Upscale Impact of Mesoscale Convective Systems and Its Parameterization in an Idealized GCM for an MJO Analog above the Equator

Qiu Yang Center for Prototype Climate Modeling, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

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Andrew J. Majda Department of Mathematics, and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York, and Center for Prototype Climate Modeling, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

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Mitchell W. Moncrieff National Center for Atmospheric Research, Boulder, Colorado

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Abstract

The Madden–Julian oscillation (MJO) typically contains several superclusters and numerous embedded mesoscale convective systems (MCSs). It is hypothesized here that the poorly simulated MJOs in current coarse-resolution global climate models (GCMs) is related to the inadequate treatment of unresolved MCSs. So its parameterization should provide the missing collective effects of MCSs. However, a satisfactory understanding of the upscale impact of MCSs on the MJO is still lacking. A simple two-dimensional multicloud model is used as an idealized GCM with clear deficiencies. Eddy transfer of momentum and temperature by the MCSs, predicted by the mesoscale equatorial synoptic dynamics (MESD) model, is added to this idealized GCM. The upscale impact of westward-moving MCSs promotes eastward propagation of the MJO analog, consistent with the theoretical prediction of the MESD model. Furthermore, the upscale impact of upshear-moving MCSs significantly intensifies the westerly wind burst because of two-way feedback between easterly vertical shear and eddy momentum transfer with low-level eastward momentum forcing. Finally, a basic parameterization of the upscale impact of upshear-moving MCSs modulated by deep heating excess and vertical shear strength significantly improves key features of the MJO analog in the idealized GCM with clear deficiencies. A three-way interaction mechanism between the MJO analog, parameterized upscale impact of MCSs, and background vertical shear is identified.

Denotes content that is immediately available upon publication as open access.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Qiu Yang, yangq@cims.nyu.edu

Abstract

The Madden–Julian oscillation (MJO) typically contains several superclusters and numerous embedded mesoscale convective systems (MCSs). It is hypothesized here that the poorly simulated MJOs in current coarse-resolution global climate models (GCMs) is related to the inadequate treatment of unresolved MCSs. So its parameterization should provide the missing collective effects of MCSs. However, a satisfactory understanding of the upscale impact of MCSs on the MJO is still lacking. A simple two-dimensional multicloud model is used as an idealized GCM with clear deficiencies. Eddy transfer of momentum and temperature by the MCSs, predicted by the mesoscale equatorial synoptic dynamics (MESD) model, is added to this idealized GCM. The upscale impact of westward-moving MCSs promotes eastward propagation of the MJO analog, consistent with the theoretical prediction of the MESD model. Furthermore, the upscale impact of upshear-moving MCSs significantly intensifies the westerly wind burst because of two-way feedback between easterly vertical shear and eddy momentum transfer with low-level eastward momentum forcing. Finally, a basic parameterization of the upscale impact of upshear-moving MCSs modulated by deep heating excess and vertical shear strength significantly improves key features of the MJO analog in the idealized GCM with clear deficiencies. A three-way interaction mechanism between the MJO analog, parameterized upscale impact of MCSs, and background vertical shear is identified.

Denotes content that is immediately available upon publication as open access.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Qiu Yang, yangq@cims.nyu.edu

1. Introduction

The Madden–Julian oscillation (MJO) is the dominant component of tropical intraseasonal variability (Zhang 2005) and dramatically impacts local weather through extreme rainfall and midlatitude atmospheric conditions by tropical–extratropical teleconnection (Zhang 2013; Stan et al. 2017; Henderson et al. 2017). Tropical convection associated with the MJO is hierarchically organized across multiple spatial and temporal scales. The MJO typically contains multiple eastward- and westward-moving superclusters of cloudiness (Nakazawa 1988; Chen et al. 1996) with numerous embedded mesoscale convective systems (MCSs; Houze 2004) and cumulus clouds on smaller scales. As the major rainfall producer in the tropics, MCSs contribute up to 50% of the rainfall in most tropical regions (Tao and Moncrieff 2009). Although the effects of large-scale atmospheric conditions on the modulation of MCSs have been well documented in observations (Lin and Johnson 1996; Chen et al. 1996; LeMone et al. 1998), a satisfactory understanding of the collective effects of MCSs on the momentum and heat budgets of the MJO is still lacking.

It is hypothesized that the poorly simulated MJOs in current coarse-resolution GCMs are related to the inadequate treatment of MCSs and their upscale impact. The essential difference between MCSs and smaller individual convective towers lies in the fact that the former typically have front-to-rear-tilted organized structures, while the latter are unorganized (Moncrieff and Klinker 1997). Typical behavior of the poorly simulated MJOs in the GCMs includes impersistent eastward propagation, unrealistic planetary–intraseasonal variability in precipitation and winds, and upright vertical structure with a negligible westerly wind burst (WWB; Jiang et al. 2015). In contrast, global cloud-system-resolving simulations that explicitly represent MCSs successfully capture some key features of the MJOs (Grabowski 2003; Miura et al. 2007) and motivated the development of the superparameterization method based on two-dimensional cloud-resolving models (CRMs; Grabowski 2001, 2004; Randall et al. 2003; Majda 2007a) and a sparse space–time technique (Xing et al. 2009). Nevertheless, the computational cost to explicitly resolve MCSs is impractical for long GCM simulations. An alternative way to address this issue is to develop new parameterizations for coarse-resolution GCMs that capture the upscale impact of unresolved MCSs on the MJO.

Several studies have assessed the upscale impact of MCSs based on observational measurement, reanalysis datasets, and cloud-resolving simulations, many of which focus on convective momentum transfer (CMT; Moncrieff 1981; LeMone 1983; Moncrieff 1992; LeMone and Moncrieff 1994). Convective-scale CMT by unorganized convection normally has frictional effects that reduce large-scale vertical shear (Zhang and McFarlane 1995). In contrast, mesoscale CMT by organized convection over 100 km in horizontal scale can have countergradient momentum transport that enhances the large-scale vertical shear (Moncrieff 1981, 1992). Tung and Yanai (2002a) concluded that CMT is, on average, downgradient over the western Pacific warm pool but upgradient during the westerly wind phase of the MJO (Tung and Yanai 2002b). Oh et al. (2015) found that the subgrid-scale and mesoscale CMT associated with the MJO has a distinctive three-layer vertical structure. Grabowski and Moncrieff (2001) demonstrated that CMT from westward-moving MCSs embedded in the eastward-moving convective envelope promotes the large-scale organization of convection. Inspired by multiscale organization and the observed statistical self-similarity of tropical convection, Majda (2007b) systematically derived multiscale asymptotic models that describe scale interactions among clusters, superclusters, and intraseasonal oscillations and highlight the crucial role of eddy transfer of momentum and temperature. Brenowitz et al. (2018) concluded that mesoscale CMT dominates the total vertical flux feedback on the planetary-scale kinetic energy budget, providing new mechanisms for the planetary-scale organization of convection.

From a theoretical perspective, several modeling studies have sought to better understand the upscale impact of MCSs on the large-scale organization of tropical convection. Majda and Stechmann (2009) utilized a simple dynamic model with features of CMT from convectively coupled gravity waves and their interactions with large-scale mean flow. Khouider et al. (2012) demonstrated that in the active region of the MJO with WWB, CMT from both convectively coupled Kelvin waves (CCKWs) and MCSs plays a significant role in accelerating the low-level westerly winds. The three-dimensional mesoscale equatorial synoptic dynamics (MESD) model, originally derived by Majda (2007b), was used as a multiscale framework to assess the upscale impact of MCSs on eastward-moving CCKWs (Yang and Majda 2018) and westward-moving 2-day waves (Yang and Majda 2019). Explicit expressions for the eddy transfer of momentum and temperature obtained from the MESD model are an essential basis for the parameterization of upscale impact of MCSs provided here. Moncrieff et al. (2017) introduced the multiscale coherent structure parameterization (MCSP) that achieved significant improvement in tropical precipitation patterns and precipitation variability in a GCM. In general, idealized models that simulate some key features of the MJO can serve as a useful test bed. Here, we refer to these MJO-like events arising from the idealized models as the MJO analog.

The goals of this paper include the following three aspects: First, use a simple multicloud model for the MJO analog and intraseasonal variability above the equator to mimic the typical behavior of GCMs with clear deficiencies. Second, assess the upscale impact of MCSs on key features of the MJO analog, including persistent propagation of a two-scale structure, realistic planetary–intraseasonal variability in precipitation and winds, and a significant WWB. Third, introduce a basic parameterization of the upscale impact of MCSs and test its effects in the idealized GCM to address deficiencies.

In general, the multicloud models represent three dominant cloud types (congestus, deep, stratiform) by using the first- and second-baroclinic vertical modes and build the life cycle of these cloud types into the convective heating closure through a switch function for midtropospheric dryness (Khouider and Majda 2006c, 2007, 2006a). The deterministic version of the multicloud models successfully captures characteristic features of CCEWs (Khouider and Majda 2008b, 2006b, 2008a) and the diurnal cycle (Frenkel et al. 2011a,b, 2013). The stochastic version captures the MJO (Khouider et al. 2010; Deng et al. 2015; Goswami et al. 2017) when coupled to the GCM. In this paper, we use a deterministic two-dimensional multicloud model for the MJO analog and intraseasonal variability above the equator (Majda et al. 2007; Harlim and Majda 2013). By reducing the magnitude of both congestus and stratiform heating, this model mimics the typical behavior of GCMs with clear deficiencies, where both convection types are underestimated (Seo and Wang 2010; Del Genio et al. 2012; Lappen and Schumacher 2012; Del Genio et al. 2015). To introduce the upscale impact of MCSs, we use explicit expressions for the eddy transfer of momentum and temperature theoretically predicted by the MESD model (Yang and Majda 2018).

The upscale impact of MCSs on the MJO analog is assessed through comparison experiments with and without adding extra eddy transfer of momentum and temperature. The modulation effects of deep heating excess on eddy transfer of momentum and temperature are considered in order to mimic the scenario that MCSs are prominent in the active convection region of the MJO (Khouider et al. 2012). The results show that the upscale impact of westward-moving MCSs promotes the eastward propagation of the MJO analog, consistent with the theoretical prediction by the MESD model (Yang and Majda 2018). The modulation effects of vertical shear are considered to mimic the observation that MCSs typically move toward the convection center (Lin and Johnson 1996; Chen et al. 1996; Moncrieff and Klinker 1997; Yanai et al. 2000; Houze et al. 2000). The results show that the upscale impact of upshear-moving MCSs leads to a significant WWB in the middle and west of the MJO analog, because of the positive feedback between large-scale easterly vertical shear and embedded eddy momentum transfer with low-level eastward momentum forcing. Finally, we provide a basic parameterization of the upscale impact of upshear-moving MCSs, where modulation effects of deep heating effects and vertical shear strength are linearly combined. Significant improvement is achieved by adding this parameterization to the idealized GCM that has clear deficiencies. A further simulation illustrates a three-way interaction mechanism between the MJO analog, parameterization of upscale impact of MCSs, and background mean flow over a long time scale. Specifically, the resulting oscillatory background mean flow resembles the QBO-like oscillation identified in cloud-resolving simulations (Held et al. 1993; Nishimoto et al. 2016) and simplified GCMs (Horinouchi and Yoden 1998).

The results of this paper are presented as follows. Section 2 summarizes the governing equations and properties of the two-dimensional multicloud model, including the realistic MJO analog above the equator and the idealized GCM that has clear deficiencies. Section 3 discusses the effects of eddy transfer of momentum and temperature from MCSs on the MJO analog. Section 4 provides a basic parameterization of upscale impact of upshear-moving MCSs under the modulating effects of deep heating excess and vertical shear strength and tests its effects in the idealized GCM that has clear deficiencies. The paper concludes with a discussion in section 5.

2. An idealized GCM for an MJO analog and intraseasonal variability above the equator

In this section, we briefly review the equations governing the multicloud model and the convective heating closure. The simple two-dimensional multicloud model used here (Majda et al. 2007; Harlim and Majda 2013) captures the MJO analog and intraseasonal variability above the equator. The goals of this section are to reproduce (i) a realistic MJO analog above the equator as a proxy for the observations and (ii) a simulation with reduced congestus and stratiform heating as an idealized GCM having clear deficiencies.

a. Governing equations and multicloud model parameterization

The multicloud models describe the life cycle of congestus, deep, and stratiform cloud types (Johnson et al. 1999) and incorporate it as a convective heating closure by using a switch function for midtropospheric dryness. Specifically, shallow congestus convection is first initialized with low-level heating and upper-level cooling, moistening the lower troposphere and preconditioning the deep convection. Then deep convection warms the whole troposphere because of extreme rainfall, followed by stratiform convection with upper-level latent heating and low-level cooling by rain evaporation (Khouider and Majda 2008b).

The governing equations and multicloud convective parameterization in dimensionless units are listed in Tables 1 and 2 and all relevant parameters in Table 3. All physical variables are nondimensionalized by the following synoptic scaling: first-baroclinic dry Kelvin wave speed for horizontal velocity, equatorial Rossby deformation radius for length, for time, for temperature, and for heating. For convenience, the moisture anomaly has the unit of temperature (K). Correspondingly, as the moisture sink, the precipitation has the unit of heating (K day−1).

Table 1.

Prognostic governing equations in the 2D multicloud model for the MJO analog and intraseasonal variability above the equator.

Table 1.
Table 2.

Diagnostic equations in the 2D multicloud model for the MJO analog and intraseasonal variability above the equator. The notation bar indicates the value of variables at RCE state. The notation represents positive value of f and vanishes when ; that is, .

Table 2.
Table 3.

Parameters and constants in the idealized GCM with clear deficiencies. The different values of parameters and constants used for the realistic MJO analog above the equator is shown in the parentheses. All the remaining values are as in Majda et al. (2007).

Table 3.

Consistent with the first-baroclinic deep heating and the second-baroclinic congestus–stratiform heating, both momentum and temperature variables in the free troposphere are truncated to the first- and second-baroclinic modes using the following Galerkin projection:
e1
e2
where the vertical coordinate z is scaled by 5 km so that in dimensionless units correspond to the surface (0 km) and top of the troposphere , respectively. Here, u is zonal velocity, p is the pressure perturbation, is eddy momentum transfer, θ is potential temperature anomaly, is heating, and is eddy heat transfer. As shown by Table 1, the first- and second-baroclinic momentum is forced by linear momentum damping mimicking boundary layer turbulent drag , Rayleigh friction , and eddy momentum transfer . The first-baroclinic potential temperature is driven by the deep heating P and the second-baroclinic potential temperature by congestus and stratiform heating . Both are further forced by radiative cooling and eddy heat transfer . These dynamical fields are coupled to a column-integrated moisture perturbation (Khouider and Majda 2006b), where both linear and nonlinear moisture advection terms are retained and precipitation and downdrafts are added as moisture sink and source, respectively. Specifically, the precipitation in dimensionless units is assumed to be equal to the total column-integrated heating, contributed by the first baroclinic mode. The boundary layer equivalent potential temperature equation shows that surface-level evaporation (E/hb) warms and moistens the boundary layer while the downdrafts (D/hb) have the opposite effects. Both congestus heating and stratiform heating are governed by linear relaxation equations. Congestus heating is triggered in the leading cold and dry midtroposphere, and stratiform heating lags the deep heating region. A switch function for midtroposphere dryness Λ is defined in Table 2. The multicloud heating closure is completed by introducing deep heating P, downdrafts D, and evaporation E.

All physical variables are imposed on the domain of the tropical belt, 0 ≤ x < 40 000 km, with periodic boundary conditions in the zonal direction. The governing equations shown in Tables 1 and 2 are solved numerically by spatially discretizing the solutions at equal-spaced grids and then temporally integrated using the fourth-order Runge–Kutta scheme. The horizontal resolution is 100 km, and each time step is 4.5 min, close to typical coarse-resolution GCMs. The moisture equation with nonlinear advection terms is solved by pseudospectral methods. To stabilize the numerical scheme and eliminate grid-scale numerical instability, a fourth-order hyper-diffusion term is added to all prognostic equations where the dimensionless value of ν, chosen as 2 × 10−5, is based on trial and error.

The radiative–convective equilibrium (RCE) state conveniently describes the linear convective instability of the multicloud model. Specifically, we consider a state where zonal velocity, , and potential temperature and moisture perturbation vanish in both the troposphere and the boundary layer, , , and . The actual values of the other variables at the RCE state are included in Table 4. Both eddy momentum transfer and eddy heat transfer are set to zero in the simulations presented in this section. A random field of moisture of a very weak magnitude (10−5 in dimensionless units) is added to the initial conditions to trigger unstable moist modes. All solutions presented in this paper refer to the equilibrium state obtained after long simulations (4000 days in sections 2 and 3 and 7000 days in section 4).

Table 4.

Value of thermodynamic variables at RCE state. The remaining variables not mentioned here are all zero. The different values of parameters and constants used for the realistic MJO analog above the equator are shown in the parentheses.

Table 4.

b. Realistic MJO analog and intraseasonal variability above the equator

We first implement the 2D multicloud model with all default parameter values as in Majda et al. (2007). The default parameters for the congestus and stratiform adjustment coefficients are and , respectively, and the background moisture stratification is 1.0. Although the typical value of in other studies based on observations is smaller (0.9), the larger value of is chosen to increase convective instability and intensify precipitation. We run the simulation for 4000 days, of which the last 1000-day output are used in the equilibrium state for interpretation purposes. Since the model output in the default parameter regime share features that resemble observations, we regard them to be a realistic MJO analog of intraseasonal variability above the equator, a proxy for observations. It is worth clarifying that by “realistic,” we refer to the good solutions with optimal parameters in this idealized framework, in contrast to the deficient solutions as shown in section 2c.

Figure 1a is the Hovmöller diagram for precipitation during the last 200 days, characterized by a two-scale structure consisting of eastward-moving planetary-scale envelopes and numerous embedded westward-moving synoptic-scale disturbances. The wavenumber-2 envelopes of period 40 days propagate eastward at 6.17 m s−1. Embedded in these planetary-scale envelopes are several synoptic-scale disturbances that propagate westward at slower speeds, resembling the observed westward-moving superclusters in the active phases of MJO over the west Pacific, (e.g., 2-day waves; Chen et al. 1996). However, this too-regular pulsing of precipitation during the eastward propagation of planetary-scale envelopes is less realistic than the more intermittent behavior of observed superclusters in the MJO. Figures 1b and 1c show the log-scale wavenumber–frequency spectra of precipitation and zonal velocity. The eastward-moving precipitation component has a dominant peak in wavenumber 2 and a period of 30 days. The spectra of zonal velocity are similar but confined to a smaller wavenumber and longer period, consistent with the observation that the dynamical circulation usually has larger spatial scales than the heating that drives it. For both precipitation and zonal velocity, the eastward-moving mode is the sum of at least three distinct harmonics with the same phase speed, thus differing from the single peak for the MJO seen in observations (Kiladis et al. 2009). The spectra of the westward-moving mode feature three regular and linear bands, according to the linearity of the dynamic core in Table 1. These three bands for westward-moving synoptic-scale disturbances in Fig. 1b are aligned such that the straight line perpendicular to them has the same slope as that across the three peaks for eastward-moving modes. This indicates a modulation of westward-moving synoptic-scale disturbances by eastward-moving envelopes. Figures 1d and 1e show the zonal and vertical profiles of the composite planetary-scale envelopes in the moving frame of reference. As shown by Fig. 1d, the precipitation peak is led by both column-integrated moisture and boundary layer equivalent potential temperature and followed by stratiform heating. This is consistent with the conceptual understanding that a moist free troposphere and boundary layer tends to precondition deep convection while stratiform convection in the form of anvil clouds forms subsequent to deep convection. Figure 1e shows the vertical cross sections of zonal velocity and potential temperature anomalies in the free troposphere. Both fields are characterized by a front-to-rear tilt with increasing height, akin to the observed MJO. The surface-level westerlies resemble the WWB of the observed MJO. It is worth mentioning that the model is invariant under changing the signs of x and u so that the solution does not have direction preference. The eastward propagation of the MJO analog in Fig. 1a is solely determined by the initial random perturbation. In a three-dimensional setup, the presence of the Coriolis force would break the zonal symmetry and favor the propagation direction of the MJO analog to be eastward.

Fig. 1.
Fig. 1.

Realistic MJO analog above the equator. Hovmöller diagram for (a) precipitation and log-scale wavenumber–frequency spectra for (b) precipitation and (c) surface-level zonal velocity . (d),(e) Vertical cross sections of composite planetary-scale envelope in the moving reference frames (6.1 m s−1), based on model output between day 3000 and day 4000. (d) Deep heating P, stratiform heating , and congestus heating with the left y axis and moisture q, and boundary layer equivalent potential temperature with the right y axis. (e) Zonal velocity u (color) and potential temperature θ (solid lines for positive values, dashed lines for negative; contour interval 0.05 K). The pink curve shows the zonal profile of precipitation anomalies with the right axis. The vertical dashed line indicates the longitude with easterly (westerly) vertical shear to its west (east). Domain-mean potential temperature is removed. The units of precipitation and zonal velocity are kelvins per day and meters per second, respectively.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Key features of the realistic MJO analog include the following three aspects: first, two-scale structure with eastward-moving planetary-scale envelope and embedded westward-moving synoptic-scale disturbances; second, spectra of precipitation and zonal velocity with dominant peaks at wavenumbers 1–3 and periods of 30–90 days in eastward-moving components and wide bands of spectra signals for westward-moving components at wavenumbers 5–15 and periods less than 30 days; and third, front-to-rear tilts in zonal velocity and potential temperature with the WWB located in the middle and west of the planetary-scale envelope. In the remaining experiments, we will focus on these three key features of the MJO analog.

c. Idealized GCM with clear deficiencies

Sensitivity experiments (not shown) show that the model solutions are quite sensitive to several key parameters, such as stratiform heating adjustment coefficient , congestus heating adjustment coefficient , and background moisture stratification . There is no guarantee that these key parameters will have optimal values in physically motivated applications, causing significant bias and poor behavior. To mimic the behavior of GCMs with clear deficiencies, we reduce the heat adjustment coefficients for congestus and stratiform convection to half as shown by Table 3. Meanwhile, the background moisture stratification is increased from 1.00 to 1.03 to give relatively stronger convective instability. Physically, this increment in the value of corresponds to a 3% increase of background moisture in the lower troposphere.

Figure 2a shows Hovmöller diagrams for precipitation after the system attains the equilibrium. Both the eastward-moving mode (wavenumber 3) and the embedded westward-moving mode (wavenumber 5) have a similar spatial scale, exhibiting significant deficiencies from the clear two-scale structure as shown in Fig. 1a. The planetary-scale mode propagates eastward at a speed of 2.4 m s−1, much slower than the typical observed MJO (5 m s−1). The maximum magnitude of precipitation is equivalent to about 8 K day−1 heating, significantly weaker than in Fig. 1. Figure 2b and Fig. 2c show the log-scale wavenumber–frequency spectra of precipitation and zonal velocity. Notably, these spectra peaks are quasi symmetric about the wavenumber-0 axis, and both are featured by the planetary-scale (about wavenumber 4) and intraseasonal (near 40 days) variability. Such eastward–westward symmetry stems from the mixture of both eastward- and westward-moving analogs. In fact, present-day GCMs suffer a similar bias in that the spectra of westward-moving planetary-scale precipitation is as significant as its eastward-moving counterpart (see Fig. 6 in Jiang et al. 2015). Because of the simplicity of the model, the spectra in this deficient model feature several discrepancies from that in conventional GCMs.

Fig. 2.
Fig. 2.

An idealized GCM with clear deficiencies. Hovmöller diagram for (a) precipitation and log-scale wavenumber–frequency spectra for (b) precipitation and (c) surface-level zonal velocity based on model output between day 3000 and day 4000. The units of precipitation and zonal velocity are kelvins per day and meters per second, respectively.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

In connection with the known biases in complex weather and climate models, contemporary GCMs have difficulty in simulating the persistent eastward propagation of the MJO (Zhang 2005), let alone the embedded westward-moving synoptic-scale disturbances. First, the coherent two-scale structure of the MJO with embedded synoptic-scale disturbances is poorly simulated in the GCMs (see Figs. 7–9 of Guo et al. 2015). Second, the GCMs also show significant discrepancies from the observations in the wavenumber–frequency spectra with unrealistic westward-moving planetary–intraseasonal variability (see Fig. 6 of Jiang et al. 2015). Third, a strong WWB is observed to extend from the surface to the midtroposphere, while in the GCMs, the WWB is typically weak and confined to lower levels near the surface (see Fig. 13 of Jiang et al. 2015).

3. Upscale impact of mesoscale convective systems on the MJO analog above the equator

In this section, we assess the upscale impact of MCSs on the MJO analog through comparison experiments with and without eddy transfer of momentum and temperature from mesoscale fluctuations. Specifically, we use the idealized GCM with clear deficiencies in Fig. 2 as the control simulation. To introduce the upscale impact of MCSs, we use the explicit expressions for eddy transfer of momentum and temperature obtained from theoretical predictions of the MESD model (Yang and Majda 2018). We consider the upscale impact of MCSs that propagate either slowly (5 m s−1) or rapidly (20 m s−1), either upshear or downshear, modulated by either deep heating excess or vertical shear strength. The observed typical propagation speed of MCSs lies within the range 5–20 m s−1 (Houze 1975, 1977, 2004). The two speeds (5 and 20 m s−1) are chosen to highlight differences between slow- and rapid-propagating scenarios. Because of the invariance of this model under changing signs of x, u, and , we need only consider the case with westward-moving MCSs because the opposite case can be inferred through counteranalogy.

We investigate how the upscale impact of MCSs improves the simulations of the MJO analog in the idealized GCM with clear deficiency by conducting several experiments with different eddy transfer of momentum and temperature. In brief, we first consider cases with eddy transfer of momentum and temperature modulated by the deep heating excess in section 3b. Two specific cases with upscale impact of MCSs propagating westward at either a slow or fast speed are investigated. We then consider cases with eddy transfer of momentum and temperature modulated by the vertical shear strength in section 3c, including three cases with upscale impact of MCSs propagating westward, upshear, or downshear at a slow speed. Details of the model setup in each experiment are shown in Table 5.

Table 5.

Summary of all experiments under the different model setup and their results in capturing key features of the MJO. In the “upscale impact of MCSs” column, “no” means no eddy is added, “westward” or “eastward” means the propagation direction of MCSs, “slow” (“fast”) corresponds to 5 (20) m s−1, and “upshear” (“downshear”) means the propagation direction of MCSs is opposite (along) vertical shear direction. The “modulation” column shows the modulation effects of deep heating excess P0 and vertical shear strength. The “key feature” column includes (i) two-scale structure of the MJO analog, (ii) wavenumber–frequency spectra of precipitation and winds with planetary–intraseasonal peaks, and (iii) westerly wind burst.

Table 5.

a. Eddy transfer of momentum and temperature predicted by the MESD model

In general, the multiscale models based on the multiscale asymptotic methods (Majda and Klein 2003; Majda 2007b) have been applied to study multiscale interactions of tropical convection such as the upscale impact of synoptic-scale fluctuations on the MJO (Majda and Biello 2004; Biello and Majda 2005, 2006), the intraseasonal impact of the diurnal cycle on the MJO (Yang and Majda 2014; Majda and Yang 2016), and ITCZ breakdown (Yang et al. 2017). In particular, the Majda (2007b) MESD model has been used to assess upscale impact of embedded MCSs on eastward-moving CCKWs (Yang and Majda 2017, 2018) and westward-moving 2-day waves (Yang and Majda 2018). In those studies, mesoscale heating is prescribed by phase-lagged first- and second-baroclinic modes to mimic the observed front-to-rear-tilt structure (Houze 2004):
e3
where points to the propagation direction of mesoscale heating, is magnitude coefficient, and k and ω are wavenumber and frequency, respectively. Here, α measures the relative strength of the second baroclinic mode and is the phase lag. The MESD model provides explicit expressions for eddy transfer of momentum and temperature:
e4
e5
where γ is the tilt angle between propagation direction of mesoscale heating and zonal direction in the horizontal plane. In the following experiments, for simplification, and are further truncated by retaining only the dominant first-baroclinic mode.

Figure 3 shows vertical profiles of mesoscale fluctuations and the eddy transfer of momentum and temperature. In particular, the red curves in Figs. 3c and 3d show the corresponding eddy transfer of momentum and temperature for eastward-propagating mesoscale systems. When the mesoscale systems propagate westward, the sign of eddy momentum transfer is reversed, while that of eddy heat transfer remains unchanged. In fact, the CRM study by Badlan et al. (2017) showed that the vertical profile of eddy momentum transfer is dominated by the first-baroclinic mode. In the simple multicloud model that resolves the first two baroclinic modes, we further truncate the vertical profiles of eddy transfer of momentum and temperature by retaining only the first-baroclinic mode. Consequently, the eddy momentum transfer has eastward (westward) momentum forcing in the lower (upper) troposphere, with maximum strength at the surface (top) of the domain. The eddy heat transfer cools throughout the troposphere, with maximum strength in the middle troposphere.

Fig. 3.
Fig. 3.

Vertical profiles of (a) zonal and vertical velocity ( and ; arrows), (b) potential temperature anomalies (; contours interval 0.06 K), (c) eddy momentum transfer , and (d) eddy heat transfer in an eastward-moving mesoscale system. The colors in (a) and (b) show mesoscale heating . The maximum magnitudes of zonal and vertical velocities are 3.73 and 0.47 m s−1, respectively. Also shown in (c) and (d) is the truncated eddy transfer of momentum and temperature with only the first-baroclinic mode. One dimensionless unit of eddy momentum transfer and eddy heat transfer is 15 m s−1 day−1 and 4.5 K day−1, respectively.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

It is straightforward to show that the ratio between and in dimensionless units is determined by propagation speed of the mesoscale heating:
e6
where c is the dimensionless value (dimensional value divided by 50 m s−1) of propagation speed of the mesoscale heating. In the following simulations, we do not need to specify exact values of parameters in the expressions of and but just specify the value of . The value of is then inferred by Eq. (6), when the propagation speed of the mesoscale heating c is specified.

b. Eddy transfer of momentum and temperature modulated by deep heating excess

Here, we consider the scenario where the eddy transfer of momentum and temperature in the first-baroclinic mode is modulated by the maximum allowable deep heating excess as follows:
e7
e8
where is the anomaly component of the maximum allowable deep heating (see Table 2) and is the corresponding RCE value. The value of the expression stays the same as if is positive and zero if it is negative. The closure for is a combination of the Betts–Miller relaxation-type parameterization and convective available potential energy (CAPE) parameterization. Physically, the maximum allowable deep heating excess resembles the effect of CAPE in modulating MCSs and the resulting CMT (Moncrieff 2004). Majda and Stechmann (2008) developed a stochastic parameterization for CMT, whose strength is modulated by the square of the maximum allowable deep heating.

Three cases are compared with and without and modulated by the effects of . The first case is the control simulation in Fig. 2. The second and third cases consider the eddy transfer of momentum and temperature from MCSs that propagate at a slow (5 m s−1) and fast speed (20 m s−1). The magnitude coefficient for eddy momentum transfer is fixed at 0.0032. The difference between the second and third cases lies in the stronger magnitude of in the case of fast propagation.

Figure 4 shows the Hovmöller diagrams for precipitation. The control simulation in Fig. 4a features both eastward- and westward-moving planetary-scale disturbances with no clear two-scale structure. Compared with the control simulation, the cases with eddy terms from westward-moving MCSs at a slow (fast) speed in Fig. 4b (Fig. 4c) respectively show a two-scale structure, where planetary-scale envelopes propagate eastward and embedded synoptic-scale disturbances propagate westward. In Fig. 4b, the maximum magnitude of precipitation is equivalent to 28 K day−1. Such intense precipitation and the promoted eastward-moving planetary-scale envelope by westward-moving MCSs is consistent with Yang and Majda (2018). In Fig. 4c, the maximum magnitude of precipitation is reduced to 12 K day−1 and convection is suppressed because of the extra cooling from eddy heat transfer, again consistent with Yang and Majda (2018). This additional cooling reduces low-level moisture convergence, resulting in a weaker growth rate of the unstable modes.

Fig. 4.
Fig. 4.

Hovmöller diagrams for precipitation [; K day−1] between day 3800 and day 4000 in the cases with and without eddy terms from westward-moving MCSs modulated by the deep heating excess . (a) The case without eddy terms. (b),(c) The case with eddy terms from (b) slowly propagating MCSs (5 m s−1) and (c) fast-propagating MCSs (20 m s−1).

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Figure 5 shows the log-scale wavenumber–frequency spectra of precipitation and zonal velocity. Compared with the control simulation in Fig. 5a, both cases with eddy terms from westward-moving MCSs in Figs. 5c–f show a clear east–west contrast in the spectra, similar to the realistic MJO analog shown in Fig. 1. For the slowly propagating MCSs, the spectra of precipitation in Fig. 5c are characterized by three discrete spectra peaks for the eastward-moving components and three bands of spectra of westward-moving components. In particular, the peak for the eastward-moving planetary-scale envelope has wavenumber 3 and a period of about 50 days. The spectra of zonal velocity in Fig. 5d resembles that in Fig. 5c, indicating close correlation between convection and the large-scale circulation. As for the faster-propagating MCSs in Figs. 5e and 5f, the associated spectra of precipitation are dominated by a planetary-scale peak for the eastward-moving component and a band of spectra for the westward-moving component.

Fig. 5.
Fig. 5.

Log-scale wavenumber–frequency spectra of (a),(c),(e) precipitation and (b),(d),(f) surface-level zonal velocity in wavenumber–frequency diagrams, based on the model output between day 3000 and day 4000 in the cases with and without eddy terms from westward-moving MCSs modulated by the deep heating excess . (top to bottom) Cases with (a),(b) no eddy, (c),(d) eddy terms from slowly propagating MCSs (5 m s−1), and (e),(f) eddy terms from fast-propagating MCSs (20 m s−1). Color bars are different for each column and are below (e) and (f).

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Figure 6 shows the vertical cross sections of the composite planetary-scale envelopes in the moving reference frame. The vertical structure of zonal velocity and potential temperature anomalies features a significant front-to-rear tilt, consistent with the in-built transition of life cycle from congestus to deep to stratiform convection. In the case with eddy terms from westward-moving MCSs at a slow speed in Fig. 6a, the maximum magnitude of zonal velocity of about 2 m s−1 occurs at the top of the domain. In the lower troposphere, the wind convergence is mostly in phase with the maximum precipitation with westerlies to the west and easterlies to the east. The WWB is negligible. The maximum magnitude of both positive and negative potential temperature are both attained in the upper troposphere. In contrast, in the case with eddy terms from westward-moving MCSs at a fast speed in Fig. 6b, both the maximum magnitude of zonal velocity, potential temperature anomalies, and precipitation anomalies in Fig. 6b are much weaker than those in Fig. 6a, indicating suppressed convection due to eddy heat transfer. The control simulation features both eastward- and westward-propagating large-scale disturbances. The corresponding composite planetary-scale envelope that is calculated only along the eastward-moving reference frame is less meaningful and thus not shown.

Fig. 6.
Fig. 6.

Vertical cross sections of composite planetary-scale envelope in the moving reference frames (s is the propagation speed), based on model output between day 3000 and day 4000 in the cases with eddy terms from westward-moving MCSs modulated by the deep heating excess . Panels show the cases with eddy terms from (a) slowly propagating MCSs (5 m s−1) and s = 3.05 m s−1 and (b) fast-propagating MCSs (20 m s−1) and s = 3.35 m s−1. Zonal velocity u (m s−1) is shown by color and potential temperature θ (K) is shown by contours (solid lines for positive value, dashed lines for negative; contour interval 0.005 K). The pink curve shows the zonal profile of precipitation anomalies using the right axis. Domain-mean potential temperature is removed.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

c. Eddy transfer of momentum and temperature modulated by vertical shear

Here, we consider the scenario when eddy transfer of momentum and temperature is modulated by the strength of vertical shear as follows:
e9
e10
where and the strength of vertical shear is defined as follows:
e11
e12
e13
Figure 7a explains the definition of vertical shear strength , which calculates the maximum possible easterly and westerly shear between the upper and lower troposphere and selects the larger one. Figure 7b describes the scenarios when the MCSs propagate upshear (along the opposite direction of vertical shear) and downshear (along the same direction of vertical shear).
Fig. 7.
Fig. 7.

A conceptual diagram for the definition of (a) vertical shear strength and (b) upshear–downshear propagation. In (a), the red (blue) bars indicate the maximum (minimum) magnitude of zonal winds in the upper and lower troposphere. The strength of vertical shear is defined as the stronger magnitude between westerly and easterly vertical shear and the direction of vertical shear is defined to be eastward (westward) if ΔU is positive (negative). (b) An eastward-moving MJO analog with wind convergence (divergence) in the lower (upper) troposphere. According to the definition of vertical shear in (a), this MJO analog is accompanied by easterly (westerly) vertical shear to the west (east), indicated by blue arrows and letter S on top. Upshear is defined as propagation along the opposite direction of vertical shear.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Four cases are compared with and without and modulated by the effects of . Besides the first cases from the control simulation in Fig. 2, the remaining three cases consider the eddy transfer of momentum and temperature from MCSs that propagate westward, upshear, and downshear at a slow speed (5 m s−1). Correspondingly, the magnitude coefficient of eddy momentum transfer is 0.0024, 0.0030, and 0.0030, respectively. The choice of a smaller value of in the second case is to obtain a more realistic precipitation intensity.

Figure 8 shows the Hovmöller diagrams for precipitation. Compared with the control simulation in Fig. 8a, the maximum magnitude of precipitation in both Figs. 8b and 8c is intensified, while that in Fig. 8d is weakened. Specifically, the maximum magnitude of precipitation in Fig. 8b reaches 25 K day−1, consistent with the Yang and Majda (2018) result that westward-moving MCSs favor the eastward propagation of convection. The pattern of spatiotemporal variability of precipitation in Fig. 8b features the two-scale structure with eastward-moving planetary-scale envelopes at wavenumber 3 and embedded shorter-wavelength westward-moving synoptic-scale disturbances. Compared with the realistic MJO analog in Fig. 1, the solutions exhibit more intermittency in precipitation intensity and spatiotemporal pattern. In Fig. 8c, the maximum precipitation also intensifies to 19 K day−1, which is associated with the strengthened low-level moisture convergence because of the positive feedback between vertical shear and eddy momentum transfer. The precipitation anomalies are dominated by both eastward- and westward-moving planetary-scale envelopes and exhibit no clear east–west contrast. Based on a similar argument, the precipitation in Fig. 8d is reduced because of the negative feedback between vertical shear and eddy momentum transfer from downshear-moving MCSs. Because of the lack of persistent propagating planetary-scale envelopes, this downshear-moving case is omitted in Figs. 9 and 10.

Fig. 8.
Fig. 8.

As in Fig. 4, but for the cases with and without eddy terms modulated by vertical shear strength . (a) The case without eddy terms. (b)–(d) The cases with eddy terms from MCSs propagating at a slow speed and moving (b) westward, (c) upshear, and (d) downshear.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Fig. 9.
Fig. 9.

As in Fig. 5, but for the cases with and without eddy terms modulated by the vertical shear strength . (top to bottom) The case with (a),(b) no eddy, (c),(d) eddy terms from westward-moving MCSs, and (e),(f) eddy terms from upshear-moving MCSs.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Fig. 10.
Fig. 10.

As in Fig. 6, but for the cases with eddy terms modulated by the vertical shear strength . Panels show the cases with eddy terms from (a) westward-moving MCSs and s = 3.075 m s−1 and (b) upshear-moving MCSs and s = 3.5 m s−1.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Figure 9 shows the log-scale wavenumber–frequency spectra of precipitation and zonal velocity for these three cases. Compared with the symmetric spectra in the control simulation, Figs. 9c and 9d are characterized by significant zonal asymmetry. Specifically, the eastward-moving components are dominated by a continuous band of spectra along a straight line across the origin, which extends from wavenumbers 3 to 10 and periods from 15 to 50 days. In this case, such continuous spectra reflect the intermittent nature of both precipitation and zonal velocity. For the case in Figs. 9e and 9f, the spectra of both precipitation and zonal velocity exhibit significant symmetry under changing sign of x, indicating the prevalence of both the eastward-moving MJO analog and westward-moving reversed MJO analog.

Figure 10 shows vertical cross sections of zonal velocity and potential temperature anomalies. Compared with the weak WWB in Fig. 10a, Fig. 10b features a WWB with a much stronger magnitude at the surface. In the case with eddy terms from westward-moving MCSs, the eddy momentum transfer induces low-level westward (upper-level eastward) momentum forcing, reducing the westerlies to the west but increasing easterlies to the east. In contrast, in the case where MCSs propagate upshear, the positive feedback between vertical shear and eddy momentum transfer tends to strengthen both westerlies (easterlies) to the west (east) at the surface [see Eq. (9)]. Because of the relatively stronger modulation by vertical shear strength to the west, the resulting surface-level westerly winds dominate. In these two cases, both zonal velocity and potential temperature fields exhibit a front-to-rear tilt, because of the in-built transition from congestus to deep to stratiform convection.

In the upshear (downshear)-propagation category, the propagation is in the opposite (same) direction as the shear vector. Upshear and downshear propagation in vertically sheared environments have fundamentally different effects on the structure and transport properties of organized deep convection. The classical MCS (Houze 2004) resides in the upshear category because its direction of propagation is opposite to the lower-tropospheric shear. The rearward tilt of the airflow with height in the classical MCS means that the convective momentum transport has opposite sign from the propagation direction. Consequently, the acceleration of the large-scale flow, which is proportional to the negative of the vertical gradient of momentum transport, is in the direction of propagation for the lower troposphere but in the opposite direction for the upper troposphere (Moncrieff 1992). Therefore, in Fig. 7b, the upshear-propagating systems to the left (right) of the deep convection will enhance the WWB (lower-tropospheric easterlies). The latter will increase moisture evaporation from the ocean surface and likely enhance the cumulus congestus. The remarkably different response to the downshear regime in Fig. 8d compared to the upshear regime in Figs. 8b and 8c may be due to the sign reversal of momentum transport associated with that regime. In summary, the upshear-propagating MCSs ahead and behind the deep convection maximizes their positive impact on the MJO.

4. Parameterization of the upscale impact of MCSs in the idealized GCM

According to section 3, the upscale impact of westward-moving MCSs under the modulation of deep heating excess produces a persistent propagating MJO analog with a two-scale structure and realistic variability of precipitation and winds. In contrast, the upscale impact of upshear-moving MCSs modulated by vertical shear produces a significant WWB. In this section, we provide a basic parameterization of the upscale impact of upshear-moving MCSs modulated by both deep heating excess and vertical shear strength. We test the improvement of key features of the MJO analog in the idealized GCM having clear deficiencies. In particular, we focus on the cases with upscale impact of MCSs propagating upshear at a slow speed, modulated by the effects of both deep heating excess and vertical shear strength. It is worthwhile mentioning that the upscale impact of MCSs is missing from the current cumulus parameterizations implemented in GCMs (Moncrieff et al. 2017), which is another important GCM deficiency besides the reduced congestus and stratiform convection (Pilon et al. 2016; Cao and Zhang 2017). By adding this proposed parameterization to the deficient GCM in Fig. 2, we investigate the improvement of the simulated MJO analog by the upscale impact of MCSs.

a. A basic parameterization of upscale impact of MCSs combining upshear momentum and deep heating excess in the GCM

In reality, the maximum allowable deep heating (conceptually similar to CAPE) should mainly influence the magnitude of mesoscale heating, while the vertical shear strength influences the vertical tilting angles of MCSs (i.e., relative location among shallow congestus and deep and stratiform convection). According to previous results based on the MESD model (Yang and Majda 2018), both conditions control the magnitude and sign of the eddy transfer of momentum and temperature. Here, we combine these two conditions by summing them linearly with a tuning coefficient α and assume that all MCSs propagate upshear.

A basic parameterization for upscale impact of MCSs (eddy transfer of momentum and temperature: and , respectively) is
e14
e15
where is the positive excess of the maximum allowable deep heating, is its RCE value, and represents the vertical shear strength, . Recall that the magnitude coefficients satisfy the relation , where c is the absolute propagation speed of the MCSs. The coefficient α controls the relative importance of and vertical shear in modulating the strength of the eddy transfer of momentum and temperature.

b. Three-way interaction between MJO analog, parameterized upscale impact of MCSs, and background vertical shear on longer time scales

Here, we test the effects of the parameterization by adding it into the idealized GCM having clear deficiencies. Four cases with various value of α in Eqs. (14) and (15) are considered. The magnitude coefficient is fixed at 0.0008 and speed of MCSs c is 0.1 (corresponds to 5 m s−1). To explore the solutions over longer time scales, we extend the integration period to 7000 days and use the last 3000-day model output for analysis. For better visualization, we perform a low-pass filtering by transforming solutions into wavenumber–frequency spectra in Fourier space and only keeping small wavenumber and frequency (large wavelength and period). Only precipitation anomalies at length scales longer than 10 000 km and time scales longer than 30 days are retained.

Figure 11 shows the Hovmöller diagrams for precipitation with various values of α. A large value of α corresponds to stronger modulation by deep heating excess , while a smaller value of α corresponds to stronger modulation by vertical shear strength. One particular interesting feature is the direction switching of the MJO analog for (Fig. 11c) and (Fig. 11d). In Fig. 11c, the MJO analog persistently propagates eastward between day 4000 and day 4500, switches to westward propagation between day 4500 and day 4800, then switches back to eastward propagation between day 5000 and day 5300, and so forth. The period between two eastward (or westward)-propagation phases is about 800 days, much longer than the intraseasonal time scale. Such a QBO-like behavior in the presence of CMT resembles Majda and Stechmann (2009), which also shows periodic direction switching of unstable CCEWs and background mean flow. It is worth clarifying that “QBO-like” means a direction switching that resembles the QBO, not the underlying dynamical processes. Compared with Fig. 11c, the solution in Fig. 11d differs in the duration of persistent propagation of the MJO analog and reversed MJO analog in each phase, exhibiting more chaotic features. For example, the persistently eastward-propagating MJO analog lasts 1200 days between day 5500 and day 6700, while that between day 4500 and day 5000 only lasts 500 days. Unlike Figs. 11c and 11d, the solutions in Figs. 11a and 11b show little QBO-like behavior. Such a clear difference among the cases with large and small values of α indicates the crucial modulation effects of deep heating excess on the eddy terms from upshear-moving MCSs. As shown by Fig. 11e, the realistic MJO analog features persistent eastward propagation over a long time period. In contrast, the solution in Fig. 11f from the deficient GCM shows a transient behavior with alternate eastward and westward propagation during day 4000 and day 5000, standing-wave pattern near day 5500, and persistent westward propagation after day 6000.

Fig. 11.
Fig. 11.

Hovmöller diagrams for planetary–intraseasonal anomalies (deviation from RCE value) of precipitation [; K day−1] between day 4000 and day 7000. (a)–(d) The cases with α = (a) 0.0, (b) 0.4, (c) 0.8, and (d) 1.0. (e),(f) The cases in Figs. 1a and 2a, respectively. The planetary–intraseasonal anomalies are obtained by using a low-pass filter, and only those on length scales larger than 10 000 km and time scales longer than 30 days are retained. The unit is kelvins per day.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

For in Fig. 12a shows a periodic direction switch between eastward-propagating MJO analog and westward-propagating reversed MJO analog. Figure 12b shows the domain-mean zonal winds in the first-baroclinic mode, which also exhibits a periodic direction switch between easterlies and westerlies. Such QBO-like behavior in the domain-mean flow also occurs in the CRM studies by Held et al. (1993). Specifically, during the phase with eastward-moving (westward-moving) MJO analog, the domain-mean zonal winds gradually increase from low-level easterlies (westerlies) to low-level westerlies (easterlies), reaching its maximum magnitude as the MJO analog switches direction. The persistently eastward (westward)-propagation phase is highly correlated with the increasing (decreasing) background zonal winds. According to the governing equations for in Table 1, domain-mean zonal winds vanish in the cases without eddy momentum transfer. Thus, the accumulating contribution by eddy momentum transfer modulated by deep heating excess associated with the MJO analog induces these nonzero domain-mean background flows. Figure 12c shows the time series of domain-mean thermodynamical fields, including first-baroclinic potential temperature, boundary layer equivalent potential temperature, and moisture. The domain-mean first-baroclinic potential temperature decreases at each phase when the MJO analog persistently propagates westward or eastward. Such cooling effects can be explained by eddy heat transfer from MCSs that accumulate in space and time as the MJO analog persistently propagates across the domain.

Fig. 12.
Fig. 12.

Time series of precipitation, zonal velocity, and thermodynamical fields between day 4000 and day 7000. (a) The Hovmöller diagram for planetary–intraseasonal anomalies of precipitation (as in Fig. 11c). (b),(c) Domain-mean zonal velocities and and thermodynamic fields (first-baroclinic potential temperature , boundary layer equivalent potential temperature , and moisture q) during the same period, respectively. Only anomalies of these thermodynamic fields (, , q) on the time scales longer than 50 days are retained by using the low-pass filter. The units of precipitation and zonal velocity are kelvins per day and meters per second, respectively.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

Figure 13a shows the zonal–vertical cross sections of zonal velocity and zonal profiles of deep heating excess and vertical shear strength in the composite eastward-moving planetary-scale envelopes. A significant WWB is produced to the west of the convection center and extends from the surface to the middle troposphere, resembling the realistic MJO analog in Fig. 1 and good GCM simulations (Jiang et al. 2015). This improved WWB of the MJO analog with deeper vertical extent and stronger wind strength by the parameterization should help fix the GCM deficiency with weak and shallow westerlies (see Fig. 13 of Jiang et al. 2015). A crucial feature is the displacement of the peak of deep heating excess to the west of the dashed line, which is consistent with the observation that the convective center of the MJO typically sits over the WWB in easterly vertical shear. Such westward displacement of the deep heating excess preferably modulates eddy momentum transfer in the trailing edge, resulting in a stronger low-level eastward momentum forcing in the trailing edge than in the leading edge. The relatively weak maximum zonal velocity compared to the realistic MJO analog in Fig. 1 is due to the intermittency of the solutions in Fig. 13b. Specifically, westerlies and easterlies are not aligned during the eastward propagation of planetary-scale envelopes and cancel each other after averaging in the eastward-moving reference frame.

Fig. 13.
Fig. 13.

An idealized GCM with clear deficiencies and extra parameterization for upscale impact of MCSs with . (a) Vertical cross sections of composite planetary-scale envelope in the moving reference frames (3.65 m s−1) based on model output between day 5750 and day 6000. Zonal velocity u is shown by color. Dimensionless value of deep heating excess is shown by the pink curve, while that of vertical shear strength is shown by the black curve. The dashed line indicates the longitude with easterly (westerly) vertical shear to its west (east). (b) The Hovmöller diagram for precipitation during this 250-day period. (c),(d) The log-scale wavenumber–frequency spectra for (c) precipitation and (d) surface-level zonal velocity . The units of precipitation and zonal velocity are kelvins per day and meters per second, respectively.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

We identify the following three-way interaction between MJO analog, parameterized upscale impact of MCSs, and background vertical shear:

  1. Eastward-moving MJO analog modulates eddy momentum transfer mainly through deep heating excess.

  2. Because of the westward displacement of the deep heating excess, the resulting eddy momentum transfer with low-level eastward momentum forcing accumulates in space and time and switches the low-level background flow from easterlies to westerlies. This explains why eastward propagation of the MJO analog matches direction switching of background winds from easterlies to westerlies.

  3. Background vertical shear with low-level westerlies favors the westward-moving reversed MJO analog. The underlying mechanism is related to eastward moisture advection, resulting in eastward-moving synoptic-scale disturbances and a westward-moving planetary-scale envelope. This explains why the background winds peaks slightly lead the direction switching of the MJO analog.

  4. Mechanisms similar to 1–3 are repeated but in opposite directions.

Figure 13b shows the Hovmöller diagram for precipitation associated with the eastward-moving MJO analog. The additional parameterization for the upscale impact of MCSs boosts the eastward-moving planetary envelope in terms of spatial size and heavier precipitation compared with the deficient model in Fig. 2a. It also reduces the spatial size of the embedded westward-moving disturbances, improving the two-scale structure of the simulated MJO analog. This improved two-scale structure of the MJO analog by the parameterization fixes the deficient simulation in Fig. 2a with no clear scale separation. However, unlike the deficient simulation here, deficient GCMs often underestimate the embedded synoptic-scale disturbances. Thus, the implication of this parameterization for deficient GCMs needs future investigation. Figures 13c and 13d show the log-scale wavenumber–frequency spectra of precipitation and zonal velocity, akin to the realistic MJO analog in Fig. 1. The spectra of both fields show a clear peak for the eastward-moving planetary-scale envelope at wavenumber 2 and a period of 50 days, with a band of extra spectra extending to higher wavenumbers and frequencies. For westward-moving components, the spectra of precipitation shows a peak at wavenumbers 5–8 and periods of 25–40 days. Extra bands of spectra occur at higher wavenumbers and frequencies, while that of zonal velocity has a more dominant peak at smaller wavenumbers. This improved spectra of the MJO analog with a dominant eastward-moving MJO mode by the parameterization should address the deficient westward-moving planetary–intraseasonal modes and weak synoptic-scale variability (see Fig. 6 of Jiang et al. 2015).

It is interesting to question why the scenario with dominant modulation effects by vertical shear does not exhibit such a QBO-like behavior, considering that the easterly vertical shear in the trailing edge is stronger than in the leading edge. Although the magnitude of westerly vertical shear in the leading edge is weaker, it spans a much broader area. After the eddy momentum transfer in both leading and trailing edges accumulate in space, the resulting background zonal winds are comparable, with no persistent direction preference. Also, the mechanism that background vertical shear with low-level westerlies favoring westward-moving reversed MJO analog differs from observations over the Indian Ocean, presumably because of the idealized two-dimensional model setup without rotation. In a three-dimensional setup, the presence of the Coriolis force would break the zonal symmetry and induce favorable propagation direction (eastward) of the MJO analog.

c. Improving other deficiencies by parameterizing the upscale impact of MCSs

It would be interesting to consider other deficiencies in this idealized GCM due to different parameter values and investigate how the upscale impact of MCSs would improve them. Here, we specifically focus on two deficiencies. The first one has almost the same parameters as the realistic simulation in Fig. 1 except for the coefficient of the second baroclinic mode in linear moisture convergence (optimal value is 0.6) and the background moisture stratification (optimal value is 1.0). This deficiency due to the reduced coupling of the second-baroclinic mode mimics the underestimated role of shallow convection in the cumulus parameterization in the GCMs (Zhang and Song 2009). The second deficiency differs from the realistic simulation in Fig. 1 only by the inverse convective buoyancy time scale of deep clouds (optimal value is 12). These two deficiencies are modified by adding the parameterization under the same configuration as Fig. 13.

Figure 14a shows the Hovmöller diagram for precipitation in the first deficiency during a 200-day period. The solution is characterized by eastward-moving precipitating events in wavenumber 5 and a period of 40 days. In contrast, the improved simulation by the parameterization in Fig. 14b shows a clear two-scale structure with eastward-moving planetary-scale envelopes and embedded westward-moving synoptic-scale disturbances. The maximum precipitation increases to 30 K day−1. Over a longer period, this improvement also shows a QBO-like behavior with direction switching in Fig. 14c, similar to Fig. 12a. Figure 14d shows the Hovmöller diagram for precipitation in the second deficiency. The solution is characterized by periodic eastward-moving events in wavenumber 5, which is much shorter than the observed MJO in wavenumbers 1–3 (Kiladis et al. 2009). After adding the parameterization, these eastward-moving events have larger spatial scales in wavenumber 3 with more intermittency in Fig. 14e. Interestingly, these planetary-scale envelopes exhibit persistent eastward propagation over the longer period in Fig. 14f, possibly because of the stronger coupling with the second-baroclinic mode.

Fig. 14.
Fig. 14.

Hovmöller diagrams for precipitation (K day−1) from deficient GCMs and improved simulations by the parameterization of upscale impact of MCSs. (a) The solutions from the deficient GCM with and between day 3800 and day 4000. (b) The improved simulation by the parameterization during the same period. (c) Planetary–intraseasonal anomalies from the improved simulation between day 4000 and day 7000 by using the same low-pass filter as Fig. 11. (d)–(f) As in (a)–(c), respectively, but for the other deficient GCM with . Panels in each column share the same color bar in the bottom.

Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0260.1

5. Concluding discussion

A simple multicloud model for the MJO analog and intraseasonal variability above the equator is studied. With reduced congestus and stratiform heating, the resulting solutions from this simple model are used as an idealized GCM having clear deficiencies. By adding eddy transfer of momentum and temperature predicted by the MESD model, we assess the upscale impact of MCSs on three key features of the MJO analog: persistent propagation of a two-scale structure, realistic planetary–intraseasonal variability in precipitation and winds, and a significant WWB. We then introduce a basic parameterization of upscale impact of upshear-moving MCSs modulated by the effects of deep heating excess and vertical shear strength and test its effects in the idealized deficient GCM.

Table 5 summarizes results reported in this paper regarding the above three key features of the MJO analog in the idealized deficient GCM. Compared to the realistic MJO analog, the idealized deficient GCM fails to reproduce these three features, thereby mimicking the significant bias of the simulated MJO in present-day GCMs. According to Khouider et al. (2012), MCSs and squall lines are prominent in the convectively active regions of the MJO envelope, indicating the modulation of the MCSs by the MJO convective center. The eddy transfer of momentum and temperature from westward-moving MCSs traveling at a slow speed (5 m s−1) improves the two-scale structure of the eastward-moving MJO analog and space–time variability of precipitation and winds but fails to strengthen the WWB. This is consistent with the basic MESD model (Yang and Majda 2018; i.e., westward-moving MCSs embedded in the large-scale convective envelope provide favorable conditions for convection to the east) that promotes the eastward-moving convective envelope. On the other hand, vertical shear plays a crucial role in organized tropical convection (Moncrieff 1992), including the influence on its front-to-rear-tilted structure and propagation directions (Moncrieff and Liu 1999; Stechmann and Majda 2009). In particular, eddy transfer of momentum and temperature from upshear-moving MCSs induces a significant WWB in the middle and west of the MJO analog. This is due to the two-way feedback between environmental easterly vertical shear and the embedded eddy momentum transfer with low-level eastward momentum forcing. The eddy transfer of momentum and temperature modulated by the effects of vertical shear strength alone fails to reproduce the two-scale structure of the MJO analog and a realistic space–time variability of precipitation and winds.

To incorporate those improvements in global models, we provide a basic parameterization of the upscale impact of upshear-moving MCSs that linearly combines the modulation effects of deep heating excess and vertical shear strength. This basic parameterization shares goals similar to the MCSP introduced by Moncrieff et al. (2017), notably, representing the upscale effects of organized tropical convection that are missing from contemporary parameterizations in GCMs. The main purpose of the Moncrieff et al. (2017) prototype version of MCSP was to demonstrate the upscale effects of top-heavy convective heating and momentum transport in the simplest possible manner in order to provide proof of concept. This was achieved by focusing on eastward propagation and a full GCM. The results of this present paper will be valuable for the future development of MCSP, because the heating and CMT (i.e., upscale impact of MCSs) have been quantified in simple ways. However, this basic parameterization differs from the MCSP in several aspects that significantly improve the feasibility and reliability of the parameterization. First, it considers both deep heating excess (a similar concept as CAPE) and vertical shear strength in modulating the upscale impact of MCSs, while the parameterized CMT in MCSP has constant magnitude over convective regions. Second, it assumes eddy transfer of momentum and temperature from MCSs that propagate upshear (opposite to vertical shear direction), allowing vertical shear to determine the propagation direction of MCSs and the sign of eddy momentum transfer. Third, it highlights the crucial contribution of the eddy transfer of temperature as predicted theoretically by the MESD model.

The implementation of this basic parameterization of upscale impact of MCSs in the idealized deficient GCM shows significant improvement in capturing key features of the MJO. A further examination of a longer-period simulation reveals a three-way interaction between the MJO analog, the parameterization of upscale impact of MCSs, and the background mean flow. The westward-displaced deep heating excess in the eastward-moving MJO analog favors eddy momentum transfer with low-level eastward (upper-level westward) momentum forcing. The effects of the eddy momentum transfer accumulate in space and time and gradually switch the direction of the background mean flow, which, in turn, alters the propagation directions of the MJO analog. Under this three-way interaction mechanism, the background mean flow exhibits a QBO-like behavior, resembling a similar phenomenon in CRM simulations (Held et al. 1993). Although in reality the Coriolis force would break the zonal symmetry, such a three-way interaction mechanism may shed light on the interactions between the eastward-moving MJO, upscale impact of MCSs, and climatological vertical shear.

The basic parameterization of upscale impact of MCSs can be elaborated in several ways and tested in a hierarchy of models. Besides the first-baroclinic mode, it is also interesting to investigate the effects of eddy transfer of momentum and temperature due to higher baroclinic modes, as shown by studies based on the MESD model (Yang and Majda 2018) and reanalysis data (Oh et al. 2015). A different scenario to assess the upscale impact of MCSs on the planetary–intraseasonal variability includes the Walker circulation over the warm pool. Furthermore, we would like to test effects of this basic parameterization of upscale impact of MCSs in more comprehensive GCMs, particularly the improvement of GCM deficiencies in simulating key features of the MJO. It should be interesting to see whether the proposed parameterization will improve the MJO mode with a clear two-scale structure in GCMs that typically fail to capture MJO-like coherent propagating structures.

Acknowledgments

This research of A.J.M. is partially supported by the office of Naval Research ONR MURI N00014-12-1-0912 and the Center for Prototype Climate Modeling (CPCM) in New York University Abu Dhabi (NYUAD) Research Institute. Q.Y. is funded as a postdoctoral fellow by CPCM in NYUAD Research Institute. M.W.M. acknowledges NASA Grant NNX13AO39G, Subcontract 49A03A with the City College of New York. NCAR is sponsored by the National Science Foundation.

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