1. Introduction
Tropical rainfall is largely controlled by convectively coupled equatorial waves (CCEWs), whose dynamical and convective morphology exhibit self-similarity across multiple spatial and temporal scales (Tao and Moncrieff 2009). Among these CCEWs, the convectively coupled Kelvin wave (CCKW) is an important component of synoptic variability, which peaks along the latitude of the intertropical convergence zone (ITCZ), Africa, the Indian Ocean, and South America (Kiladis et al. 2009). The early observational studies about CCKWs date back to the 1970s (Wallace and Chang 1972; Zangvil 1975), when satellite-derived data on cloud brightness were utilized to define the dominant scales of motion in the tropics. The dynamical fields associated with CCKWs are characterized by low-level wind convergence leading upper-level wind divergence in a front-to-rear tilt (Yang et al. 2007a). Such horizontal and vertical structures of CCKWs are explained by stratiform instability mechanism (Mapes 2000; Majda and Shefter 2001) and also simulated by the multicloud model (MCM; Khouider and Majda 2006 a,b,c, 2008a,b; Khouider et al. 2010, 2011). Besides governing a large fraction of tropical rainfall, CCKWs are also known to interact strongly with the Madden–Julian oscillation (MJO; Straub et al. 2006) and link synoptic-scale variation of the Atlantic ITCZ with precipitation anomalies in South America (Wang and Fu 2007).
Instead of organizing on the synoptic scale alone, the hierarchical structure of CCKWs was identified by Nakazawa (1988) and further explained as an eastward-moving synoptic-scale convective envelope (a supercluster) with embedded westward-moving mesoscale disturbances (cloud clusters). During the 1997 Pan American Climate Studies (PACS) Tropical East Pacific Process Studies (TEPPS), it was observed that the large-scale convective envelope of a CCKW in the eastern Pacific ITCZ consists of many smaller-scale, westward-moving convective elements (Straub and Kiladis 2002). Similar multiscale coherent structures of tropical convection are also observed in westward-propagating 2-day waves (Chen et al. 1996). These small-scale convective elements are categorized as mesoscale convective systems (MCSs), the dominant heavy-rain producers in the tropics and subtropics (Tao and Moncrieff 2009). Squall-line systems are one particular type of MCS and propagate at various speeds and directions (Houze 1975, 1977, 2004). In general, the multiscale coherent structure of CCKWs with embedded mesoscale disturbances are illustrated in the conceptual diagram in Fig. 1.
Conceptual diagram for a CCKW with embedded mesoscale disturbances. (left) An eastward-moving CCKW (blue) on the synoptic scale, where the rectangular cuboid denotes a mesoscale domain. (right) An MCS propagating at a tilt angle γ in the mesoscale domain (zoom in of the rectangular cuboid in the left panel).
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
In spite of such progress in the observational studies, simulating multiscale coherent structures of CCKWs with embedded mesoscale disturbances is still a challenging problem. With the development of computing resources and cloud modeling, several attempts have been done to reproduce these multiscale features by using cloud-resolving models (CRMs) in two-dimensional model setup. For example, in the trade wind regime with a strong easterly background flow, large-scale organization of tropical deep convection with numerous MCSs is investigated in idealized two-dimensional cloud-resolving simulations of Grabowski and Moncrieff (2001). The convective momentum transport (CMT) from mesoscale disturbances is identified as a key process responsible for the large-scale organization of convection. In contrast, in the state-of-rest regime with zero mean flow, upscale transport of horizontal momentum by coherent eddy circulations is found to be small in the cloud-resolving simulations of Tulich and Mapes (2008). Besides, the evidence of energy exchange through momentum transport between mesoscale disturbances and synoptic-scale propagating waves is also presented in the Weather Research and Forecasting (WRF) Model (Khouider and Han 2013). There is still no clear understanding about scale interactions between synoptic-scale circulation and mesoscale disturbances. Particularly, how do mesoscale disturbances that propagate at various speeds and directions impact synoptic-scale circulation? Answering this question cannot only improve our understanding about multiscale coherent structure of tropical convection but also provide valuable intuition for convective parameterization in global climate models (GCMs).
Because of limited computing resources, it is a huge challenge for present-day GCMs in coarse resolutions to explicitly resolve those mesoscale disturbances inside large-scale organization of convection (Jiang et al. 2015). One hypothesis to explain the significant discrepancies of precipitation in GCMs is the inadequate treatment of mesoscale disturbances and their upscale impact on the large-scale organization of convection. In fact, much progress about parameterization of organized tropical convection in GCMs has already been made. Considering the fact that countergradient vertical transport of horizontal momentum by organized convection increases wind shear and transports kinetic energy upscale, Moncrieff et al. (2017) set the archetypal dynamical models of slantwise overturning (Moncrieff 1981, 1992) into a parameterization for organized convection and its upscale effects on the resolved large-scale circulation. However, since the slantwise overturning is modeled in a two-dimensional framework, it is unclear how to parameterize the associated vertical transport of horizontal momentum if organized tropical convection has a complete three-dimensional structure and propagates at various speeds and directions. Also, the vertical structure of eddy transfer of temperature and its relative significance to impact synoptic-scale circulation is not well understood. Interestingly, the MCM (Khouider and Majda 2006c,b,a, 2008b; Khouider et al. 2010, 2011) based on three cloud types (congestus, deep, and stratiform) simulates realistic features of shear-parallel MCSs in a three-dimensional structure (Khouider and Moncrieff 2015), which are commonly observed in the ITCZ. Furthermore, the stochastic multicloud model (SMCM) successfully captures the variability due to multiscale organized convective systems, especially synoptic and intraseasonal variability (Goswami et al. 2017).
The goals of this paper are as follows: first, using a simple multiscale model to capture multiscale structures of CCKWs with embedded mesoscale disturbances and assess the associated upscale impact of mesoscale disturbances through eddy transfer of momentum and temperature; second, theoretically predicting the upscale impact of mesoscale disturbances propagating at various tilt angles and speeds on the mean-heating-driven Kelvin waves in terms of favorability for convection in a moist environment and characteristic morphology; third, exploring whether the front-to-rear-tilted vertical structure of CCKWs can still be induced by eddy transfer of momentum and temperature in the presence of upright mean heating; and last, providing a useful framework to explain CMT and synoptic-scale circulation as simulated in CRMs.
The simple multiscale model used here is the mesoscale equatorial synoptic-scale dynamics (MESD) model, originally derived by Majda (2007). The MESD model can be used to model cluster–supercluster interactions across mesoscale and synoptic scale and incorporate them together in a simple multiscale framework. In fact, the two-dimensional version of the MESD model has already been used to model scale interactions across mesoscale and synoptic scale (Yang and Majda 2017) and concluded several crucial results as follows. It successfully reproduces many key features of synoptic-scale circulation response in a front-to-rear tilt and compares well with results from a two-dimensional CRM (Grabowski and Moncrieff 2001). In the presence of elevated upright mean heating, the tilted vertical structure of synoptic-scale circulation can still be induced by the upscale impact of mesoscale disturbances. When the large-scale convective envelope propagates faster, the upscale impact becomes less important, and mean-heating-driven circulation response dominates. Such a result successfully explains discrepancies of numerical results in CRMs. Specifically, the simulations by Grabowski and Moncrieff (2001) in the trade wind regime with slowly propagating large-scale organization of convection feature significant CMT, while those by Tulich and Mapes (2008) in the state-of-rest regime with fast-propagating wave packets conclude that the upscale transport of horizontal momentum by coherent eddy circulations is small. When the westward-propagating mesoscale heating has an unrealistic westward tilt with height, positive potential temperature anomalies are induced in the leading edge, suppressing shallow convection in a moist environment.
In this paper, several crucial results are achieved by using the three-dimensional version of the MESD model. First, explicit expressions for eddy momentum transfer (EMT) and eddy heat transfer (EHT) are obtained. Second, when mesoscale disturbances propagate at tilt angles of 110°–250°, negative potential temperature anomalies are induced in the leading edge, providing favorable conditions for shallow convection. Third, in the presence of both top-heavy and bottom-heavy upright mean heating, when the mesoscale heating propagates at tilt angles of 120°–240°, the front-to-rear-tilted vertical structure of synoptic-scale circulation can still be induced by eddy terms. Fourth, in the case with fast-propagating mesoscale heating, positive potential temperature anomalies are induced in the lower troposphere, suppressing convection in a moist environment. Last, by considering slowly eastward-propagating mesoscale disturbances driven by baroclinic mesoscale heating and barotropic momentum forcing, the MESD model successfully reproduces the vertical profile of CMT and CCKWs as simulated in a WRF simulation (Khouider and Han 2013).
The rest of this paper is organized as follows. Section 2 summarizes properties of the MESD model. Section 3 discusses the prescribed mesoscale heating propagating at a tilt angle and mesoscale fluctuations of flow field and the associated eddy transfer of horizontal momentum and temperature. Section 4 shows the synoptic-scale circulation response to the eastward-propagating mean heating with embedded mesoscale heating propagating at a tilt angle. Sections 5 and 6 consider two different scenarios with upright mean heating and fast-propagating mesoscale heating, respectively. In section 7, the MESD model is used to directly compare with a WRF simulation for CCKWs in terms of CMT and large-scale circulation response. The paper ends with a concluding discussion.
2. Properties of the MESD model
In general, the multi-spatial-scale, multi-time-scale simplified asymptotic models are derived systematically from the equatorial primitive equations, providing a useful framework to understand multiscale phenomenon (Majda and Klein 2003; Majda 2007; Yang and Majda 2014; Majda and Yang 2016). In particular, the MESD model, originally derived by Majda (2007), describes the multitime, multispace interaction from the mesoscale to the synoptic scale, which is useful for modeling CCEWs with embedded mesoscale disturbances. Specifically, the MESD model consists of two groups of equations, one of which governs mesoscale gravity waves and the other one of which governs synoptic-scale equatorial waves including Kelvin waves, Rossby waves, mixed Rossby–gravity waves, and gravity waves in the baroclinic mode, as well as barotropic Rossby waves (Majda 2003).
The MESD model is derived systematically from the primitive equations on an equatorial β plane by following the multiscale asymptotic procedure (Majda and Klein 2003). The derivation details can be found in Majda (2007). Equations (1) describe mesoscale fluctuations driven by some momentum and thermal forcing, while Eqs. (2) describe synoptic-scale circulation driven by some momentum and thermal forcing and momentum damping as well as eddy transfer of momentum and temperature. The eddy transfer of momentum and temperature 
Physical parameters and dimensional scaling in the MESD model.
a. Mesoscale gravity waves in the baroclinic modes
b. Mesoscale fluctuations driven by barotropic momentum forcing
c. Synoptic-scale equatorial waves
The governing equations for synoptic-scale circulation in Eqs. (2) are linear primitive equations on an equatorial β plane, forced by eddy transfer of momentum and temperature, momentum forcing, and thermal forcing. Under the rigid-lid boundary conditions, the resulting equatorial waves arising from the linear primitive equations have been well studied (Matsuno 1966; Majda 2003) and also used as a methodology to isolate horizontal and vertical structures of CCEWs (Yang et al. 2007a,b,c). In spite of moist processes, these solutions share crucial features of horizontal structures and dispersion characteristics of CCEWs observed in nature (Kiladis et al. 2009).
3. Mesoscale disturbances propagating at an angle to the zonal direction
In the tropics, it is frequently observed that numerous small-scale convective elements are embedded in CCEWs such as Kelvin waves (Straub and Kiladis 2002) and 2-day waves (Haertel and Kiladis 2004). These small-scale disturbances, categorized as MCSs (Houze 2004), are typically characterized by cloud clusters and release a large amount of latent heat during tropical precipitation. In fact, the multicloud models based on three types of cloudiness (congestus, deep, and stratiform) have successfully simulated multiscale features of CCEWs in the tropics (Khouider and Majda 2006c,a, 2007, 2008a).
Squall-line systems are one particular type of MCS and consist of a squall line forming the leading edge of the system and a trailing anvil cloud region. It has been recognized for a long time that there is a life cycle of three types of clouds from congestus to deep convective to stratiform in a squall-line system. Moreover, precipitation falling from the trailing anvil cloud was stratiform and accounts for 40% of the total rain from the squall-line system (Houze 1977). Unlike eastward- and westward-moving equatorial waves, squall-line systems actually propagate at arbitrary tilt angles (Houze 1977) and various speeds of 5–20 m s−1 (Houze 1975).
In this section, the equations for mesoscale fluctuations in Eqs. (1) are used to model the mesoscale disturbances embedded in the synoptic-scale convective envelope. The rigid-lid boundary condition is imposed at the surface and top of the troposphere. The solutions are assumed to be periodic in the horizontal domain and have finite extent in the vertical direction.
a. Mesoscale heating propagating at a tilt angle
Vertical profile of mesoscale heating in the new reference frame. (a) The normal reference frame is denoted by the x axis (east) and y axis (north) in solid lines. The new reference frame with the x′ axis and y′ axis in dashed lines is derived by rotating the normal reference frame counterclockwise by an angle γ. The red thick arrow shows the propagation direction of mesoscale heating. (b) The vertical profile of mesoscale heating in the new reference frame. The dimensional unit is 100 K day−1.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
b. Mesoscale velocity and potential temperature anomalies
Figure 3a shows vertical profiles of zonal and vertical velocity along the propagation direction. Upward motion prevails in heating regions, and downward motion prevails in cooling regions. Such a deep slantwise ascending layer is considered to be crucial for maintaining a mature MCS (Moncrieff 1978, 1981; Crook and Moncrieff 1988; Moncrieff 1992). Besides, the maximum zonal and vertical velocity occurs in the upper troposphere where the maximum magnitude of mesoscale heating is reached. In addition, at the lower troposphere, wind divergence (convergence) is located in the mesoscale cooling (heating) regions, while such a relation is reversed in the upper troposphere.
Vertical profiles of (a) zonal velocity and vertical velocity (arrows) and (b) potential temperature anomalies (contours; interval: 0.1 K) along the propagation direction of mesoscale heating. The colors in both panels show mesoscale heating. The maximum magnitudes of zonal and vertical velocities are 3.72 and 0.47 m s−1, respectively. The dimensional unit of mesoscale heating is 100 K day−1.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
Figure 3b shows the vertical profile of potential temperature anomalies along the propagation direction. Similarly, potential temperature anomalies also have a front-to-rear tilt. Besides, the vertical structure of potential temperature anomalies is significantly dominated by the second baroclinic mode. In heating regions such as the longitude 1.9 × 102 km, positive anomalies are sitting on top of negative anomalies, resembling the observation that in an MCS, latent heat is released on top because of stratiform precipitation, and cooling effects are induced below because of rain evaporation (Houze 2004).
c. Eddy momentum transfer and eddy heat transfer
Figure 4a shows the vertical profile of eddy zonal momentum flux 
Vertical profiles of (a) eddy zonal momentum transfer (blue) and the associated eddy flux (red) and (b) eddy heat transfer (blue) and the associated eddy flux (red). One dimensionless unit of eddy zonal momentum and eddy heat transfer is 15 m s−1 day−1 and 10 K day−1, respectively.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
Figure 4b shows the vertical profile of eddy potential temperature flux 
4. Convectively coupled Kelvin waves with embedded mesoscale disturbances
As discussed in section 3, mesoscale disturbances of tropical convection in a front-to-rear tilt tend to generate eddy transfer of horizontal momentum and temperature, further driving the synoptic-scale circulation. In this section, the synoptic-scale circulation response to both the upscale impact of mesoscale fluctuations and mean heating is discussed, in terms of low-tropospheric potential temperature anomalies and horizontal velocity and temperature at various levels.
Here, the equations for synoptic-scale circulation in Eqs. (2) are used. As for boundary conditions, the solutions are assumed to be periodic in the zonal direction and decay as the latitude increases. The rigid-lid boundary condition is imposed at the surface and top of the troposphere. The actual numerical simulations are implemented in the domain (longitude, latitude, height) = (
a. Synoptic-scale mean heating and mesoscale heating modulated by a large-scale envelope
Vertical profile of mean heating at the equator: (a) tilted mean heating, (b) top-heavy upright mean heating, and (c) bottom-heavy upright mean heating. The dimensional unit of mean heating is 10 K day−1.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
b. Potential temperature anomalies in the lower troposphere
In a moist environment, negative potential temperature anomalies in the lower troposphere provide favorable conditions for convection through decreasing saturation rate of vapor, CIN, and increasing convective available potential energy (CAPE). As a counterpart of that, positive anomalies provide unfavorable conditions for convection. Here, lower-tropospheric potential temperature anomalies induced by mean heating and eddy terms (EZMT, EMMT, and EHT) at various tilt angles are discussed. The goal here is to understand the upscale impact of mesoscale disturbances that propagate at various tilt angles on lower-tropospheric potential temperature and interpret the associated favorability for convection in a moist environment. Considering the fact that flow fields will just be mirror symmetric if the tilt angle is reflected about the equator, the cases at tilt angles 
Figure 6a shows the horizontal profile of lower-tropospheric potential temperature anomalies induced by mean heating at 2.62 km, which is characterized by warm anomalies in the middle and cold anomalies to the east and west. Figures 6b–h show horizontal profiles of lower-tropospheric potential temperature anomalies induced by eddy terms. As summarized by Fig. 6i, the upscale impacts of mesoscale disturbances that propagate at various tilt angles are divided into three categories. In the blue region (110°–250°) such as Figs. 6b and 6c, eddy terms induce negative lower-tropospheric potential temperature anomalies in the leading edge of the convective envelope. In a moist environment, such lower-tropospheric negative anomalies provide favorable conditions for convection, initializing new shallow convection in the leading edge and preconditioning deep convection as the whole convective envelope propagates eastward. In the pink region (70°–110° and 250°–290°) such as Figs. 6d–f, eddy terms induce positive lower-tropospheric potential temperature anomalies off the equator in the leading edge, providing unfavorable conditions for shallow convection and resulting in an asymmetric meridional profile of the convective envelope. In the red region (0°–70° and 290°–360°) such as Figs. 6g and 6h, eddy terms induce positive lower-tropospheric potential temperature anomalies in the leading edge. In a moist environment, such strong positive anomalies provide unfavorable conditions for convection, suppressing shallow convection and further destroying coherent structures of CCKWs. This result explains the fact that most of the mesoscale disturbances in CCKWs propagate westward in nature (Nakazawa 1988; Straub and Kiladis 2002) instead of eastward.
Horizontal profiles of potential temperature anomalies in the lower troposphere (2.62 km) in longitude–latitude diagrams. (a) Potential temperature anomalies induced by tilted mean heating. (b)–(h) Potential temperature anomalies induced by eddy terms at tilt angles of 180°, 135°, 110°, 90°, 70°, 45°, and 0°, respectively. (i) Favorability of convection in different tilt-angle cases (blue: favorable; pink: unfavorable, asymmetric; red: unfavorable). The dimensional unit of potential temperature anomalies is kelvins.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
c. Horizontal velocity and pressure perturbation at different levels
Here, horizontal velocity and pressure perturbation induced by mean heating and eddy terms at various tilt angles are discussed and interpreted in terms of their impact on characteristic morphology of CCKWs, favorability for tropical cyclogenesis, and moisture transport in a moist environment. The goal is to understand how the upscale impact of mesoscale disturbances that propagate at different tilt angles modifies the mean-heating-driven circulation.
Figure 7a shows the horizontal profile of mean-heating-driven horizontal velocity and pressure perturbation at the surface, which are characterized by zonal wind convergence and an east–west dipole of pressure perturbation. By comparing the flow fields induced by eddy terms with the mean-heating-driven circulation, several crucial results are obtained. In the cases with tilt angles of 180° and 135° in Figs. 7b and 7c, the westerlies induced by eddy terms tend to strengthen (weaken) the westerlies (easterlies) from the mean-heating-driven circulation, pushing the longitude of wind convergence farther east. Such strengthened westerlies in the convection region led by wind convergence to the east resemble the typical wind field associated with CCKWs at the surface (Yang et al. 2007a). Meanwhile, eddy terms induce negative pressure perturbation in the leading edge, resulting in convergence of winds and moisture and providing favorable conditions for tropical cyclogenesis. In the cases with tilt angles of 110°, 90°, and 70° in Figs. 7d–f, northeasterly winds are induced by eddy terms in the Northern Hemisphere, introducing meridional asymmetry of mean-heating-driven circulation with strengthened easterlies off the equator. In the cases with tilt angles of 45° and 0° in Figs. 7g and 7h, significant easterlies induced by eddy terms tend to weaken (strengthen) the westerlies (easterlies) from the mean-heating-driven circulation. Also, positive pressure perturbation induced by eddy terms provides unfavorable conditions for tropical cyclogenesis.
Horizontal profiles of horizontal velocity (arrows) and pressure perturbation (colors) at the surface in longitude–latitude diagrams. (a) Flow field induced by mean heating. (b)–(h) Flow field induced by eddy terms at tilt angles of 180°, 135°, 110°, 90°, 70°, 45°, and 0°, respectively. The dimensional units of horizontal velocity and pressure perturbation are meters per second and 100 m2 s−2 per mass. The maximum magnitude of horizontal velocity is shown in above each panel.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
Figure 8 shows horizontal profiles of horizontal velocity and pressure perturbation in the lower troposphere. In the cases with tilt angles of 180° and 135° in Figs. 8b and 8c, the lower-tropospheric easterlies induced by eddy terms tend to strengthen the inflow of mean-heating-driven circulation in the leading edge, bringing moisture into the convective envelope and preconditioning deep convection in a moist environment. In the cases with tilt angles of 110°, 90°, and 70° in Figs. 8d–f, eddy terms induce significant westerlies in the Northern Hemisphere with positive pressure perturbation, resulting in meridional asymmetry of dynamical fields. In the cases with tilt angles of 45° and 0° in Figs. 8g and 8h, the strong westerlies and positive pressure perturbation induced by eddy terms tend to destroy the mean-heating-driven circulation.
As in Fig. 7, but in the lower troposphere (5.24 km).
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
Figure 9 shows horizontal profiles of horizontal velocity and pressure perturbation in the upper troposphere. In particular, the flow fields induced by eddy terms at tilt angles of 180° and 135° in Figs. 9b and 9c are characterized by significant westerly winds in the upper troposphere, which tend to strengthen the outflow in the leading edge, result in strong vertical shear of zonal winds between the lower and upper troposphere, and provide favorable conditions for convection (Moncrieff 1978). Figure 10 shows horizontal profiles of horizontal velocity and pressure perturbation at the top. In particular, in the cases with tilt angles of 180° and 135° in Figs. 10b and 10c, easterlies and negative pressure perturbation induced by eddy terms tend to strengthen the easterly winds in the mean-heating-driven circulation in the trailing edge but weaken the westerly winds in the leading edge.
As in Fig. 7, but in the upper troposphere (10.47 km).
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
As in Fig. 7, but at the top of the troposphere (15.70 km).
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
5. Upright mean heating
In contrast to tilted mean heating in Fig. 5a, upright mean heating is characterized by upright vertical profile of diabatic heating, which is released from deep and congestus/stratiform convection with no phase lag between them. The goal of this section is to explore whether the upward-/westward-tilted vertical structure of zonal velocity, potential temperature anomalies can still be induced by eddy terms in the presence of upright mean heating. Specifically, both top-heavy and bottom-heavy upright mean heating shown in Figs. 5b and 5c are considered. All model setup is exactly the same as section 4. It turns out that the relative location between the mean heating and convective envelope for mesoscale heating plays an important role here; thus, it is carefully chosen below.
a. Top-heavy upright mean heating
Figure 11a shows the vertical profile of potential temperature anomalies induced by top-heavy upright mean heating. Although the upright mean heating has only significant anomalies in the upper troposphere, the resulting potential temperature anomalies feature significant second baroclinic mode to the east and cold upper-tropospheric anomalies to the west. Figures 11c–f show vertical profiles of potential temperature anomalies induced by eddy terms at various tilt angles. The resulting anomalies are dominated by significant third baroclinic mode, whose signs change as the tilt angle switches from westward to eastward. Figures 11g–j show vertical profiles of total potential temperature anomalies induced by mean heating and eddy terms. As summarized in Fig. 11b, all these cases at various tilt angles are divided into two categories. In the blue region (120°–240°) such as Figs. 11g and 11h, the tilted vertical structure of potential temperature anomalies can still be induced by eddy terms in the presence of top-heavy upright mean heating. The corresponding total zonal velocity in Figs. 12f and 12g resembles the zonal winds in large-scale organization of convection as simulated in the cloud-resolving model (Grabowski and Moncrieff 2001). In the red region (0°–120° and 240°–360°) such as Figs. 11i, 11j, 12h, and 12i, no tilted vertical structure of potential temperature anomalies and zonal velocity are induced by eddy terms in the presence of top-heavy upright mean heating. The upper-tropospheric zonal velocity induced by eddy terms in Figs. 12d and 12e tends to strengthen upper-tropospheric easterlies and westerlies in the leading edge from the mean-heating-driven circulation.
Vertical profiles of potential temperature anomalies at the equator in longitude–height diagrams. (a) Potential temperature anomalies (colors) induced by top-heavy upright mean heating (contours). (c)–(f) Potential temperature anomalies induced by eddy terms at tilt angles of 180°, 135°, 90°, and 0°, respectively. (g)–(j) Total anomalies induced by both mean heating and eddy terms at tilt angles of 180°, 135°, 90°, and 0°, respectively. (b) The upscale impact of mesoscale fluctuations at different tilt angles on the tilted vertical structure (blue: tilted; red: destroyed). The dimensional unit of potential temperature anomalies is kelvins.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
Vertical profiles of zonal velocity at the equator in longitude–height diagrams. (a) Zonal velocity (colors) induced by top-heavy upright mean heating (contours). (b)–(e) Zonal velocity induced by eddy terms at tilt angles of 180°, 135°, 90°, and 0°, respectively. (f)–(i) Total zonal velocity induced by both mean heating and eddy terms at tilt angles 180°, 135°, 90°, and 0°, respectively. The dimensional unit of zonal velocity is meters per second.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
b. Bottom-heavy upright mean heating
Figure 13a shows the vertical profile of potential temperature anomalies induced by bottom-heavy mean heating. The resulting potential temperature anomalies share the similar spatial pattern as Fig. 11a but in the opposite sign. As summarized in Figs. 13b and 14b, all these cases at various tilt angles are divided into two categories. In the blue region (120°–240°) such as Figs. 13g, 13h, 14g, and 14h, the tilted vertical structure of zonal velocity and potential temperature anomalies can still be induced by eddy terms in the presence of bottom-heavy upright mean heating. Specifically, Figs. 13g and 13h show tilted positive potential temperature anomalies with its maximum value in the lower troposphere, while Figs. 14g and 14h show an upward/westward inflow layer with easterlies. Easterly winds are also noted near the top. However, in the red region (0°–120° and 240°–360°) such as Figs. 13i, 13j, 14i, and 14j, no tilted vertical structure of zonal velocity and potential temperature anomalies are induced by eddy terms.
As in Fig. 11, but for bottom-heavy upright mean heating case.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
As in Fig. 12, but for bottom-heavy upright mean heating case.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
6. Faster-propagating mesoscale heating
The early observation about MCSs such as tropical squall lines dates back to 1970s. For example, during phase III of GATE, four squall lines passed over the U.S. NOAA ship Researcher (Houze 1975). According to Houze (1975), propagating speeds of squall line systems vary from 5 to 20 m s−1. In section 4, the propagation speed of mesoscale heating is set as 5 m s−1. According to Eq. (21), such a slow propagation speed of mesoscale heating means that the synoptic-scale circulation response to EMT is much stronger than that to EHT. In this section, a faster-propagating mesoscale heating (15 m s−1) is considered so that the synoptic-scale circulation response to EHT dominates. The goal is to understand the upscale impact of fast-propagating mesoscale disturbances on synoptic-scale potential temperature anomalies.
Figure 15a shows the vertical profile of potential temperature anomalies induced by mean heating. Similar to mean heating, the resulting potential temperature anomalies are also characterized by a front-to-rear tilt. As shown in Fig. 15b, potential temperature anomalies induced by EHT are dominated by the third baroclinic mode with cold anomalies in the middle troposphere and warm anomalies in both upper and lower troposphere. Potential temperature anomalies induced by EZMT at the tilt angles of 180° and 0° are manifested by the third baroclinic mode but in the opposite signs in Figs. 15c and 15d. Figures 15e and 15f show total potential temperature anomalies induced by eddy terms at the tilt angles of 180° and 0°. In these two cases, the both anomalies are characterized by warm anomalies in the lower troposphere, providing unfavorable conditions for shallow convection in a moist environment. Specifically, potential temperature anomalies induced by EHT in Fig. 15b compete with those induced by EZMT in Fig. 15c. Thus, in the case with westward-propagating mesoscale heating in Fig. 15e, the total anomalies induced by eddy terms have weak magnitude in the trailing edge. In contrast, potential temperature anomalies induced by EHT in Fig. 15b and those induced by EZMT in Fig. 15d strengthen each other. Thus, in the case with eastward-propagating mesoscale heating in Fig. 15f, the total potential temperature anomalies induced by eddy terms have strong magnitude and are mostly located in the leading edge. Last, according to Fig. 15e, the maximum magnitude of potential temperature anomalies induced by EHT and EZMT increases as mesoscale heating propagates faster, while their relative strength decreases, consistent to the result in Eq. (21). The threshold propagating speed when they have equal strength is around 12 m s−1.
Vertical profiles of potential temperature anomalies at the equator in the fast-propagating mesoscale heating case (15 m s−1) in longitude–height diagrams. (a) Potential temperature anomalies (colors) induced by tilted mean heating (contours; interval: 1.5 K day−1). (b) Potential temperature anomalies induced by eddy heat transfer. (c),(d) Potential temperature anomalies induced by eddy momentum transfer at the tilt angles of 180° and 0°, respectively. (e),(f) Total anomalies induced by eddy terms at the tilt angles of 180° and 0°, respectively. (g) Maximum magnitude of potential temperature anomalies induced by eddy terms at different propagation speeds of mesoscale heating. The dimensional unit of potential temperature anomalies is kelvins.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
7. Comparison with a WRF simulation for convectively coupled Kelvin waves
In Khouider and Han (2013), idealized simulations of CCKWs are implemented in the WRF Model, which reproduces a coherent eastward-propagating CCKW with many common features as observed in nature. Furthermore, the evidence of energy exchange, through momentum transport, between small-scale circulation due to mesoscale convection and the propagating synoptic-scale waves is also included. In this section, the MESD model is set in the same model setup as that in Khouider and Han (2013). The goals are to explain the vertical profile of CMT and reproduce the total synoptic-scale circulation as simulated in Khouider and Han (2013), including zonal velocity and potential temperature anomalies.
a. Barotropic momentum forcing and baroclinic heating at the mesoscale
Mesoscale heating thermally drives mesoscale fluctuations of velocity, pressure perturbation, and potential temperature anomalies in the free tropical atmosphere. In reality, mesoscale fluctuations can also be impacted by momentum forcing through the boundary layer dynamics such as the orographic effects (McFarlane 1987) and sea surface temperature gradient (Lindzen and Nigam 1987; Wang and Li 1993). For example, the barotropic mode of the boundary layer dynamics was considered in a multiscale model for the Madden–Julian oscillation (Biello and Majda 2006).
Figure 16a shows the vertical profile of zonal velocity induced by mesoscale heating. The resulting zonal velocity is characterized by a front-to-rear tilt. In contrast, the zonal velocity induced by the barotropic momentum forcing is upright with an alternate zonal profile in Fig. 16b. As shown in Fig. 16c, the total zonal velocity still has a significant upward-/westward-tilted vertical structure, resembling the typical zonal winds associated with MCSs. Meanwhile, the total vertical velocity also has a front-to-rear tilt in an alternate zonal profile in Fig. 16c.
Vertical profiles of zonal velocity in longitude–height diagrams. (a) The colors show zonal velocity induced by mesoscale heating, and the contours show mesoscale heating (interval: 65 K day−1). (b) The colors show zonal velocity induced by mesoscale barotropic momentum forcing, and the contours show mesoscale barotropic momentum forcing (interval: 22.5 m s−1 day−1). (c) The colors show total zonal velocity, and the contours show vertical velocity (interval: 0.1 m s−1). The dimensional unit of zonal velocity is meters per second.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
b. Eddy momentum transfer and eddy heat transfer
Figure 17a shows the vertical profile of EZMT, which is characterized by the third baroclinic mode with alternate value at different levels. Such a vertical profile of EZMT resembles that from the WRF simulation of Khouider and Han (2013) in Fig. 17b, where positive value of CMT is found at the lower troposphere and top and negative value of CMT is found at the surface and the upper troposphere. The EHT in Fig. 17c has much weaker magnitude but the same profile as Fig. 4b.
Vertical profiles of eddy zonal momentum transfer and eddy heat transfer induced by slowly eastward-propagating mesoscale heating and mesoscale barotropic momentum forcing. (a) Eddy zonal momentum transfer. (b) Adjusted from Fig. 11c of Khouider and Han (2013). (c) Eddy heat transfer. The dimensional units of eddy zonal momentum transfer and eddy heat transfer are 15 m s−1 day−1 and 10 K day−1, respectively.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
c. Zonal velocity and potential temperature anomalies on the synoptic scale
In this section, the synoptic-scale circulation response to EMT and EHT from the MESD model, including zonal velocity and potential temperature, is directly compared with those as simulated in Khouider and Han (2013). Two central questions are addressed here: whether the total circulation response induced by mean heating and eddy terms resembles those from Khouider and Han (2013) and what the upscale impact of CMT on the synoptic-scale circulation is.
Figure 18 shows vertical profiles of total zonal velocity induced by mean heating and eddy terms at the equator. As shown in Fig. 18a, the mean-heating-driven zonal velocity has a front-to-rear tilt with zonal wind convergence (divergence) at the surface (top) in heating regions. In contrast, the zonal velocity induced by eddy terms in Fig. 18b features significant third baroclinic mode with its maximum value at the top. When compared with a mean-heating-driven zonal velocity in Fig. 18a, the zonal velocity induced by eddy terms tends to strengthen mean-heating-driven westerlies at the top, lift up the easterlies at the middle troposphere, and weaken the westerlies at the surface. As shown by Fig. 18c, the total zonal velocity resembles many features of zonal velocity from the WRF simulation in Fig. 18d, such as the strong westerlies at the 250-hPa level and the easterlies at the 400-hPa level.
Vertical profiles of zonal velocity at the equator. (a) The colors show zonal velocity induced by mean heating, and the contours show mean heating (interval: 1.25 K day−1). (b) Zonal velocity induced by eddy terms. (c) Total zonal velocity. (d) Adjusted from Fig. 11d of Khouider and Han (2013). The dimensional unit of zonal velocity is meters per second.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
Figure 19 shows vertical profiles of potential temperature anomalies induced by mean heating and eddy terms at the equator. The mean-heating-driven potential temperature anomalies are upward/westward tilted in Fig. 19a. The anomalies induced by eddy terms feature a significant third baroclinic mode in Fig. 19b. It turns out that the anomalies induced by eddy terms tend to weaken mean-heating-driven negative anomalies at lower troposphere and positive anomalies in the middle troposphere but add extra positive anomalies in the upper troposphere. The resulting total potential temperature anomalies share several common features as those from the WRF simulation in Fig. 19d, such as the two positive maximum anomalies at both the lower and upper troposphere and negative anomalies in the trailing edge.
Vertical profiles of potential temperature anomalies at the equator. (a) The colors show potential temperature anomalies induced by mean heating, and the contours show mean heating (interval: 1.25 K day−1). (b) Potential temperature anomalies induced by eddy terms. (c) Total anomalies. (d) Adjusted from Fig. 9d of Khouider and Han (2013). The dimensional unit of potential temperature anomalies is kelvins.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0178.1
8. Concluding discussion
The goals of this paper include the following four aspects: first, using a simple multiscale model to capture multiscale structures of CCKWs with embedded mesoscale disturbances and assess the upscale impact of mesoscale disturbances through the eddy transfer of momentum and temperature; second, theoretically predicting the upscale impact of mesoscale disturbances that propagate at various tilt angles and speeds on the mean-heating-driven Kelvin waves in terms of favorability for convection in a moist environment and characteristic morphology; third, exploring whether the front-to-rear-tilted vertical structure of CCKWs can still be induced by eddy transfer of momentum and temperature in the presence of upright mean heating; and last, providing a useful framework to explain CMT and synoptic-scale circulation as simulated in CRMs.
The simple multiscale model used here is the MESD model, originally derived by Majda (2007). It consists of two groups of equations on mesoscale and synoptic scale, respectively. Specifically, mesoscale fluctuations of flow field are directly driven by a prescribed mesoscale heating in a front-to-rear tilt in the first and second baroclinic modes. The resulting EMT and EHT are expressed in an explicit form and further interpreted as the upscale impact of mesoscale fluctuations on the synoptic-scale circulation. Such explicit expressions for eddy transfer of momentum and temperature should be useful to improve parameterization of the upscale impact of mesoscale tropical convection in the GCMs. In connection with the minimalist second baroclinic convective momentum transport as implemented in Moncrieff et al. (2017), the EMT from the MESD model shares similar vertical profile in the interior but has vanishing value at the surface and top. Meanwhile, the MESD model shows that eddy transfer of horizontal momentum is along the same direction as the propagation direction of mesoscale heating, providing a simple way to generalize CMT parameterization for both zonal and meridional momentum. The direction of EMT is determined by the tilt angle of mesoscale heating, which may further depend on the large-scale background flow or wind shear. Also, the EHT dominated by the third baroclinic mode could be another important component in the parameterization of organized tropical convection in the GCMs. The MESD model shows that the relative strength of EHT and EMT in dimensionless units depends on the propagating speed of mesoscale heating, highlighting the dominant magnitude of EMT in the slowly propagating mesoscale heating cases.
By focusing on low-tropospheric potential temperature anomalies, the MESD model theoretically predicts that the upscale impact of mesoscale disturbances favors shallow convection in the leading edge at tilt angles of 110°–250°, while it suppresses shallow convection at tilt angles (less than 70° or larger than 290°). Such a result explains the observation that most mesoscale disturbances propagate westward in CCKWs and few of them propagate eastward (Straub and Kiladis 2002). In the remaining tilt angles, the MESD model shows that the upscale impact of mesoscale disturbances provides unfavorable conditions for shallow convection off the equator, explaining the meridional asymmetry of convection as CCKWs propagate eastward along the equator. At tilt angles of 135°–180°, the upscale impact of mesoscale disturbances is found to strengthen the westerlies at the surface, the inflow at the lower troposphere, and the outflow at the upper troposphere. However, it tends to destroy coherent structures of CCKWs in the remaining tilt-angle cases.
It is frequently observed that vertical structures of tropical convection are characterized by a front-to-rear tilt, which shows self-similarity across multiple spatial and temporal scales (Houze 2004; Kiladis et al. 2009). It is important to understand how much of tilted vertical structures of tropical convection is induced by the upscale impact of mesoscale fluctuations instead of mean heating. The MESD model shows that the synoptic-scale circulation in a front-to-rear tilt can still be induced by eddy terms at tilt angles of 120°–240° in the presence of upright mean heating, indicating the significant contribution of the upscale impact of mesoscale disturbances on characteristic morphology of CCKWs.
In the case with fast-propagating mesoscale heating, the MESD model shows that the synoptic-scale circulation response to EHT dominates and induces positive potential temperature anomalies in the lower troposphere, providing unfavorable conditions for shallow convection in a moist environment. Such a result explains the observation that most mesoscale disturbances inside the convective envelope of CCKWs propagate slowly in reality.
To compare with results from the WRF simulation by Khouider and Han (2013), slowly eastward-propagating mesoscale disturbances driven by baroclinic mesoscale heating and barotropic momentum forcing are considered along with the front-to-rear-tilted mean heating. The MESD model successfully reproduces the vertical profile of CMT in the third baroclinic mode and the total synoptic-scale circulation, providing encouraging evidence for validating this simple multiscale model. Nevertheless, such a theoretical explanation about the results from a WRF simulation requires more validation by cloud-resolving simulations in various model setups. One essential motivation of this paper is to inspire more detailed examination on the spatial pattern of mesoscale disturbances and the associated CMT in WRF simulations for CCKWs.
The MESD model could also be used to model many other multiscale phenomenon such as westward-propagating 2-day waves (Haertel and Kiladis 2004) and easterly waves in the ITCZ (Toma and Webster 2010a,b). Meanwhile, it can be elaborated and generalized in various ways. The first interesting research direction is to couple boundary layer dynamics with that in the free troposphere, in a similar way as Biello and Majda (2006). The augmented model should be useful to capture more realistic features of CCEWs in the equatorial regions such as the ITCZ. The second research direction is to introduce a two-way feedback between the synoptic-scale circulation and mesoscale heating. For instance, the tilt angle in which direction mesoscale heating propagates could also be influenced by large-scale winds. Such a two-way feedback may come up with an instability mechanism for CCEWs in the tropics. The third research direction is to couple the MESD model with an active heating function such as the MCM (Khouider and Majda 2006c,b,a, 2008b, 2008a; Khouider et al. 2010, 2011). The resulting model allows two-way feedbacks between circulation and heating, providing a simple test bed to study convective instability. Last, it would be also interesting to diagnose the upscale impact of mesoscale disturbances on CCEWs by filtering observational data into different wavenumber–frequency domains (Wheeler and Kiladis 1999), isolating mesoscale disturbances from synoptic-scale waves, and quantifying their relationship through momentum and heat budgets.
Acknowledgments
This research of A.J.M. is partially supported by the office of Naval Research ONR MURI N00014-12-1-0912, and Q.Y. is supported as a graduate research assistant on this grant and partially funded as a postdoctoral fellow by the Center for Prototype Climate Modeling (CPCM) in New York University Abu Dhabi (NYUAD) Research Institute.
APPENDIX
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