1. Introduction
The Madden–Julian oscillation (MJO) is the largest weather disturbance on our planet, but was only discovered in 1971 by Madden and Julian (Madden and Julian 1971) based on observations of winds and pressure between Singapore and Canton Island in the west-central equatorial Pacific. [A study published in Chinese in 1963 also documented a 40–50-day oscillation in the Asian monsoon region (Xie et al. 1963; Li et al. 2018).] Today we know that even though the MJO is a tropical disturbance, it influences weather all around the globe, modulating hurricanes, atmospheric rivers, and other phenomena (Zhang 2013).
Wheeler and Kiladis (1999) and Kiladis et al. (2009) find, based on wavenumber–frequency power spectra of outgoing longwave radiation (OLR), that the strongest disturbance in OLR in the tropical atmosphere is the MJO. This disturbance propagates eastward with planetary wavenumbers l = 1, 2 with a period of about 30–90 days or, for the planetary wavenumber l = 1, with phase speeds in the range from 5 to 15 m s−1.
Theoretical modeling of large-scale disturbances in the tropics started with Matsuno (1966), who developed an analytical shallow-water model for the adiabatic atmosphere, i.e., with no convective coupling. The only modes that propagate eastward are the Kelvin mode, higher-order gravity modes, and eastward branch of mixed Rossby–gravity mode in the antisymmetric component of OLR. Researchers after Matsuno (Wheeler and Kiladis 1999; Raymond and Fuchs 2007; Kuang 2008; Kiladis et al. 2009) found that the slower Kelvin waves, when convectively coupled, have largest growth rates for wavenumbers five to seven and that they have vertical structure that differs from that of the MJO. Since the MJO has different scale and structure from the Kelvin mode, it appears not to have an equivalent in a dry atmosphere. This implies that the MJO can exist only when there is moist convection. The modeled mode, if it is the MJO, thus needs to be a convectively coupled mode that has the greatest instability for largest wavelengths and it needs to propagate eastward. This might seem simple, but five decades of research have shown that these are difficult conditions to meet.
Many theoretical models that have tried to explain the mechanisms behind the MJO are well summarized in review papers by Zhang (2005) and by Jiang et al. (2020). Jiang et al. (2020) identifies seven contemporary theories that explain the slow eastward propagation and planetary scale of the MJO. Those are the weak temperature gradient approximation (WTG) moisture mode (Sobel and Maloney 2012, 2013; Adames and Kim 2016), wind-induced surface heat exchange (WISHE) moisture mode (Fuchs and Raymond 2005, 2017; Fuchs-Stone 2020), boundary layer quasi-equilibrium (BLQE) model (Khairoutdinov and Emanuel 2018; Emanuel 2020), trio interaction (Wang and Rui 1990; Wang et al. 2016), skeleton (Majda and Stechmann 2009; Thual et al. 2014), gravity wave (Yang and Ingersoll 2013, 2014), nonlinear solitary wave (Yano and Tribbia 2017; Rostami and Zeitlin 2019), and large-scale convective vortex (Hayashi and Itoh 2017) theories. Four of those theories (skeleton, WTG–moisture mode, gravity wave, and trio-interaction theory) are discussed and compared in detail in a review paper by Zhang et al. (2020). The moisture-mode theory is further discussed in a review paper by Adames and Maloney (2021). Readers are encouraged to read those papers to gain insights into the extensive research on the MJO over the past 50 years.
In this paper we aim to find a common framework between the WISHE-moisture mode and the BLQE models by developing the simplest possible linear analytical model with minimal differences. Those theoretical models are the ones from Fuchs and Raymond (2005, 2017) and Fuchs-Stone (2020) on the one hand and Khairoutdinov and Emanuel (2018) and Emanuel (2020) on the other. The first we will call the FR model and second the KE model. The value of linear analytical models is in their ability to clearly identify the physical mechanism responsible for the instability and propagation mechanism of the MJO. The disadvantage is that the simple analytical models cannot capture nonlinearity or other complexities of the real world.
Two processes that are crucial to the WISHE–moisture mode and the BLQE theories of tropical intraseasonal variability are surface enthalpy fluxes and cloud–radiation effects. We therefore focus on those two mechanisms in comparing the models.
In formulating this paper we take advantage of the fact that there are easterly global mean zonal winds always present in the tropics (Peixoto and Oort 1992; Sentić et al. 2020). We do not view the global (wavenumber 1) MJO as a disturbance occurring only over the warm pool where there are often westerlies (Sobel and Maloney 2012, 2013; Adames and Kim 2016), but rather assume mean easterlies in formulating wind-induced perturbations to the surface enthalpy flux, a process known as WISHE. The WISHE parameterization goes back to Emanuel (1987), Neelin et al. (1987), and Yano and Emanuel (1991) and relies on the fact that mean surface easterly anomalies result in enhanced surface fluxes in the presence of mean easterly winds. Under this paradigm WISHE is responsible for the eastward propagation of the MJO. Sentić et al. (2020) showed the importance of WISHE for the reanalysis-derived climatological MJO through analysis of zonal wind, moisture, and moisture tendency anomalies from reanalyses. The WISHE assumption is challenged by studies that show the surface latent heat fluxes to the east of enhanced MJO convection to be suppressed due to mean westerlies in warm pool (Zhang 1996; Kiranmayi and Maloney 2011; Sobel et al. 2014; de Szoeke et al. 2015). Things get more complicated when we look at the Shi et al. (2018) and Khairoutdinov and Emanuel (2018) aquaplanet simulations that find that suppressing WISHE eliminates the MJO, while the same is not shown by simulations with realistic boundary conditions (Kim et al. 2011; Ma and Kuang 2016).
WTG moisture mode theories of the MJO with emphasis on the warm pool mentioned above (Sobel and Maloney 2012; Sobel and Maloney 2013; Adames and Kim 2016) hold that the MJO propagation and maintenance can be explained by the evolution of moisture alone. Sobel and Maloney (2012) assume mean westerlies that produce a westward-propagating disturbance with maximum growth rate near planetary wavenumber 7. Sobel and Maloney (2013) modified this model by introducing an empirical value based on reanalysis and general circulation model (GCM) moist static budgets that is the sum of evaporation anomalies, horizontal advection, and frictionally driven boundary layer convergence. This resulted in moistening east of the convection, thus overcoming the effect of the westerly WISHE and produced eastward propagation. Adames and Kim (2016) elaborated on Sobel and Maloney (2013) by treating the meridional and vertical structure of the MJO explicitly. They adjusted several parameters and produced disturbances with more realistic characteristics, i.e., their phase speed is in agreement with the observations. They found that the eastward propagation of the MJO is explained through horizontal moisture advection, frictional convergence, and modulation of surface fluxes. Adames and Kim (2016) find that the cloud-radiative feedback at the MJO time scale is the main driver of instability while moisture advection is the primary propagation mechanism. The propagation mechanism also results in a westward group velocity.
We hold that cloud–radiation interactions (CRI) are important in MJO physics. The CRI mechanism states that deep clouds block OLR, so that the radiative cooling rate is smaller in cloudy regions than in clear ones (Albrecht and Cox 1975). Cloud–radiation interactions were shown to be of potentially great significance to large-scale motions in the tropics (Slingo and Slingo 1988; Slingo and Slingo 1991; Randall et al. 1989; Sherwood et al. 1994; Raymond 2000b, 2001; Bony and Emanuel 2005; and others). Via the CRI mechanism, the net effect of radiation reinforces the existing upward motion, which further enhances precipitation. Clear regions reinforce downward motion.
Section 2 reviews the FR and KE theories and presents a derivation of “minimal difference” models. Results for “minimal difference” models are discussed in section 3, while conclusions are given in section 4.
2. Analytical models
In these equations v = (u, υ, w) is fluid velocity field, ρ is the mass density, p is the pressure, Ω is Earth’s rotation vector, while g is gravity, the sum of gravitational and centripetal accelerations. F is any other force that can act on the flow. In the thermodynamic equations, s is the dry entropy with S being its source and sink term and sm is the moist entropy with source and sink term Sm. In the simplest form dry entropy is defined as s = Cpln(θ/θ0) and moist entropy as sm = Cpln(θ/θ0) where θ is the potential temperature, θe is the equivalent potential temperature, Cp is the specific heat of dry air at constant pressure, and θ0 is a reference temperature. For full definitions of entropies and their source terms see Raymond (2013).
Numerical values of parameters.
In the original KE model x was nondimensionalized using the mean radius of Earth a. Here we have adjusted the equations in the FR model by using the same nondimensionalization as KE in x. Note that this changes values of the original FR nondimensional parameters. One of the scaling differences between the KE and FR model that is still present comes through different time scaling. KE use
The momentum and mass continuity equations of FR and KE are identical though they were originally arrived at using somewhat different reasoning. KE assume that the temperature lapse rate is always moist adiabatic, which filters all but the barotropic and first baroclinic modes while FR assume first baroclinic mode structure at the outset. The barotropic mode was neglected in both models. The differences between these models arise from different treatments of moist convection and cloud–radiation interaction. FR assume that precipitation and the associated latent heating are proportional to column moisture, while KE use a mass flux approach in which the net convective mass flux is determined through an assumption of boundary layer quasi equilibrium. Although these assumptions are quite different, they lead to similar thermodynamic equations, though with some important differences, as we shall see. Another difference is that FR assume that CRI depends on column moisture whereas KE assume that it depends on column moist entropy. The differences are subtle but do lead to slightly different forms of the linear equations.
Note that we are neglecting horizontal advection of moisture that others (Sobel and Maloney 2012, 2013; Adames and Kim 2016) found could be important, because we are comparing two models that neglected it.
a. Fuchs and Raymond model
The source terms in the thermodynamic equations, Eqs. (10) and (11), come from the assumptions that the heating S is a result of the precipitation and cloud–radiation interactions, while moist entropy source Sm depends on surface evaporation and cloud–radiation interactions. The precipitation perturbation is parameterized to depend on the precipitable water. FR assume that it increases linearly with the vertically integrated mixing ratio perturbation q = sm − s (Raymond 2013). This increase is proportional to a moisture relaxation rate α (Raymond 2000a). The CRI mechanism, which is known to drive self-aggregation of convection in idealized, convection-permitting models, is represented here as sources terms for both dry and moist entropy that scale with their difference, with the proportionality constant ε. Surface sensible heat flux is assumed to be zero as over the tropical oceans it is very small. Surface latent heat flux is parameterized via the WISHE mechanism wherein stronger winds increase surface latent heat fluxes. WISHE comes into the Eq. (11) via Λu, where the WISHE parameter Λ is positive for imposed mean westerlies and negative for mean easterlies. The perturbations in gustiness and boundary layer mixing ratio are neglected.
The conceptual picture behind the FR model is that surface fluxes increase moisture in a vertical column, most likely over a period of several weeks, which results in an increase of water vapor in the atmospheric column and therefore the precipitation. This increase in moisture then acts via the heating budget equation aided with cloud–radiation interactions that become more significant for a more humid, and hence cloudier environment.
For the higher meridional numbers n the dispersion relations are more complex and can be found in Fuchs and Raymond (2017).
b. Khairoutdinov and Emanuel model
The nondimensional constants used by Khairoutdinov and Emanuel (2018) and Emanuel (2020) in the thermodynamic equations, Eqs. (13) and (14), with above substitution are the WISHE parameter λ, the cloud–radiation interaction parameter ϵ, damping effect of boundary layer entropy perturbations on surface fluxes χ′ and D′, diffusion parameter d, and scaling constant γ that accounts for the different ways that s and sm have been scaled. In the original papers of KE model (Khairoutdinov and Emanuel 2018; Emanuel 2020) there was an additional constant κ, multiplying sm in the second term on the left side of (14), that one of us (Emanuel) has since discovered should always be one and is therefore omitted here. Note that definitions of the WISHE and CRI parameters in the KE model (λ and ϵ) differ from the corresponding parameters in the FR model (Λand ε) due to different scaling and parameterization and are presented using different symbols.
In comparing (13) and (14) to the FR thermodynamic equations, (10) and (11), we observe several structural differences. First, there is a term involving the zonal velocity perturbation u in (13) that is missing from (10). This is a WISHE term that arises from the convective mass flux closure used by KE: perturbations in surface wind speed increase the surface enthalpy flux, which increases the moist convective mass flux, which increases temperature. Second, the terms involving dry and moist entropy, (s and sm) appear individually, not simply as their difference, which is the case in FR. This arises from the fact that the cloud–radiation effect, represented here by the coefficient ϵ, is here taken to be proportional to sm rather than the column moisture, which can be expressed in terms of sm − s. Also, in KE the mass flux closure has a direct dependence on sm, with mass flux increasing as sm increases. This is responsible for the “1” in (13). Finally, the last term in (14) is a scale-dependent damping of moist entropy and was added by KE as a rough mimic of the effect of wave radiation into the stratosphere.
For the higher meridional numbers n the dispersion relations are more complex and can be found in Emanuel (2020).
c. FR and KE “minimal difference” model
The similarities between FR and KE models come from the fact that they both rely on WISHE and CRI to drive unstable modes. The main difference is that FR assume that precipitation is proportional to column moisture, while KE make use of boundary layer quasi equilibrium, producing a contribution to the mass flux proportional to the moist entropy perturbation. If one considers the moist entropy perturbation as essentially a moisture perturbation, this difference should not lead to very different results. Another difference is that FR assume that CRI depends on column moisture whereas KE assume that it depends on column moist entropy. It is important to note that the FR system has four independent specifiable coefficients, while the KE system has eight. These differences lead to different forms of the linear thermodynamic equations.
We now wish to develop a “minimal difference” model between FR and KE models, where the two models are brought most closely into alignment without losing the main physics, i.e., the WISHE and CRI mechanisms. In comparing the two systems, we should try to separate differences that arise from different parameter choices from differences that arise from structural differences in the equations.
Equations (23) and (24) are structurally almost identical to (21) and (22) with the only difference in an additional term, the first term, on the right-hand side of (23). This WISHE term arises from the convective mass flux closure used by KE.
We shall explore the differences between the FR and KE systems for this “minimal difference” case for the eastward-propagating MJO-like mode.
3. Results
We present the results for the FR and KE “minimal difference” models for υ = 0 or n = −1 case as well as for higher meridional numbers n for the same nondimensional parameters, given in Table 1: Γ = 0.1, the WISHE parameter Λ = λ = −0.62 and the CRI parameter ε = ϵ = 0.4. The reader is invited to further explore the parameter space of both models.1
Figure 1 shows the eastward-propagating MJO-like mode nondimensional phase speed in the top plot, and nondimensional growth rate in the bottom plot as a function of planetary wavenumber. The red line shows results obtained from the dispersion relation (25) for the FR model. The blue line shows results obtained from the dispersion relation (26) for the KE model. The phase speeds and growth rates of the modes are fairly similar. Most importantly, both models have the highest instability at planetary wavenumber l = 1. Without WISHE (Λ = λ = 0), the modes from FR and KE model are identical, stationary with the same growth rate at all wavenumbers. Without the CRI mechanism (ε = ϵ = 0), the modes from both models are unstable only for the planetary wavenumber l = 1. WISHE is therefore responsible for eastward propagation, while both WISHE and CRI produce instability. However, WISHE is needed to produce maximum instability at planetary wavenumber l = 1. Note that the CRI results in instability for all wavelengths. Yano and Emanuel (1991) found that replacing the rigid lid assumption by a realistic stratosphere has the effect of strongly damping the short waves.
(top) Eastward-propagating MJO-like mode phase speed, and (bottom) growth rate for υ = 0 case as a function of planetary wavenumber for the FR model (red) and the KE model (blue).
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
(top) Eastward-propagating MJO-like mode phase speed, and (bottom) growth rate for υ = 0 case as a function of planetary wavenumber for the FR model (red) and the KE model (blue).
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
(top) Eastward-propagating MJO-like mode phase speed, and (bottom) growth rate for υ = 0 case as a function of planetary wavenumber for the FR model (red) and the KE model (blue).
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
The explanation for the phase speed for l = 1 being larger than the observed OLR propagation speeds over the warm pool for both analytical models (ω/k = 15–18 m s−1, k = l/a) was given in Raymond and Fuchs (2018). In that paper, a simple linear superposition of the warm pool rainfall and the rain from the eastward-propagating WISHE-moisture mode led to a pattern of combined rainfall moving eastward with a speed of order 5 m s−1. The resulting westerly wind burst started to the west of the rainfall maximum, but ended up overrunning it as the MJO progressed across the warm pool. The strongest values of surface evaporation occurred to the east of rainfall, with an enhancement in evaporation as the MJO moved over the warm pool.
Figure 2 shows solutions of the dispersion relations for the FR “minimal difference” model and KE “minimal difference” model with the same nondimensionalized parameters given in Table 1 (WISHE and CRI included) for n = −1 as well as for the higher meridional numbers n = 1, 2, 3 for eastward- and westward-propagating unstable modes. Figure 2 shows the similarities of FR and KE “minimal difference” models across a span of zonal wavenumber and meridional mode number. KE growth rates are a little smaller at low wavenumber but do not fall off as fast with wavenumber, owing to the fact that WISHE affects the dry as well as moist entropy budgets. Across all meridional numbers the greatest instability is at the planetary wavenumber l = 1 for the eastward-propagating mode, giving us confidence that it represents the MJO mode. KE frequencies are a little bigger than FR ones at low wavenumber. The westward-propagating mode is the Rossby wave as seen by Fuchs-Stone et al. (2019) and Emanuel (2020).
Solutions of the dispersion relation for (a) the FR model and (b) the KE model. Diameters of the circles are proportional to the growth rates.
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
Solutions of the dispersion relation for (a) the FR model and (b) the KE model. Diameters of the circles are proportional to the growth rates.
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
Solutions of the dispersion relation for (a) the FR model and (b) the KE model. Diameters of the circles are proportional to the growth rates.
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
Figure 3 shows the wave structure for n = 1, l = 1 case with convergent winds along the equator, in phase with the vertical velocity and a quadruple vortex structure in entropy. This figure is similar to Fig. 3 from Emanuel (2020) and is in good agreement with Fig. 2a from Hendon and Liebmann (1994) and Fig. 2 from Kiladis et al. (2005).
Eigenfunction of the n = 1, l = 1 mode for the FR minimal difference model. The colors show the vertical velocity, the arrows show the low-level perturbations horizontal winds, and the black contours show the distribution of entropy.
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
Eigenfunction of the n = 1, l = 1 mode for the FR minimal difference model. The colors show the vertical velocity, the arrows show the low-level perturbations horizontal winds, and the black contours show the distribution of entropy.
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
Eigenfunction of the n = 1, l = 1 mode for the FR minimal difference model. The colors show the vertical velocity, the arrows show the low-level perturbations horizontal winds, and the black contours show the distribution of entropy.
Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0165.1
4. Conclusions
The value of analytical models is that they are able to clearly investigate and identify the physical mechanisms behind modeled weather disturbances. In a complex environment such as the tropical atmosphere this task is not easy. Decades of research have led the authors to narrow their focus to two models that seem to capture the essential physics behind the largest weather disturbance on our planet: the MJO. These are also highly consistent in their behavior with cloud-permitting simulations (Khairoutdinov and Emanuel 2018).
Unlike many convectively coupled waves, such as Kelvin and Rossby waves, the MJO has no analog in the dry atmosphere, which makes the starting point of research more complex. The diabatic sources that govern the heating and/or moisture budget were sought in order to model the mode that slowly propagates eastward and has the greatest instability at planetary wavenumber one as required by observations of the MJO. In this paper we present two models that accomplish this task: the FR model and the KE model. The FR model originates with Fuchs and Raymond (2002), who first modeled the moisture mode. The KE model was developed from Emanuel (1987), in which the idea of WISHE was introduced. Not until three decades later did it capture the global character of the MJO by also accounting for the effects of cloud–radiation interactions.
The FR and KE models have different parameterizations of convection. While they both include the WISHE and CRI mechanisms in their closure, the way WISHE is included is different. The FR model uses a moisture budget equation to parameterize the effect of surface fluxes, while the KE model directly parameterizes surface fluxes and precipitation efficiency into the heating budget equation. The KE model also includes damping effects of boundary layer entropy perturbations on surface fluxes and Fickian diffusion, which the FR model lacks.
In this paper, we have developed “minimal difference” models from FR and KE models, the ones that closest align to each other. The motivation behind this paper is to find a common framework between the two MJO theories, the WISHE-moisture mode theory (FR model) and the BLQE theory (KE) model. The value of this common framework is to bring two theories closer together to advance our understanding of the MJO.
The dispersion relations and their solutions for “minimal difference” FR and KE models show remarkable similarities for the υ = 0 case as well as for higher meridional numbers. Both models obtain the eastward-propagating mode when WISHE and CRI are included, they both exhibit the greatest instability for largest wavelengths and their eigenfunctions agree with the observed horizontal structure of the MJO. The phase speeds for largest wavelengths are larger than the observed OLR speeds over the warm pool. However, Raymond and Fuchs (2018) offer an explanation for that: when the eastward-propagating mode from FR model is combined with the warm pool rainfall via linear superposition the combined pattern leads to phase speed on the order of 5 m s−1.
Much of Fuchs and Raymond’s previous analytical model work focused on the υ = 0 case, which does not account for the observed meridional flows associated with the MJO. The more recent work discussed in this paper rectifies this by showing that CRI makes possible unstable n = 1 solutions which exhibit meridional flow. Unchanged with this extension is the global scale of the modes and the role of WISHE in their eastward propagation.
The FR and KE theories do not explain the multiscale structure of the MJO or its seasonal cycle. They do not include the roles of nonlinear effects, strong mean westerlies at certain longitudes, or horizontal moisture advection. The FR and KE models formulate WISHE assuming mean planetary easterlies while many other studies focus on the fact that there are mainly westerlies present in the warm pool region. Fuchs and Raymond (2017) reported that incorporating the horizontal advection of moisture in their model did not lead to major modifications in their results. This might be because of the differences in the model setups between WTG moisture models and the FR model. It is interesting to note that both FR and KE theories as well as the WTG moisture-mode theory predict a westward group velocity of the MJO in agreement with Janiga et al. (2018), but in disagreement with some studies based on observations (Chen and Wang 2018). For more discussion and observational evidence of MJO westward group velocity see Adames and Kim (2016).
In spite of their differences both the FR and KE theories rely on tropical mean easterly low-level flows that make the WISHE mechanism possible together with cloud–radiation interactions. Both theories thus provide maximally simplified models of the MJO that agree well with observations and cloud-permitting simulations. Further evaluation is needed from observations and GCMs.
The reader interested in exploring the parameter space of both models may download a MATLAB script to do so, from ftp://texmex.mit.edu/pub/emanuel/scripts/FR_and_KE_model_scripts/.
Acknowledgments.
We thank David Raymond for valuable conversations and insightful comments. Željka Fuchs-Stone was supported by the NSF Grant AGS-2034817 to NMT. Kerry Emanuel was supported by the NSF Grant AGS-1906768 to MIT.
APPENDIX
FR and KE Model Nondimensionalization
Here we show nondimensionalization and scaling for the variables in FR and KE models used in Eqs. (5)–(9). Mean radius of Earth is a while β is the meridional gradient of the Coriolis parameter at the equator.
a. FR model
b. KE model
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