in the zonally nonuniform basic-state cases. (c),(d) The horizontal and vertical structures of the zonal wavenumber k = 1 perturbation used in the zonally uniform basic-state cases. (e)–(h) As in (c), but for k = 2, 3, 4, and 5. The horizontal structure is at σ = 0.8, and the vertical structure is averaged for 5°S–5°N.
Citation: Journal of Climate 38, 1; 10.1175/JCLI-D-24-0141.1
REFERENCES
Adames, A. F., and D. Kim, 2016: The MJO as a dispersive, convectively coupled moisture wave: Theory and observations. J. Atmos. Sci., 73, 913–941, https://doi.org/10.1175/JAS-D-15-0170.1.
Adames, A. F., D. Kim, S. K. Clark, Y. Ming, and K. Inoue, 2019: Scale analysis of moist thermodynamics in a simple model and the relationship between moisture modes and gravity waves. J. Atmos. Sci., 76, 3863–3881, https://doi.org/10.1175/JAS-D-19-0121.1.
Ahmed, F., 2021: The MJO on the equatorial beta plane: An eastward-propagating Rossby wave induced by meridional moisture advection. J. Atmos. Sci., 78, 3115–3135, https://doi.org/10.1175/JAS-D-21-0071.1.
Ahmed, F., A. F. Adames, and J. D. Neelin, 2020: Deep convective adjustment of temperature and moisture. J. Atmos. Sci., 77, 2163–2186, https://doi.org/10.1175/JAS-D-19-0227.1.
Ahn, M.-S., D. Kim, K. R. Sperber, I.-S. Kang, E. Maloney, D. Waliser, and H. Hendon, 2017: MJO simulation in CMIP5 climate models: MJO skill metrics and process-oriented diagnosis. Climate Dyn., 49, 4023–4045, https://doi.org/10.1007/s00382-017-3558-4.
Ahn, M.-S., and Coauthors, 2020: MJO propagation across the Maritime Continent: Are CMIP6 models better than CMIP5 models? Geophys. Res. Lett., 47, e2020GL087250, https://doi.org/10.1029/2020GL087250.
Bellenger, H., and J. P. Duvel, 2012: The event-to-event variability of the boreal winter MJO. Geophys. Res. Lett., 39, L08701, https://doi.org/10.1029/2012GL051294.
Betts, A. K., and M. J. Miller, 1986: A new convective adjustment scheme. Part II: Single column tests using GATE wave, BOMEX, ATEX and Arctic air-mass data sets. Quart. J. Roy. Meteor. Soc., 112, 693–709, https://doi.org/10.1002/qj.49711247308.
Betts, A. K., and M. J. Miller, 1993: The Betts-Miller scheme. The Representation of Cumulus Convection in Numerical Models, K. A. Emanuel and D. J. Raymond, Eds., Amer. Meteor. Soc., 107–121.
Beucler, T., and T. W. Cronin, 2016: Moisture-radiative cooling instability. J. Adv. Model. Earth Syst., 8, 1620–1640, https://doi.org/10.1002/2016MS000763.
Bretherton, C. S., M. E. Peters, and L. E. Back, 2004: Relationships between water vapor path and precipitation over the tropical oceans. J. Climate, 17, 1517–1528, https://doi.org/10.1175/1520-0442(2004)017<1517:RBWVPA>2.0.CO;2.
Chen, G., and B. Wang, 2019: Dynamic moisture mode versus moisture mode in MJO dynamics: Importance of the wave feedback and boundary layer convergence feedback. Climate Dyn., 52, 5127–5143, https://doi.org/10.1007/s00382-018-4433-7.
Chikira, M., 2014: Eastward-propagating intraseasonal oscillation represented by Chikira–Sugiyama cumulus parameterization. Part II: Understanding moisture variation under weak temperature gradient balance. J. Atmos. Sci., 71, 615–639, https://doi.org/10.1175/JAS-D-13-038.1.
Chou, C., and J. D. Neelin, 1996: Linearization of a longwave radiation scheme for intermediate tropical atmospheric models. J. Geophys. Res., 101, 15 129–15 145, https://doi.org/10.1029/96JD01015.
Copernicus Climate Change Service, 2017: ERA5: Fifth generation of ECMWF atmospheric reanalyses of the global climate. Copernicus Climate Change Service Climate Data Store (CDS), accessed 26 April 2019, https://cds.climate.copernicus.eu/cdsapp#!/home.
DeMott, C. A., N. P. Klingaman, W.-L. Tseng, M. A. Burt, Y. Gao, and D. A. Randall, 2019: The convection connection: How ocean feedbacks affect tropical mean moisture and MJO propagation. J. Geophys. Res. Atmos., 124, 11 910–11 931, https://doi.org/10.1029/2019JD031015.
Emanuel, K. A., 1987: An air-sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci., 44, 2324–2340, https://doi.org/10.1175/1520-0469(1987)044<2324:AASIMO>2.0.CO;2.
Emanuel, K. A., J. D. Neelin, and C. S. Bretherton, 1994: On large-scale circulations in convecting atmospheres. Quart. J. Roy. Meteor. Soc., 120, 1111–1143, https://doi.org/10.1002/qj.49712051902.
Hayashi, M., and H. Itoh, 2017: A new mechanism of the slow eastward propagation of unstable disturbances with convection in the tropics: Implications for the MJO. J. Atmos. Sci., 74, 3749–3769, https://doi.org/10.1175/JAS-D-16-0300.1.
Hersbach, H., and Coauthors, 2023a: ERA5 monthly averaged data on single levels from 1940 to present. Copernicus Climate Change Service (C3S) Climate Data Store (CDS), accessed 26 April 2019, https://doi.org/10.24381/cds.f17050d7.
Hersbach, H., and Coauthors, 2023b: ERA5 monthly averaged data on pressure levels from 1940 to present. Copernicus Climate Change Service (C3S) Climate Data Store (CDS), accessed 26 April 2019, https://doi.org/10.24381/cds.6860a573.
Houze, R. A., Jr., 2004: Mesoscale convective systems. Rev. Geophys., 42, RG4003, https://doi.org/10.1029/2004RG000150.
Huang, B., and Coauthors, 2017: Extended Reconstructed Sea Surface Temperature, version 5 (ERSSTv5): Upgrades, validations, and intercomparisons. J. Climate, 30, 8179–8205, https://doi.org/10.1175/JCLI-D-16-0836.1.
Jiang, X., and Coauthors, 2011: Vertical diabatic heating structure of the MJO: Intercomparison between recent reanalyses and TRMM estimates. Mon. Wea. Rev., 139, 3208–3223, https://doi.org/10.1175/2011MWR3636.1.
Jiang, X., and Coauthors, 2015: Vertical structure and physical processes of the Madden-Julian oscillation: Exploring key model physics in climate simulations. J. Geophys. Res. Atmos., 120, 4718–4748, https://doi.org/10.1002/2014JD022375.
Jiang, X., Á. F. Adames, M. Zhao, D. Waliser, and E. Maloney, 2018: A unified moisture mode framework for seasonality of the Madden–Julian oscillation. J. Climate, 31, 4215–4224, https://doi.org/10.1175/JCLI-D-17-0671.1.
Jiang, X., H. Su, and D. E. Waliser, 2019: A damping effect of the Maritime Continent for the Madden-Julian oscillation. J. Geophys. Res. Atmos., 124, 13 693–13 713, https://doi.org/10.1029/2019JD031503.
Jiang, X., and Coauthors, 2020: Fifty years of research on the Madden-Julian oscillation: Recent progress, challenges, and perspectives. J. Geophys. Res. Atmos., 125, e2019JD030911, https://doi.org/10.1029/2019JD030911.
Jin, F.-F., L.-L. Pan, and M. Watanabe, 2006: Dynamics of synoptic eddy and low-frequency flow interaction. Part I: A linear closure. J. Atmos. Sci., 63, 1677–1694, https://doi.org/10.1175/JAS3715.1.
Kang, D., D. Kim, M.-S. Ahn, and S.-I. An, 2021: The role of the background meridional moisture gradient on the propagation of the MJO over the Maritime Continent. J. Climate, 34, 6565–6581, https://doi.org/10.1175/JCLI-D-20-0085.1.
Kikuchi, K., 2021: The Boreal Summer Intraseasonal Oscillation (BSISO): A review. J. Meteor. Soc. Japan, 99, 933–972, https://doi.org/10.2151/jmsj.2021-045.
Kiranmayi, L., and E. D. Maloney, 2011: Intraseasonal moist static energy budget in reanalysis data. J. Geophys. Res., 116, D21117, https://doi.org/10.1029/2011JD016031.
Klingaman, N. P., and S. J. Woolnough, 2014: The role of air–sea coupling in the simulation of the Madden–Julian oscillation in the Hadley Centre model. Quart. J. Roy. Meteor. Soc., 140, 2272–2286, https://doi.org/10.1002/qj.2295.
Li, W., C. Schumacher, and S. A. McFarlane, 2013: Radiative heating of the ISCCP upper level cloud regimes and its impact on the large-scale tropical circulation. J. Geophys. Res. Atmos., 118, 592–604, https://doi.org/10.1002/jgrd.50114.
Ling, J., C. Zhang, S. Wang, and C. Li, 2017: A new interpretation of the ability of global models to simulate the MJO. Geophys. Res. Lett., 44, 5798–5806, https://doi.org/10.1002/2017GL073891.
Louis, J.-F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17, 187–202, https://doi.org/10.1007/BF00117978.
Ma, D., and Z. Kuang, 2011: Modulation of radiative heating by the Madden-Julian Oscillation and convectively coupled Kelvin waves as observed by CloudSat. Geophys. Res. Lett., 38, L21813, https://doi.org/10.1029/2011GL049734.
Madden, R. A., and P. R. Julian, 1971: Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci., 28, 702–708, https://doi.org/10.1175/1520-0469(1971)028<0702:DOADOI>2.0.CO;2.
Madden, R. A., and P. R. Julian, 1972: Description of global-scale circulation cells in the tropics with a 40–50 day period. J. Atmos. Sci., 29, 1109–1123, https://doi.org/10.1175/1520-0469(1972)029<1109:DOGSCC>2.0.CO;2.
Majda, A. J., and S. N. Stechmann, 2009: The skeleton of tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA, 106, 8417–8422, https://doi.org/10.1073/pnas.0903367106.
Mayta, V. C., and Á. F. Adames Corraliza, 2023: Is the Madden-Julian oscillation a moisture mode? Geophys. Res. Lett., 50, e2023GL103002, https://doi.org/10.1029/2023GL103002.
Mayta, V. C., Á. F. Adames, and F. Ahmed, 2022: Westward-propagating moisture mode over the tropical Western Hemisphere. Geophys. Res. Lett., 49, e2022GL097799, https://doi.org/10.1029/2022GL097799.
Miller, M. J., A. C. M. Beljaars, and T. N. Palmer, 1992: The sensitivity of the ECMWF model to the parameterization of evaporation from the tropical oceans. J. Climate, 5, 418–434, https://doi.org/10.1175/1520-0442(1992)005<0418:TSOTEM>2.0.CO;2.
Nakano, M., and K. Kikuchi, 2019: Seasonality of intraseasonal variability in global climate models. Geophys. Res. Lett., 46, 4441–4449, https://doi.org/10.1029/2019GL082443.
Neelin, J. D., and J.-Y. Yu, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part I: Analytical theory. J. Atmos. Sci., 51, 1876–1894, https://doi.org/10.1175/1520-0469(1994)051<1876:MOTVUC>2.0.CO;2.
Pohl, B., and A. J. Matthews, 2007: Observed changes in the lifetime and amplitude of the Madden–Julian oscillation associated with interannual ENSO sea surface temperature anomalies. J. Climate, 20, 2659–2674, https://doi.org/10.1175/JCLI4230.1.
Raymond, D. J., 2001: A new model of the Madden–Julian oscillation. J. Atmos. Sci., 58, 2807–2819, https://doi.org/10.1175/1520-0469(2001)058<2807:ANMOTM>2.0.CO;2.
Reichler, T., M. Dameris, and R. Sausen, 2003: Determining the tropopause height from gridded data. Geophys. Res. Lett., 30, 2042, https://doi.org/10.1029/2003GL018240.
Roxy, M. K., P. Dasgupta, M. J. McPhaden, T. Suematsu, C. Zhang, and D. Kim, 2019: Twofold expansion of the Indo-Pacific warm pool warps the MJO life cycle. Nature, 575, 647–651, https://doi.org/10.1038/s41586-019-1764-4.
Rushley, S. S., D. Kim, C. S. Bretherton, and M.-S. Ahn, 2018: Reexamining the nonlinear moisture-precipitation relationship over the tropical oceans. Geophys. Res. Lett., 45, 1133–1140, https://doi.org/10.1002/2017GL076296.
Sobel, A., and E. Maloney, 2012: An idealized semi-empirical framework for modeling the Madden–Julian oscillation. J. Atmos. Sci., 69, 1691–1705, https://doi.org/10.1175/JAS-D-11-0118.1.
Sobel, A., and E. Maloney, 2013: Moisture modes and the eastward propagation of the MJO. J. Atmos. Sci., 70, 187–192, https://doi.org/10.1175/JAS-D-12-0189.1.
Sobel, A. H., J. Nilsson, and L. M. Polvani, 2001: The weak temperature gradient approximation and balanced tropical moisture waves. J. Atmos. Sci., 58, 3650–3665, https://doi.org/10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2.
Sperber, K. R., H. Annamalai, I.-S. Kang, A. Kitoh, A. Moise, A. Turner, B. Wang, and T. Zhou, 2013: The Asian summer monsoon: An intercomparison of CMIP5 vs. CMIP3 simulations of the late 20th century. Climate Dyn., 41, 2711–2744, https://doi.org/10.1007/s00382-012-1607-6.
Stuecker, M. F., A. Timmermann, F.-F. Jin, S. McGregor, and H.-L. Ren, 2013: A combination mode of the annual cycle and the El Niño/Southern Oscillation. Nat. Geosci., 6, 540–544, https://doi.org/10.1038/ngeo1826.
Suematsu, T., and H. Miura, 2018: Zonal SST difference as a potential environmental factor supporting the longevity of the Madden–Julian oscillation. J. Climate, 31, 7549–7564, https://doi.org/10.1175/JCLI-D-17-0822.1.
Timmermann, A., and Coauthors, 2018: El Niño–Southern Oscillation complexity. Nature, 559, 535–545, https://doi.org/10.1038/s41586-018-0252-6.
Tulich, S. N., 2023: How important are horizontal eddy moisture transports for the MJO’s eastward propagation? Geophys. Res. Lett., 50, e2023GL103944, https://doi.org/10.1029/2023GL103944.
Waliser, D., and Coauthors, 2009: MJO simulation diagnostics. J. Climate, 22, 3006–3030, https://doi.org/10.1175/2008JCLI2731.1.
Wang, B., and H. Rui, 1990: Dynamics of the coupled moist Kelvin–Rossby wave on an equatorial β-plane. J. Atmos. Sci., 47, 397–413, https://doi.org/10.1175/1520-0469(1990)047<0397:DOTCMK>2.0.CO;2.
Wang, B., and G. Chen, 2017: A general theoretical framework for understanding essential dynamics of Madden–Julian oscillation. Climate Dyn., 49, 2309–2328, https://doi.org/10.1007/s00382-016-3448-1.
Wang, B., G. Chen, and F. Liu, 2019: Diversity of the Madden-Julian oscillation. Sci. Adv., 5, eaax0220, https://doi.org/10.1126/sciadv.aax0220.
Wang, S., and A. H. Sobel, 2022a: A unified moisture mode theory for the Madden–Julian oscillation and the boreal summer intraseasonal oscillation. J. Climate, 35, 1267–1291, https://doi.org/10.1175/JCLI-D-21-0361.1.
Wang, S., and A. H. Sobel, 2022b: A filtered model for the tropical intraseasonal moisture mode. Geophys. Res. Lett., 49, e2022GL098320, https://doi.org/10.1029/2022GL098320.
Watanabe, M., and M. Kimoto, 2000: Atmosphere-ocean thermal coupling in the North Atlantic: A positive feedback. Quart. J. Roy. Meteor. Soc., 126, 3343–3369, https://doi.org/10.1002/qj.49712657017.
Watanabe, M., and F.-F. Jin, 2003: A moist linear baroclinic model: Coupled dynamical–convective response to El Niño. J. Climate, 16, 1121–1139, https://doi.org/10.1175/1520-0442(2003)16<1121:AMLBMC>2.0.CO;2.
Wei, Y., and H.-L. Ren, 2019: Modulation of ENSO on fast and slow MJO modes during boreal winter. J. Climate, 32, 7483–7506, https://doi.org/10.1175/JCLI-D-19-0013.1.
Yang, D., and A. P. Ingersoll, 2013: Triggered convection, gravity waves, and the MJO: A shallow-water model. J. Atmos. Sci., 70, 2476–2486, https://doi.org/10.1175/JAS-D-12-0255.1.
Yu, J.-Y., and J. D. Neelin, 1997: Analytic approximations for moist convectively adjusted regions. J. Atmos. Sci., 54, 1054–1063, https://doi.org/10.1175/1520-0469(1997)054%3C1054:AAFMCA%3E2.0.CO;2.
Zhang, C., 2005: Madden-Julian oscillation. Rev. Geophys., 43, RG2003, https://doi.org/10.1029/2004RG000158.
Zhang, C., and M. Dong, 2004: Seasonality in the Madden–Julian oscillation. J. Climate, 17, 3169–3180, https://doi.org/10.1175/1520-0442(2004)017<3169:SITMO>2.0.CO;2.
Zhang, C., and J. Ling, 2017: Barrier effect of the Indo-Pacific Maritime Continent on the MJO: Perspectives from tracking MJO precipitation. J. Climate, 30, 3439–3459, https://doi.org/10.1175/JCLI-D-16-0614.1.
Zhang, G. J., 2002: Convective quasi-equilibrium in midlatitude continental environment and its effect on convective parameterization. J. Geophys. Res., 107, 4220, https://doi.org/10.1029/2001JD001005.
Zhang, G. J., and N. A. McFarlane, 1995: Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian climate centre general circulation model. Atmos.–Ocean, 33, 407–446, https://doi.org/10.1080/07055900.1995.9649539.
Zhang, G. J., and M. Mu, 2005: Simulation of the Madden–Julian oscillation in the NCAR CCM3 using a revised Zhang–McFarlane convection parameterization scheme. J. Climate, 18, 4046–4064, https://doi.org/10.1175/JCLI3508.1.
