## 1. Introduction

One of the most intriguing features of tropical convection that remains unexplained resides in its organization into a hierarchy of cloud clusters and superclusters of various temporal and spatial scales that are embedded in each other and propagate along the equator at different speeds and in both directions (Nakazawa 1988; Hendon and Liebmann 1994; Takayabu 1994a,b; Wheeler and Kiladis 1999; Kiladis et al. 2009). Of particular interest, especially from the climate dynamics perspective, is the intraseasonal planetary-scale disturbance, known as the Madden–Julian oscillation (MJO) (Madden and Julian 1972), that appears as an envelope of mesoscale to synoptic-scale westward inertio-gravity (WIG) waves with periods of approximately two days, thus known as two-day waves (Takayabu 1994b; Haertel and Johnson 1998; Haertel and Kiladis 2004; Haertel et al. 2009), and Kelvin waves that move eastward (Straub and Kiladis 2002; Roundy 2008). Accordingly, throughout this paper the phrase “two-day waves” refers to synoptic-scale convectively coupled westward inertio-gravity waves with time periods of roughly two days. The MJO envelope propagates in the Indian Ocean/western Pacific warm pool toward the east at a speed of 5–10 m s^{−1} while the embedded synoptic-scale waves move with speeds ranging roughly from 12 to 20 m s^{−1}.

It is now widely recognized that the MJO affects tropical and extratropical climate and weather patterns (see, e.g., Bond and Vecchi 2003; Jones et al. 2004; Zhang 2005; Straub et al. 2006). However, despite the recent progress in computational power and continued theoretical efforts, today’s general circulation models poorly represent the MJO (Lin et al. 2006, 2008; Liu et al. 2009), and its complex dynamics and interactions with higher-frequency convectively coupled waves (CCWs) remain unexplained. Nevertheless, there is observational evidence that Kelvin waves behave differently in different phases of the MJO (Roundy 2008) and there is a clear distinction between geographical locations of eastward moving superclusters and westward moving disturbances (Dunkerton and Crum 1995) relative to the MJO. An important factor that can impact significantly the behavior of CCWs over the tropical ocean is the variations in the background state, such as sea surface temperature, midtropospheric moisture, and the wind shear. Such variations can occur due to differences induced by various climatological cycles and/or geographical locations as well as the MJO dynamics. For example, Roundy (2008) found that, depending on its phase (active or inactive), the MJO modulates differently both the strength of deep convection and the phase speed of the embedded convectively coupled Kelvin waves.

While it is somewhat intuitive how variations in sea surface temperature and atmospheric thermodynamics would influence convective activity and CCWs (weaker moisture supply will yield weaker convective forcing and thus weaker waves), the effects of wind shear is less obvious. This last issue has been extensively addressed in many research papers for the cases of both meridional and vertical shears (e.g., Webster and Holton 1982; Webster and Chang 1988; Zhang and Webster 1989; Kasahara and Silva Dias 1986; Wang and Xie 1996; Biello and Majda 2003; Stechmann and Majda 2009; Majda and Stechmann 2009; Ferguson et al. 2009). Nonhomogeneity in the thermodynamic background consisting of precipitating and nonprecipitating regions was considered in a series of papers by various researchers (Frierson et al. 2004; Khouider and Majda 2005; Stechmann and Majda 2006; Pauluis et al. 2008; Dias and Pauluis 2009). However, owing to the complexity of moist dynamics, these studies were either restricted to the dry case or to the idealized situation where the warm core baroclinic dynamics were reduced to a single vertical mode and the convective forcing was based on a simple parameterization of Betts–Miller type (Frierson et al. 2004).

In a series of papers, Khouider and Majda introduced and analyzed a simple multicloud model for organized tropical convection based on recent observations (Khouider and Majda 2006; 2008a,b—hereafter KM06, KM08a, and KM08b). As a compromise between simplicity and realism, the model warm core (i.e., the baroclinic flow that is directly forced by latent heating) comprises the first and second baroclinic modes of vertical structure, which are believed to be of central importance for the dynamics of convectively coupled equatorial waves. Consistently, the multicloud model uses three vertical heating modes: 1) a deep convective mode associated with cumulonimbus clouds that heat the entire tropospheric depth and 2) congestus and 3) stratiform modes, associated with cumulus congestus and stratiform cloud decks that heat the lower troposphere and cool the upper troposphere and cool the lower troposphere and heat the upper troposphere, respectively. The multicloud model is coupled to the sea surface through a purely thermodynamic boundary layer due to parameterized downdrafts and updrafts. The boundary layer dynamics that can interact (linearly) and induce a barotropic mode via surface convergence were considered in Waite and Khouider (2009).

The multicloud model exhibits linear instabilities, on the synoptic scales, of zonally propagating modes corresponding to convectively coupled Kelvin and inertio-gravity waves, including two-day and *n* = 0 eastward inertio-gravity (EIG) waves, also known as eastward mixed Rossby–gravity (MRG) waves. Not only do those waves have scale-selective instabilities resembling the spectral peaks associated with CCWs in outgoing longwave radiation (OLR) data (Takayabu 1994a; Wheeler and Kiladis 1999) but they also have physical and dynamical features that are similar to observations including the famous front to rear vertical tilts of temperature, velocity, and heating fields (Wheeler and Kiladis 1999; KM08b; Kiladis et al. 2009). However, convectively coupled westward MRG and Rossby waves are missing in the results of KM08b due to the lack of a barotropic mode, as conjectured in Majda et al. (2004) and in KM08b according to observations (Wheeler et al. 2000; Majda et al. 2004; KM08b; Yang et al. 2007). Yang et al. (2007) observed that the westward MRG and *n* = 1 Rossby waves are highly connected with each other and that the latter is more or less barotropic in structure, especially in the Western Hemisphere, consistent with earlier works (Wheeler et al. 2000; Kiladis et al. 2009).

Throughout this paper, the eastward branch of the MRG waves is called the *n* = 0 EIG wave, following earlier work (e.g., Wheeler and Kiladis 1999; Kiladis et al. 2009) and because of its resemblance to gravity waves, whereas the westward branch is referred to as westward MRG or simply MRG when there is no confusion.

Here we are interested in the effects of a background meridional and vertical shear flow on the dynamics and structure of CCWs in the multicloud model of Khouider and Majda on an equatorial beta plane (KM08b). Some emphasis is put on the case of a barotropic shear to test the aforementioned hypothesis about the instability of westward MRG and Rossby waves.

The rest of the paper is organized as follows. In section 2, we derive the linearized multicloud equations in a background flow involving both a meridional/barotropic and a vertical/baroclinic shear and we “Galerkin-project” them in the meridional direction on the first few parabolic cylinder functions. The details of the Galerkin projection, involving advection terms by the meridional shear, are reported in the appendix. Section 3 presents linear theory results for various vertical/baroclinic shear flows; the case of the meridional/barotropic shear, considered in section 4 interestingly exhibits instabilities of westward MRG and Rossby waves. The structure of the CCWs that are most deformed by the meridional shear (Zhang and Webster 1989) and in particular of the westward MRG and Rossby waves, which constitutes one of the novelties here, is also shown and analyzed in section 4. In addition to changes in the stability diagram of convectively coupled equatorial waves, large shear values induce instabilities of moisture modes, recognizd by their large moisture component, that are not trapped in the vicinity of the equator but exhibit Rossby gyres located some distance away from it. Those are discussed in section 5. Section 6 looks at the stability of the background in the dry case without convective–moisture coupling. This last study has especially confirmed that the emergence of the westward MRG and Rossby waves is due to a pure shear instability: the meridional/barotropic shear background. A discussion is given in section 7 and a concluding summary is given in section 8.

## 2. The multicloud model with advective nonlinearities

### a. Model formulation

**V**is the (truncated) horizontal velocity,

*W*is the vertical velocity,

*P*is pressure, and Θ is the total potential temperature. By fixing the vertical structure functions, the problem reduces to that of tracking the dynamics of the lowercase variables above representing the various projection components. The standard nondimensionalization units of synoptic-scale equatorial dynamics are adopted. The Rossby deformation radius

*L*≈ 1500 km is the length scale, the gravity wave speed

_{e}*c*= 50 m s

^{−1}is the velocity scale, and the eddy turnover time

*T*=

*L*/

_{e}*c*≈ 8.33 h is the time scale. The temperature unit is set to

*α*

*y*

**v**

_{i}^{⊥},

*i*= 0, 1, 2 is the Coriolis force with (

*u*,

*υ*)

^{⊥}= (−

*υ*,

*u*) and the parameter beta has been normalized to one according to the nondimensionalization units above;

*S*

_{j}^{v}and

*S*represent, respectively, momentum damping and convective heating and cooling terms. Note that the hydrostatic and continuity equations give the relations

_{j}^{θ}*θ*) and to the congestus (

_{eb}*H*) and stratiform (

_{c}*H*) heating fields, which otherwise are governed by simple ordinary differential equations (ODEs) (KM08a). It is important to note here that without these advective terms in the stratiform, congestus, and

_{s}*θ*equations, the multicloud model with shear exhibits “catastrophic instabilities” at small scales, that is., non-scale-selective instabilities that do not decay with wavenumber. Before we derive the PDEs for these fields, we recall that the boundary layer potential temperature in the multicloud model represents the average over the subcloud mixed layer with a fixed height

_{eb}*h*= 500 m and the stratiform/congestus mode mimics cloud decks in the upper/lower troposphere that heat the upper/lower troposphere and cool the lower/upper troposphere. Therefore it is meaningful to assume a vertical structure function for the stratiform/congestus cloud type, confined to the upper/lower troposphere:

*F*,

_{x}*x*=

*s*,

*c*are the production functions of stratiform and congestus, respectively, are projected on the basis functions

*ϕ*(

*z*) and −

*ϕ*(

*π*−

*z*), respectively, to obtain

*f*stands for the projection of

_{x}*F*.

_{x}### b. Linearization in a background shear flow

*θ*

_{j}/∂

*y*=

*y*

*u*

_{j},

*j*= 1, 2 are neglected here. These are very small terms that basically vanish at the equator and therefore expected to have no or very minor effects on the results.

### c. Galerkin truncation in the meridional direction

To perform linear wave analysis in terms of waves propagating along the equator, we further need to eliminate the *y* dependence and the *y* derivative by resorting to Galerkin projection in the meridional direction (Majda and Khouider 2001; KM08b). Except for the terms involving the barotropic/meridional shear *u*_{0}, the projection procedure is identical to what is done in KM08b and we refer the interested reader to that paper. The details involving the new barotropic advection terms are reported in the appendix.

**U**is the state vector formed by the meridional components

*u*

_{0},

*u*

_{1}, … ,

*u*

_{N−1}of all the prognostic variables:

*u*,

_{j}*υ*,

_{j}*θ*,

_{j}*j*= 1, 2,

*θ*,

_{eb}*q*,

*H*,

_{s}*H*that are retained. Note that the meridional dependence of the solution vector

_{c}*U*is given through the linear combinations of the parabolic cylinder functions

*ϕ*(

_{j}*y*) in (A.3) defined on the unbounded domain (−∞, +∞), which implies vanishing boundary conditions at infinity consistent with the finite-energy principle. However, the meridional extent of such solutions is limited by the validity of the equatorial beta-plane approximation, to within a few thousand kilometers from the equator (Gill 1980).

*k*is the zonal wavenumber and

*ω*the generalized phase. This ansatz leads to a linear (matrix) eigenvalue problem (where

**U**is the eigenvector and

*ω*the associated eigenvalue), which is solved numerically on a desktop computer. Our dimensional setup is such that

*k*= 1 corresponds to a wavelength that is equivalent to the equatorial circumference of the globe of 40 000 km. The real part of

*ω*is the frequency and its imaginary part yields the growth rate.

Next, we present such solutions in various configurations for the background shear flow (2.10). We begin with the case of a purely vertical/baroclinic shear.

## 3. Case of a vertical shear

### a. Shear prototypes

Here we assume a purely vertical shear profile with *u*_{0}(*y*) = 0 and consider the following four prototype baroclinic shear flows:

pure first baroclinic shear,

*u*_{1}≠ 0,*u*_{2}= 0, consisting of a westerly (easterly) wind at the surface capped by an easterly (westerly) wind aloft when*u*_{1}> 0(*u*_{1}< 0);pure second baroclinic shear,

*u*_{1}= 0,*u*_{2}≠ 0, consisting of a westerly (easterly) wind at the surface, an easterly (westerly) jet in the middle of the troposphere, and a westerly (easterly) wind aloft when*u*_{2}> 0(*u*_{2}< 0);a combination of first and second baroclinic shears with

*u*_{1}=*u*_{2}≠ 0, producing a strong surface westerly (easterly) wind when*u*_{1}> 0(*u*_{1}< 0) and quiet conditions aloft;a combination of first and second baroclinic shears with

*u*_{1}= −*u*_{2}≠ 0, producing a strong easterly (westerly) wind at the top of troposphere when*u*_{1}> 0(*u*_{1}< 0) and quiet conditions at the surface,

^{−1}to a moderate value of 7.5 m s

^{−1}, within the range of the MJO winds. In this respect the present study can be applied to high frequency/synoptic-scale CCWs evolving within the MJO envelope. Note that the background zonal wind in Fig. 1 is a purely vertical/baroclinic shear that has no barotropic component and no meridional dependence whereas the barotropic/meridional shear in Fig. 7, which is used in section 4 below, has no baroclinic component/vertical dependence.

### b. Frequency and instability diagrams

Here we present the results of stability analysis for the linearized and meridionally truncated multicloud model equations presented in the previous section for the different vertical shear backgrounds shown in Fig. 1. The multicloud parameters used here are listed in Table 2. Notice that this set of parameters is only slightly different from the standard regime of KM08b. In fact, here we use the slightly less aggressive value of *α _{s}* = 0.16 instead of the 0.25 value in KM08b. In Fig. 2, we present the frequency and growth rate diagram for the control case with no background shear,

*u*

_{0}=

*u*

_{1}=

*u*

_{2}= 0. The symmetric and antisymmetric dispersion relations, showing the frequency or phase in cycles per day (cpd), are plotted on the left and right panels, respectively. The unstable modes are highlighted by filled circles. The diameters of the circles are proportional to the growth rates of the corresponding modes and the associated maximum growth rate, corresponding to the largest circle, is displayed on each panel. The frequency diagrams show instabilities at the synoptic scale of eastward moving Kelvin and

*n*= 0 EIG waves (eastward MRG).

In Figs. 3 through 6 we present the stability and frequency diagrams corresponding to the four vertical shear prototypes presented above. We see from Fig. 3 that the main effect of the vertical shear with *u*_{1} > 0, *u*_{2} = 0 (westerly on bottom and easterly aloft) is to destabilize westward moving waves while the Kelvin and *n* = 0 EIG wave instability remain qualitatively unchanged. Interestingly, the maximum instability for the *n* = 0 EIG waves decreases slightly for the weak shear, of 2.5 m s^{−1}, and then increases again up with the shear strength, while the Kelvin instability grows monotonically from the beginning. The case *u*_{1} < 0, *u*_{2} = 0 (easterlies on bottom and westerlies aloft), on the other hand, shows a more subtle behavior. In this case, westward waves remain stable (or only weakly destabilized for *u*_{1} = −7.5 m s^{−1}) while the eastward ones amplify and higher meridional index waves are destabilized. A similar behavior is observed in Fig. 4 when *u*_{1} = 0 and *u*_{2} is varied instead. However, the case *u*_{2} < 0 (easterlies on top and bottom) induces instabilities of westward waves only and *u*_{2} > 0 (westerlies on top and bottom) produces instabilities of eastward waves only. Moreover, for *u*_{2} = −7.5 m s^{−1}, a slow moving (moisture) mode, reminiscent of the congestus mode reported in KM08b, becomes unstable. Note, however, that, while the congestus mode in KM08b is due to a dry RCE (*θ*_{eb} − *θ*_{em} ≥ 14K), here it is the strong (top and bottom) easterly shear that produces the moisture mode. As we will see below (see also KM08b), these modes have a zonal structure that is not confined to the vicinity of the equator and carry a large moisture or congestus anomaly. Thus, they are referred to as moisture or congestus modes, accordingly.

One important common feature of the background flows, in Figs. 3 and 4, which produce eastward and westward wave instabilities, is that in both cases they present westerlies and easterlies, respectively, at either the top or the bottom of the troposphere. According to this model the stability of CCWs in a background vertical shear is likely to be controlled by the direction of the flow in the upper or the lower troposphere so that only waves moving in the direction of the wind are unstable. Note, however, that, as shown in Fig. 3, while easterlies over westerlies also produce eastward waves, namely Kelvin and *n* = 0 EIG, westerlies over easterlies do not destabilize westward waves much. Bottom easterlies combined with high-level westerlies do not seem to have a destabilizing effect on westward moving waves.

To explore this further, we consider in Figs. 5 and 6 the shear profiles in Fig. 1 (profiles 3.a, 3.b, 4.a, and 4.b), combining the first and second baroclinic shears to form a total shear that is, either approaching zero at the top of the troposphere and has strong surface easterlies or westerlies (Fig. 5) or approaching zero at the surface and has strong easterly or westerly wind aloft (Fig. 6). The destabilization of CCWs that propagate in the direction of the strongest winds, at the top or bottom of the troposphere, is obvious. Interestingly, the Kelvin and *n* = 0 EIG seem to resist the high-level easterlies in Fig. 6 (top panel) but not to the low-level easterlies at bottom of Fig. 5. This is perhaps due to the residual lower troposphere easterly wind (see Fig. 1, profile 4.a), consistent with the results in Fig. 3 for the case in profile 1.a. Therefore, Kelvin and *n* = 0 EIG seem to be more sensitive to low-level winds compared to high-level winds. In addition, the case with strong surface winds and zero flow at the top (Fig. 1, profiles 3.a and 3.b), displayed in Fig. 5, has catastrophic instabilities of a nonequatorially trapped, moisture-type, mode that moves roughly with the speed of the wind at some steering level (inducing some kind of a Doppler shift) in the lower troposphere, consistent with the fact that moisture is bottom heavy and, as such, it responds directly to low-level winds. Their structure (not shown here) displays high oscillations in the meridional direction that expand to higher latitudes, similar to the congestus mode reported in KM08b, and are dominated by the moisture component.

An interesting application of the present findings concerning the effect of low- and high-level shear on the stability of CCWs in the MJO context is suggested in section 7 and Fig. 18. However, besides the selection of waves according to their direction of propagation and except for the nontrapped (moisture) modes, reminiscent of the congestus mode in KM08b, no other equatorially trapped waves were obtained (became unstable) through the interactions of the multicloud model with the baroclinic/vertical shears considered here. The effect of a meridional barotropic shear that both induces significant deformation to the wave structure and generates new CCWs (new in the context of the multicloud model), namely Rossby and westward MRG waves, is addressed next. Moreover, the purely vertical shear does not seem to affect significantly the zonal and vertical structure of the CCWs (not shown here), which remain fairly similar to those in the shear-free case reported in KM08b.

## 4. Case of a meridional shear

*u*

_{0}, which is varied from 5 to 20 m s

^{−1}, fixes the strength of the wind at the equator as shown in Fig. 7.

### a. Waves and instabilities

In Figs. 8 and 9 we plot the frequency and growth rate (filled circles) diagrams in the presence of the easterly and westerly meridional shears in (4.1) and Fig. 7, respectively. The maximum growth rate and the corresponding phase speed associated with each unstable branch are displayed on the panels.

A transition pattern is apparent in the instability diagrams for both easterly and westerly shear cases. First, at the small value of *u*_{0} = 5 m s^{−1}, except for a Doppler shift (which rotates the frequency diagram in one direction or the other), both easterly and westerly shears destabilize the same new waves when compared to the bare case without a background wind shown in Fig. 2, namely symmetric eastward and westward (two-day) inertia–gravity (*n* = 1 EIG and *n* = 1 WIG) waves. Beyond this threshold the corresponding pictures differ substantially. At *u*_{0} = 10 m s^{−1}, while the easterly shear case develops an antisymmetric gravity branch instability at the expense of the WIG branch, the westerly shear develops an antisymmetric moisture mode. For *u*_{0} ≥ 15 m s^{−1}, the westerly shear continues to destabilize eastward moving waves, consistent with the cases of low- or high-level westerly shears in Figs. 3 –6, and ultimately develops a catastrophic instability of the moisture mode at small scales in its symmetric part as well. At this limit the easterly shear case becomes more interesting. As it continues to destabilize eastward moving waves, it also destabilizes low-frequency westward CCWs, namely mixed Rossby–gravity and Rossby waves, which are new to the multicloud model with a barotropic wind shear background, and it does not develop a catastrophic instability, at small scales, of a moisture type. This confirms the conjecture stating that convectively coupled Rossby and westward MRG waves require a barotropic flow component to develop (Majda et al. 2004; KM08b) and is more or less consistent with observations (Wheeler et al. 2000; Yang et al. 2007). Notice that the catastrophic instability of a moisture mode, in the case of a westerly wind shear with *u*_{0} = 20 m s^{−1}, is not surprising. A shear with such a large westward equatorial wind is in fact unrealistic; it induces a midlatitude jet of about 70 m s^{−1}! This is consistent with the fact that the moisture mode is not trapped at the equator but expands strongly to the extratropics where the background jet is very large.

In addition to generating westward MRG and Rossby waves, the meridional shear significantly deforms the dispersion relations and the horizontal structure of CCWs in the multicloud model. This problem has been addressed extensively in the literature during the last few decades for the case of dry equatorially trapped waves (see Zhang and Webster 1989, and references therein). As we can see from Figs. 8 and 9, the meridional shear induces a Doppler shift that either accelerates or decelerates the equatorial waves. Typically, equatorial easterlies tend to decelerate eastward-moving equatorially trapped waves and accelerate westward waves, while equatorial westerlies do just the opposite. However, this acceleration and/or deceleration appears to be more significant in some cases than in others. For instance the symmetric (*n* = 1) Rossby waves, on the bottom left panel in Fig. 8 with *u*_{0} = 20 m s^{−1}, seem to have a much higher frequency than its antisymmetric (*n* = 2) counterpart on the right. This is perhaps due to the complex interactions of the waves with the meridional shear and also with moisture and convection, as it is often the case that convective coupling slows down the equatorial waves. This may affect different waves differently. Nonetheless, our classification of these waves into *n* = 1 and an *n* = 2 Rossby waves here is based on their meridional structure, shown in Figs. 12 and 13, in comparison with the dry analogs.

### b. Wave structure and deformation by shear

Here we present the dynamical and physical structures of the CCWs that are significantly deformed by the shear and those that are new to the multicloud model with a meridional shear background, that is, those that are otherwise stable in the case of a zero background wind in KM08b, namely the Rossby and the westward MRG waves.

In Figs. 10 –14 we display the Kelvin, westward inertia–gravity, westward MRG, and the symmetric (*n* = 1) and antisymmetric (*n* = 2) Rossby waves, respectively. The structures of the eastward gravity waves (including the *n* = 0 EIG) are less affected by the meridional shear and therefore not reproduced here. The Kelvin and WIG waves correspond to the case of an equatorial easterly wind with *u*_{0} = 5 m s^{−1} while the westward MRG and the Rossby waves are those obtained with the same easterly wind but with *u*_{0} = 15 m s^{−1} and *u*_{0} = 20 m s^{−1}, respectively. The bottom right panel shows the structure of the analog dry equatorially trapped wave of Matsuno (1966) associated with the second baroclinic mode (equivalent depth 75 m) of vertical structure (Majda 2003). Note that we chose to compare the horizontal structures of the CCWs here with those of their dry and free analogs corresponding to the second baroclinic mode and not to the first one because they appear to have comparable trapping distances: The structures of the first baroclinic mode waves have a much wider north–south extent.

On the top panels of Figs. 10 and 11, we display the magnitude bar diagrams (see KM08b for details), showing the strength of each one of the physical variables *u _{j}*,

*υ*,

_{j}*θ*(

_{j}*j*= 1, 2),

*q*,

*θ*,

_{eb}*H*, and

_{s}*H*grouped in terms of their meridional components (top left) and the zonal structure of the meridional and zonal divergence for each baroclinic mode velocity component. One of the most striking features seen here is in the structure of the Kelvin wave, bottom left panel of Fig. 10. Interestingly, it exhibits a nontrivial meridional wind somewhat more consistent with observations than its homogeneous counterpart in KM08b. Note that the implied meridional convergence (divergence) is in phase with the zonal convergence; thus, it contributes positively to the total convergence (divergence). Even though the meridional velocity is relatively weak (see the top left panel of Fig. 10), the meridional convergence/divergence, computed as the ratio max

_{c}_{x,y}|

*υ*|/max

_{y}_{x,y}|

*u*+

_{x}*υ*|, contributes about 30% to the total convergence/divergence and increases significantly with the shear strength; it exceeds 70% at

_{y}*u*

_{0}= 20 m s

^{−1}(results not shown here).

As demonstrated in Ferguson et al. (2009), this exhibition of a meridional wind for the Kelvin wave, as seen here, is due to the interaction of the wave with the barotropic wind shear in its background, and it is shown there to increase with both the strength of the meridional shear, ∂_{y}u_{0}(*y*), and the zonal wavenumber of the wave. Nonetheless, a similar behavior is reported by Dias and Pauluis (2009) for a Kelvin wave evolving in a nonhomogeneous background where a narrow moist region is separated from the dry environment by stationary “precipitation fronts” that are parallel to the equator, mimicking the intertropical convergence zone (ITCZ). The meridional shear does not seem to have a significant effect on the trapping or zonal structure of potential temperature (or equivalently pressure or geopotential) and zonal velocity for the Kelvin wave, when this later is compared to its dry second baroclinic analog shown on the bottom right panel. This is consistent with the seminal work of Zhang and Webster (1989) but, curiously, they did not report about the meridional wind, which presumably they assumed to be zero.

As a consistency check the meridional convergence is also shown for the westward inertia–gravity (two-day) wave, top right of Fig. 11. As expected, the meridional convergence for the two-day wave is relatively stronger but, similar to the Kelvin case, the contribution from the second baroclinic (i.e., low level) meridional convergence is almost in phase with the zonal convergence and leads deep convection, thus converging moisture from the extratropics to fuel the convection. The structure of the two-day wave in the shear background adopts a crescentlike shape bending backward to the direction of propagation, probably a result of the distortion, by shear, of its effective phase speed (Ferguson et al. 2009).

The structure of the (westward) mixed Rossby–gravity wave is shown in Fig. 12. A quick comparison with the structure of the dry second baroclinic MRG, bottom right corner, suggests that the meridional shear makes the MRG less trapped at the equator, consistent with the findings of Zhang and Webster (1989): it has cyclonic and anticyclonic gyres located (antisymmetrically) some distance away from the equator and strong cross-equatorial flows consistent with the dry MRG, right panel, and reflected by the meridional velocity components, which dominate the bar diagram in the top left panels. Interestingly, however, the convectively coupled MRG wave exhibits a strong front to rear tilt in temperature and wind structure, consistent with observations (Kiladis et al. 2009).

The symmetric and antisymmetric convectively coupled Rossby waves that are unstable due to the strong background barotropic shear are reported in Figs. 13 and 14, respectively. The trapping of the Rossby waves seems to be affected differently for the symmetric and antisymmetric Rossby waves. The symmetric wave in Fig. 13 exhibits long tails extending to the extratropics and thus appears to be overall less trapped, and the antisymmetric wave in Fig. 14 is significantly more trapped owing to the barotropic shear. Also, while the symmetric Rossby wave has a front to rear tilted vertical structure, the antisymmetric wave is tilted in the backward direction, perhaps due to its slow speed of propagation combined with the more pronounced extension of its horizontal structure to the extratropics where the advection by the barotropic wind is actually to the east, resulting in an eastward effective advection (i.e., Doppler shift).

## 5. The moisture modes

As pointed out in Figs. 4, 8, and 9, in some extreme limits of the vertical and meridional shear backgrounds, a few instability branches of so-called moisture modes emerge. They are classified into three categories: a low-frequency moisture mode in the purely baroclinic vertical shear, Figs. 4 and 5, and both moderate and high frequency modes in the case of the meridional–barotropic wind shear, Figs. 8 and 9.

The dispersion relations of the low-frequency moisture mode for the symmetric and antisymmetric system in Figs. 4 and 5 are somewhat similar to those of the congestus mode reported by KM08b, which emerges in the homogeneous multicloud model (with a zero background flow) when the RCE is pushed toward a dry regime. But as shown here, for the case in Fig. 4, their physical and dynamical structures are somewhat different. In Fig. 15, we plot the bar diagrams (left panels) and the horizontal structures (contours of vertical velocity and horizontal velocity arrows; right panels) for one representative from each of these three moisture modes. One of their common features is that the strength of the moisture and to some extent the boundary layer *θ _{e}* in the bar diagrams is relatively larger than the other variables but, unlike the congestus mode in KM08b, the congestus heating is weak. At the exception of the high frequency mode, they also have a relatively weak stratiform structure and therefore weak deep convection as well. More importantly, unlike the CCWs addressed above, the moisture modes have longer trapping distances from the equator and are characterized by strong off-equatorial gyres, implying convergence of moisture toward the equator in a way similar to the congestus mode (KM08b).

The moisture mode (due essentially to the moisture equation) emerges in many simple models for tropical convection (Sobel et al. 2001; Neelin and Yu 1994; Fuchs and Raymond 2002; KM06; Sugiyama 2009). It is often seen as a standing mode with a non-scale-selective instability; typically, the growth rate increases with the wavenumber. Some authors believe the moisture mode is a surrogate for a “MJO mode” (Neelin and Yu 1994; Fuchs and Raymond 2002; Sugiyama 2009). However, as shown here, its full 2D zonal structure is clearly not consistent with that of the MJO, which is more centered at the equator. In fact, their structure lies mostly within and/or beyond the location of the subtropical jet at *y* = ±3000 km, that is, beyond the domain of validity of the equatorial beta-plane approximation, which limits their applicability to the real world. In fact, the only apparent effect of the background temperature gradient terms (when activated; results not shown here) is to increase the growth of these moisture modes consistent with their off-equatorial extent. Nevertheless, the moisture modes may play an important role in the model dynamics of preconditioning the troposphere prior to deep convection, much like the congestus mode in KM08b, by “sucking in” moisture from the surrounding latitudes toward the equator.

## 6. The dry case or pure shear instabilities

It is clear from our discussion above that the background shear, whether barotropic or baroclinic, has a nontrivial impact on the stability features of the multicloud model either by modifying the stability behavior of preexisting CCWs or by destabilizing new waves. To help understand to what extent the background shear by itself creates such instabilities, we perform here linear stability analysis for a dry model consisting of the multicloud dynamical core only, in which all the effects of convection on the rhs of (2.11) coupling the fluid dynamics to moisture and boundary layer *θ _{eb}* are ignored.

Typically, the dry system is stable when the maximum background wind is below roughly 5 m s^{−1} and becomes unstable when the maximum wind is 5 m s^{−1} and larger. In Fig. 16, we plot the stability diagrams corresponding to three different extreme cases: (top) a purely first baroclinic/vertical shear with a maximum westerly wind at the surface of *u*_{1} = 7.5 m s^{−1}, (middle) a mixed first and second baroclinic/vertical shear with a maximum westerly wind at the top of the troposphere of 5 m s^{−1} with *u*_{2} = −*u*_{1} = 2.5 m s^{−1}, and (bottom) a purely barotropic/meridional shear with *u*_{0} = 20 m s^{−1}. Important differences in the instability behavior for the vertical and meridional shears can be distinguished from Fig. 16. While the vertical shear cases, top and middle panels, are dominated by an instability of a fast-moving (165 m s^{−1}) large-scale mode, isolated in both wavenumber and frequency, the case of the meridional shear has instability bands along the mixed Rossby–gravity and Rossby wave branches as well as some gravity waves.

Plots of the detailed physical and dynamical wave structure (Fig. 17) show that the most unstable modes in the baroclinic/vertical shear case exhibit a highly oscillatory meridional structure extending to high latitudes whereas for the barotropic/meridional shear case the most unstable modes appear to be genuine equatorially trapped waves, though deformed by the background shear. Precisely, the modes corresponding to the Rossby and mixed Rossby–gravity branches are, in fact, second baroclinic mode Rossby and mixed Rossby–gravity waves. The structure of the typical most unstable mode, associated with the vertical shear, and the second baroclinic MRG and Rossby waves, which are destabilized by the meridional shear, are shown on the top, middle, and bottom panels of Fig. 17.

From the bar diagram on the top left panel of Fig. 17, we can see that the vertical shear-induced instability mode has both first and second baroclinic components due to the two baroclinic modes being strongly coupled through the background vertical shear [see (2.11)]. Such linear coupling is not allowed without the effects of moisture and convection in the barotropic case. This decoupling explains in part why the most unstable waves in the meridional shear case are purely second baroclinic waves. The nontrapped and oscillatory structure of the vertical-shear-induced waves is apparent from the structure of the horizontal velocity and pressure shown in the top right corner of Fig. 17. This particular result demonstrates that a vertical shear alone (without convective forcing) cannot trigger convectively coupled equatorial waves but can be a precursor for highly nontrapped waves that can interact with midlatitude, providing a plausible mechanism for tropical–extratropical interactions in the presence of a vertical shear. Nevertheless, the barotropic/meridional shear destabilizes second baroclinic westward MRG and Rossby waves that can trigger and sustain convection, perhaps through low-level convergence of moisture and/or temperature perturbations, as in bore waves (Tulich and Mapes 2008; Stechmann et al. 2008; Stechmann and Majda 2009). Consistent with observations that show that convectively coupled westward MRG and Rossby waves have an important barotropic component (Kiladis et al. 2009), the present work suggests a mechanism for MRG and Rossby waves (i.e., through meridional shear instability).

## 7. Discussion

Sensitivity tests (not shown here) performed by artificially removing, one by one, the advective terms on the rhs of (2.11) demonstrate clearly that, in this work, the importance of low-level shear is due mainly to the moisture advection terms, *u*_{1}*q _{x}* +

*α̃*

*u*

_{2}

*q*, while the importance of high-level winds is not tied to one particular term. Both the advection of stratiform clouds and of potential temperature is revealed to be important, as well as the advection of the velocity components. While removing the stratiform advection terms reduces significantly the growth rates, the deletion of the

_{x}*θ*

_{1},

*θ*

_{2}advection terms gives rise to the instability of waves moving in the opposite direction. Note that both

*θ*

_{1}and

*θ*

_{2}have a direct effect on the stratiform heating through the closure assumptions. The advection of velocity, on the other hand, affects both moisture and potential temperature through convergence, that is, vertical transport anomalies. Thus, the importance of both low-level and high-level winds is consistent with the importance of both high-level stratiform clouds and low-level moisture and congestus preconditioning for the instability of convectively coupled waves in the multicloud model (KM06; KM08b). A schematic of a plausible theory, based on these findings in the context of the multicloud model, for the propagation of synoptic-scale waves within the MJO envelope is shown in Fig. 18.

These results are consistent with those of Majda and Stechmann (2009), who showed that a low-level westerly wind burst drives squall line–like waves that move with the wind speed at their steering level in the lower troposphere. However, while sensitivity to the low-level wind is consistent with squall-line dynamics (Robe and Emanuel 2001; Majda and Stechmann 2009), although the propagation speed of the CCWs is much faster than the maximum low-level background wind, the impact of the high-level wind brings to mind the numerical simulations of Shige and Satomura (2001). Shige and Satomura found that westward propagating streaks of convection are generated in the wake of a convective band (initiated by a thermal bubble) evolving in a vertical-shear environment with easterlies at high levels and westerlies below (see their Fig. 3). When the easterly wind aloft was set to zero, they noticed that the convection weakened significantly (their Fig. 15). Moreover, Majda and Stechmann (2009) found that, when the multicloud model for a slab atmosphere without rotation interacts with a mean vertical shear forced by a simple convective momentum transport model, the direction of propagation of the moist gravity wave packets progressively transitions and changes from eastward to westward as the mean shear, which is characterized by surface westerlies and high-level easterlies, strengthens and develops significantly large easterlies aloft (their Fig. 6b). This is consistent with the destabilization of westward waves by high-level easterlies here. It is argued that the westward propagation of synoptic-scale waves is triggered by the development of smaller-scale squall lines that move according to their steering level wind speed, in the lower troposphere, for which the synoptic wave packets are the envelopes that move in the opposite direction. The present study suggests a different mechanism, namely the westward (direct or indirect) advection of stratiform clouds and temperature anomalies by high-level easterlies competing with the eastward advection of moisture, congestus, and/or boundary layer *θ _{eb}* by low-level westerlies. Consistently, Lin and Mapes (2004) have pointed out that wind shear has a significant impact on stratiform anvils in the wake of deep convection.

We also note that, for a given direction of destabilization, the maximum growth rate always decreases with the meridional index of unstable waves. So, among the eastward moving waves, the Kelvin wave is always the one with the largest maximum growth, followed by the *n* = 0 EIG, and, among the westward waves, the *n* = 1 WIG (two-day) wave has the largest maximum growth, and so on. This theoretical result for the multicloud model is somewhat consistent with the persistence of Kelvin waves in both the active (surface westerly) phase and the suppressed (surface easterlies) phase of the MJO (e.g., Roundy 2008) and the fact that two-day waves are observed to dominate the active phase of the MJO, as this latter propagates eastward in the Indian Ocean/western Pacific warm pool region (Dunkerton and Crum 1995; Takayabu 1994b; Haertel and Kiladis 2004). Recall that the front of the MJO is characterized by vertical shear in the zonal direction with easterlies at the surface and westerlies aloft compensated in the back of the envelope by westerlies at the surface and easterlies aloft to provide the leading low-level convergence and the strong divergent flow observed aloft, in phase with deep convection (Lin and Johnson 1996; Biello and Majda 2005; Zhang 2005; Kiladis et al. 2005; Haertel et al. 2009). The low-level westerly wind in the back of the MJO envelope is often referred to as the westerly wind burst (WWB), which has westerly winds starting at the surface and slowly migrating upward halfway between the surface and the midtroposphere (Lin and Johnson 1996; Biello and Majda 2005). The background flow, bottom left panel in Fig. 1, combines the first and second baroclinic shears in order to mimic this WWB. According to the results in Fig. 5, without the topping high-level easterlies the active phase of the MJO would have only eastward moving waves (i.e., Kelvin and *n* = 0 EIG).

The barotropic/meridional shear case is interesting as well but for different reasons. Slowly propagating (low frequency and large scale) Rossby and westward MRG waves play a central role in the dynamics of organized tropical convection. They contribute a significant amount to the power spectrum of OLR data from the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) and other observational records (e.g., Takayabu 1994a; Wheeler and Kiladis 1999; Kiladis et al. 2009). As such, they are as important as other CCWs to study and to reproduce in numerical models. One of the major differences between the linear results from the (original) multicloud model without a shear background in KM08b and observations (e.g., Wheeler and Kiladis 1999) is that the Rossby and westward MRG waves were missing—they are stable in the multicloud model without a meridional shear. However, observational experts have readily noticed that convectively coupled Rossby waves evolve in an environment with significant barotropic flow (Wheeler et al. 2000; Majda et al. 2004; Kiladis et al. 2009) and are often connected to barotropic Rossby waves that propagate toward or propagated from high latitudes (Yang et al. 2007; Roundy 2008; G. Kiladis 2009, personal communication). Thus, the lack of Rossby and westward MRG waves was attributed in KM08b to the lack of a barotropic mode component in the multicloud model equations used there. Interestingly, the new results presented here with a simple meridional shear background of barotropic structure exhibit Rossby and westward MRG waves instabilities for sufficiently large shear strength. Linear stability analyses for the dry case (without convective heating or cooling effects and without coupling to moist thermodynamics) were conducted in section 6. They reveal instabilities of second baroclinic Rossby and westward MRG waves due to the meridional shear that are consistent with the convectively active case. The dry instability of Rossby and westward MRG waves by the meridionally varying barotropic shear is consistent with the finding of Xie and Wang (1996: see also Wang and Xie 1996), who showed that a meridional shear impacts both the activity of Rossby and westward MRG waves but has almost no effect on Kelvin waves.

Other important effects of the barotropic/meridional shear background on linear stability results of the multicloud model concern the deformation of the horizontal structure of the CCWs. The convectively coupled Kelvin wave acquires a nontrivial meridional velocity component that contributes (positively) a significant amount to the horizontal convergence—a key physical mechanism in organized tropical convection. This is consistent with the results in Ferguson et al. (2009) in which the interaction of a dry Kelvin wave with a background barotropic flow is shown to produce a meridional wind component proportional to the meridional gradient (shear) of the background flow and the zonal wavenumber of the wave itself. Such effects of meridional shear on Kelvin waves are commonly observed in nature. They are qualified as non-Kelvin aspects of CCWs by G. N. Kiladis (2009, personal communication) because of the deviation from the linear free Kelvin waves (Matsuno 1966). Similar wind distortion and formation of the meridional wind are also present in a Kelvin wave evolving in a nonhomogeneous background consisting of precipitating and nonprecipitating regions, mimicking the ITCZ (Dias and Pauluis 2009).

In addition to the results listed above, strong background vertical and meridional shears destabilize a family of “moisture modes” characterized by a strong moisture component, as seen on the bar diagrams in Fig. 15. The moisture mode is common in simple convective parameterization models (Neelin and Yu 1994; Fuchs and Raymond 2002; KM06). It occurs as a standing, or nearly standing, unstable or neutrally stable mode. Recall that in KM06 the moisture mode emerges when the low-level moisture convergence parameter *λ̃* is set to zero. The moisture mode is often thought to be a surrogate to a MJO mode. However, according to the pictures in Fig. 15, the moisture modes obtained here are clearly far from being a “MJO mode” or any other equatorial wave because they are not trapped near the equator: their dynamical and convective activity is located a considerable distance away from the equator, where the most prominent cyclonic and anticyclonic gyres are at work. They probably have more in common with the congestus mode reported in KM08b, which is characterized by congestus cloud activity at some latitude on both sides of the equator, and is believed to play a central role in preconditioning and moistening the tropical middle troposphere prior to deep convection. Nevertheless, congestus cloud decks are often observed on the north and south flanks of the MJO envelope (Lin and Johnson 1996) and, in the more general context of mean-seasonal convective variability in the extratropics, where congestus cloudiness is not accompanied with deep convection (Takayabu et al. 2010). These are two distinct examples from observations that somehow support the theory for such nontrapped congestus/moisture modes.

Sensitivity to parameters that are proper to the multicloud parameterization are documented in KM06, KM08a, and KM08b. It is found there that key parameters such as congestus and stratiform fractions *α _{c}*,

*α*,

_{s}*μ*and convective time scale parameters such as

*a*

_{0},

*a*

_{1},

*a*

_{2}, as well as the background climatology

*θ*

_{eb}−

*θ*

_{em}, are all important. However, besides increasing/decreasing the strength of the instability of convectively coupled (gravity, including Kelvin and

*n*= 0 EIG) waves or the moisture and/or congestus modes, changes in such parameters do not influence the direction of propagation of the unstable waves—other than that low-index gravity-like, and thus eastward moving (i.e., Kelvin and

*n*= 0 EIG) modes, are always the first to be destabilized, followed immediately by the

*n*= 1 WIG (two-day) waves: they are unable to destabilize slow-moving (nongravity) equatorial waves (i.e., Rossby and MRG waves).

## 8. Summary and conclusions

A linear stability analysis in a sheared environment on an equatorial beta plane, for the multicloud model for convectively coupled equatorial waves introduced recently by Khouider and Majda (2006, 2008a, 2008b), is presented here. The linearized equations about a radiative–convective equilibrium (RCE) are projected in the meridional direction using the Galerkin projection procedure utilized in KM08b, which is a common technique in tropical meteorology (Webster 1972; Gill 1980; Webster and Chang 1988; Zhang and Webster 1989; Yang et al. 2003). Vertical/baroclinic and meridional/barotropic shear flows were considered separately in a parameter regime for which the shear-free multicloud model exhibits synoptic-scale instability of Kelvin and *n* = 0 EIG waves only, with moderate growth rates not exceeding 0.2 day^{−1}. When the background shear is added, the maximum growth rates increase significantly with the strength of the mean wind, and new wave instabilities appear and/or disappear dependent on the strength and type of the wind shear.

It is found here (in section 3) that the baroclinic/vertical shear does not destabilize new CCWs besides those that can be destabilized with respect to some parameter changes in the convective parameterization for the original multicloud model without mean shear background (see KM08b). However, the type of CCWs that are destabilized depend on whether they propagate in the direction of the wind shear at high and/or low levels. Specifically, both high- and low-level westerlies are seen to favor the instability of eastward propagating CCWs, and high-level easterlies favor the instability of westward propagating CCWs while the waves moving in the opposite direction are typically stabilized. This is consistent with the fact that the multicloud instability mechanism is due to both (high-level) stratiform heating and (low-level) moisture and congestus processes (KM06; KM08a). Interestingly, in the event of competing high-level easterlies and low-level westerlies, the model exhibits instability of both eastward and westward waves, though the growth of the easterly waves is relatively small; however, when the easterlies are below westerlies, only eastward waves are unstable.

Moreover, the barotropic/meridional zonal wind shear is found to destabilize westward MRG and Rossby waves due to pure shear instability of the dry dynamics. Effects on the deformation of the structure and changes in trapping distance of CCWs by the imposed meridionally varying background wind are also reported. In addition to the appearance of a meridional velocity component in the Kelvin wave, the structures of the two-day and Rossby waves seem to be the most affected by the meridional shear. The two-day waves adapt a crescentlike shape in reaction to the meridional shear, while the Rossby waves develop elongated tails that are slanted toward the rear of the wave and away from the equator, thus providing a possible feedback to the midlatitudes. The MRG wave and, to some extent, the symmetric Rossby and WIG waves shown here are less trapped in the vicinity of the equator when compared to their second baroclinic dry analogs, although the horizontal gyres are of comparable sizes while the antisymmetric Rossby is significantly more trapped.

## Acknowledgments

The research of B. K. is sponsored in part by a grant from the Natural Sciences and Engineering Research Council of Canada and a grant from the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS). Y. H. is a postdoctoral fellow supported through B. K.’s CFCAS grant. We thank the three anonymous referees for their insightful and helpful comments.

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,*J. Atmos. Sci.***56****,**374–399.Wheeler, M., G. N. Kiladis, and P. J. Webster, 2000: Large-scale dynamical fields associated with convectively coupled equatorial waves.

,*J. Atmos. Sci.***57****,**613–640.Xie, X., and B. Wang, 1996: Low-frequency equatorial waves in vertically sheared zonal flow. Part II: Unstable waves.

,*J. Atmos. Sci.***53****,**3589–3605.Yang, G. Y., B. Hoskins, and J. Slingo, 2003: Convectively coupled equatorial waves: A new methodology for identifying wave structures in observational data.

,*J. Atmos. Sci.***60****,**1637–1654.Yang, G. Y., B. Hoskins, and J. Slingo, 2007: Convectively coupled equatorial waves. Part III: Synthesis structure and their forcing and evolution.

,*J. Atmos. Sci.***64****,**3438–3451.Zhang, C., 2005: Madden-Julian oscillation.

,*Rev. Geophys.***43****,**RG2003. doi:10.1029/2004RG000158.Zhang, C., and P. Webster, 1989: Effects of zonal flows on equatorially trapped waves.

,*J. Atmos. Sci.***46****,**3632–3652.

## APPENDIX

### Galerkin Truncation in the Meridional Direction

Here we present some details of the Galerkin truncation procedure used in section 2 for the multicloud equations with a background meridional shear. The general meridional truncation methodology, as applied, is discussed in Majda and Khouider (2001) and Majda et al. (2004) and the detailed application to the multicloud model is presented in KM08b. Therefore, it is not repeated here, and we will focus only on the advective terms by the meridional shear that are new to this paper.

**U**= (

*u*

_{1},

*u*

_{2},

*υ*

_{1},

*υ*

_{2},

*θ*

_{1},

*θ*

_{2},

*q*,

*θ*,

_{eb}*H*,

_{s}*H*) represents a perturbation about the RCE solution and 𝗔

_{c}_{1}, 𝗔

_{2}, 𝗥

_{1}, 𝗥

_{2}, 𝗥

_{3}, 𝗕 are 10 × 10 matrices representing the different forcing terms in the multicloud equations. To solve for linear waves along the zonal (

*x*) direction, the Galerkin truncation procedure using

*N*= 15 meridional basis functions consisting of the parabolic cylinder functions is applied to (A.1) as in KM08b. Because of the meridional shear background,

*x*,

*t*):

**U**in (A.1) is approximated via the Galerkin truncation

**U**is used in (A.2) to represent the solution

**U**(

*x*,

*t*,

*y*) and the vector of meridional components

**U**= (

*U*)

^{n}_{0≤n≤15}. The symbols 𝘆

*and ∂/∂𝘆*

_{N}*, whose precise definitions are given in Majda and Khouider (2001), are linear operators (matrices), in the finite dimension subspace spanned by the 15 parabolic cylinder basis functions that approximate respectively the multiplication by*

_{N}*y*and the

*y*derivative of an arbitrary function of

*y*expanded in terms of the parabolic cylinder functions. The operators

*y*and

_{l}*H*

_{l}are respectively the abscissas and weights of the Hermite–Gauss quadrature and

*a*=

_{l}*H*

_{l}

*u*

_{0}(

*y*),

_{l}*ũ*= ∂

_{kx}*Ũ*/∂

_{k}*x*. So,

_{2}(

*u*

_{0}∂

**U**/∂

*x*)

^{n}is given by

_{3}(

**U**∂

*u*

_{0}/∂

*y*)

^{n}is obtained in the same fashion and is given by

Dispersion diagrams and growth rates (filled circles) for the control case when the background shear is zero. Each dotted line corresponds to a dispersion relation, counting all the equatorial modes associated with the first and second baroclinic modes up to meridional index *n* = 13 plus additional modes due to the four extra equations (*q*, *θ _{eb}*,

*H*,

_{s}*H*). (left) The symmetric and (right) the antisymmetric modes. The unstable modes are indicated by the thick circles whose diameters are proportional to the growth rate. The overall maximum growth rate, corresponding to the largest circle, is displayed on top of each panel and the maximum growth rate and the corresponding phase speed are displayed next to the wave name. The associated multicloud parameters are reported in Table 2.

_{c}Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Dispersion diagrams and growth rates (filled circles) for the control case when the background shear is zero. Each dotted line corresponds to a dispersion relation, counting all the equatorial modes associated with the first and second baroclinic modes up to meridional index *n* = 13 plus additional modes due to the four extra equations (*q*, *θ _{eb}*,

*H*,

_{s}*H*). (left) The symmetric and (right) the antisymmetric modes. The unstable modes are indicated by the thick circles whose diameters are proportional to the growth rate. The overall maximum growth rate, corresponding to the largest circle, is displayed on top of each panel and the maximum growth rate and the corresponding phase speed are displayed next to the wave name. The associated multicloud parameters are reported in Table 2.

_{c}Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Dispersion diagrams and growth rates (filled circles) for the control case when the background shear is zero. Each dotted line corresponds to a dispersion relation, counting all the equatorial modes associated with the first and second baroclinic modes up to meridional index *n* = 13 plus additional modes due to the four extra equations (*q*, *θ _{eb}*,

*H*,

_{s}*H*). (left) The symmetric and (right) the antisymmetric modes. The unstable modes are indicated by the thick circles whose diameters are proportional to the growth rate. The overall maximum growth rate, corresponding to the largest circle, is displayed on top of each panel and the maximum growth rate and the corresponding phase speed are displayed next to the wave name. The associated multicloud parameters are reported in Table 2.

_{c}Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but for (top) _{1} = 2.5, _{2} = 0 m s^{−1} and (bottom) _{1} = 7.5, _{2} = 0 m s^{−1} on left page [case of westerlies on bottom and easterlies aloft (Fig. 1, profile 1.a)] and for (top) _{1} = −2.5 m s^{−1}, _{2} = 0 and (bottom) _{1} = −7.5 m s^{−1}, _{2} = 0 on right page [case of easterlies on bottom and westerlies aloft (Fig. 1, profile 1.b)].

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but for (top) _{1} = 2.5, _{2} = 0 m s^{−1} and (bottom) _{1} = 7.5, _{2} = 0 m s^{−1} on left page [case of westerlies on bottom and easterlies aloft (Fig. 1, profile 1.a)] and for (top) _{1} = −2.5 m s^{−1}, _{2} = 0 and (bottom) _{1} = −7.5 m s^{−1}, _{2} = 0 on right page [case of easterlies on bottom and westerlies aloft (Fig. 1, profile 1.b)].

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but for (top) _{1} = 2.5, _{2} = 0 m s^{−1} and (bottom) _{1} = 7.5, _{2} = 0 m s^{−1} on left page [case of westerlies on bottom and easterlies aloft (Fig. 1, profile 1.a)] and for (top) _{1} = −2.5 m s^{−1}, _{2} = 0 and (bottom) _{1} = −7.5 m s^{−1}, _{2} = 0 on right page [case of easterlies on bottom and westerlies aloft (Fig. 1, profile 1.b)].

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = 0, _{2} = 7.5 m s^{−1} and (bottom) _{1} = 0, _{2} = −7.5 m s^{−1}. (top) Westerlies on top and bottom (Fig. 1, profile 2.a); (bottom) easterlies on top and bottom (Fig. 1, profile 2.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = 0, _{2} = 7.5 m s^{−1} and (bottom) _{1} = 0, _{2} = −7.5 m s^{−1}. (top) Westerlies on top and bottom (Fig. 1, profile 2.a); (bottom) easterlies on top and bottom (Fig. 1, profile 2.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = 0, _{2} = 7.5 m s^{−1} and (bottom) _{1} = 0, _{2} = −7.5 m s^{−1}. (top) Westerlies on top and bottom (Fig. 1, profile 2.a); (bottom) easterlies on top and bottom (Fig. 1, profile 2.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = _{2} = 7.5 m s^{−1} and (bottom) _{1} = _{2} = −7.5 m s^{−1}. (top) Surface westerlies only (Fig. 1, profile 3.a); (bottom) surface easterlies only (Fig. 1, profile 3.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = _{2} = 7.5 m s^{−1} and (bottom) _{1} = _{2} = −7.5 m s^{−1}. (top) Surface westerlies only (Fig. 1, profile 3.a); (bottom) surface easterlies only (Fig. 1, profile 3.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = _{2} = 7.5 m s^{−1} and (bottom) _{1} = _{2} = −7.5 m s^{−1}. (top) Surface westerlies only (Fig. 1, profile 3.a); (bottom) surface easterlies only (Fig. 1, profile 3.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = −_{2} = 7.5 m s^{−1} and (bottom) _{1} = −_{2} = −7.5 m s^{−1}. (top) High-level easterlies only (Fig. 1, profile 4.a); (bottom) high-level westerlies only (Fig. 1, profile 4.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = −_{2} = 7.5 m s^{−1} and (bottom) _{1} = −_{2} = −7.5 m s^{−1}. (top) High-level easterlies only (Fig. 1, profile 4.a); (bottom) high-level westerlies only (Fig. 1, profile 4.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but with (top) _{1} = −_{2} = 7.5 m s^{−1} and (bottom) _{1} = −_{2} = −7.5 m s^{−1}. (top) High-level easterlies only (Fig. 1, profile 4.a); (bottom) high-level westerlies only (Fig. 1, profile 4.b).

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Structure of the basic barotropic shear flow with different strengths. The *x* axis shows the value of the wind in nondimensional units of 50 m s^{−1}: Equatorial (a) easterly and (b) westerly wind.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Structure of the basic barotropic shear flow with different strengths. The *x* axis shows the value of the wind in nondimensional units of 50 m s^{−1}: Equatorial (a) easterly and (b) westerly wind.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Structure of the basic barotropic shear flow with different strengths. The *x* axis shows the value of the wind in nondimensional units of 50 m s^{−1}: Equatorial (a) easterly and (b) westerly wind.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but for the case of meridional–barotropic shear background with an easterly wind at the equator. Left page: (top) *u*_{0} = 5 m s^{−1}, (bottom) *u*_{0} = 10 m s^{−1}. Right page: (top) *u*_{0} = 15 m s^{−1}, (bottom) *u*_{0} = 20 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but for the case of meridional–barotropic shear background with an easterly wind at the equator. Left page: (top) *u*_{0} = 5 m s^{−1}, (bottom) *u*_{0} = 10 m s^{−1}. Right page: (top) *u*_{0} = 15 m s^{−1}, (bottom) *u*_{0} = 20 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 2 but for the case of meridional–barotropic shear background with an easterly wind at the equator. Left page: (top) *u*_{0} = 5 m s^{−1}, (bottom) *u*_{0} = 10 m s^{−1}. Right page: (top) *u*_{0} = 15 m s^{−1}, (bottom) *u*_{0} = 20 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 8 but for the case of meridional–barotropic shear background with a westerly wind at the equator. Note change in scale on the *x* and *y* axes for the left bottom panel on right page.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 8 but for the case of meridional–barotropic shear background with a westerly wind at the equator. Note change in scale on the *x* and *y* axes for the left bottom panel on right page.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 8 but for the case of meridional–barotropic shear background with a westerly wind at the equator. Note change in scale on the *x* and *y* axes for the left bottom panel on right page.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

The structure of the convectively coupled Kelvin wave in a meridional shear background for the case of a weak equatorial easterly wind with *u*_{0} = 5 m s^{−1}. (top left) Bar diagram showing the relative strength of each physical variable (*u*_{1}, *u*_{2}, *υ*_{1}, *υ*_{2}, *θ*_{1}, *θ*_{2}, *q*, *θ _{eb}*,

*H*, or

_{s}*H*) for the given eigenmode. Each variable,

_{c}*X*, is represented by a group of bars corresponding to its different meridional components,

*X*,

^{l}*l*= 0, … ,

*N*(counting only the even or odd values according to the wave and variable) with the index

*l*increasing from left to right (see KM08b for details). (top right) Zonal structure of the zonal and meridional divergence terms (div

*y*and div

*x*). The quantities div1 and div2 on top of the panel are the relative contributions of the meridional convergence for the first and second baroclinic velocity components. (bottom) Zonal structure of the potential temperature (shaded) and heating contours at 8 km and the surface horizontal velocity (arrows) of the (left) convectively coupled Kelvin wave compared to the potential temperature and horizontal velocity profiles of the (right) second baroclinic dry Kelvin wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

The structure of the convectively coupled Kelvin wave in a meridional shear background for the case of a weak equatorial easterly wind with *u*_{0} = 5 m s^{−1}. (top left) Bar diagram showing the relative strength of each physical variable (*u*_{1}, *u*_{2}, *υ*_{1}, *υ*_{2}, *θ*_{1}, *θ*_{2}, *q*, *θ _{eb}*,

*H*, or

_{s}*H*) for the given eigenmode. Each variable,

_{c}*X*, is represented by a group of bars corresponding to its different meridional components,

*X*,

^{l}*l*= 0, … ,

*N*(counting only the even or odd values according to the wave and variable) with the index

*l*increasing from left to right (see KM08b for details). (top right) Zonal structure of the zonal and meridional divergence terms (div

*y*and div

*x*). The quantities div1 and div2 on top of the panel are the relative contributions of the meridional convergence for the first and second baroclinic velocity components. (bottom) Zonal structure of the potential temperature (shaded) and heating contours at 8 km and the surface horizontal velocity (arrows) of the (left) convectively coupled Kelvin wave compared to the potential temperature and horizontal velocity profiles of the (right) second baroclinic dry Kelvin wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

The structure of the convectively coupled Kelvin wave in a meridional shear background for the case of a weak equatorial easterly wind with *u*_{0} = 5 m s^{−1}. (top left) Bar diagram showing the relative strength of each physical variable (*u*_{1}, *u*_{2}, *υ*_{1}, *υ*_{2}, *θ*_{1}, *θ*_{2}, *q*, *θ _{eb}*,

*H*, or

_{s}*H*) for the given eigenmode. Each variable,

_{c}*X*, is represented by a group of bars corresponding to its different meridional components,

*X*,

^{l}*l*= 0, … ,

*N*(counting only the even or odd values according to the wave and variable) with the index

*l*increasing from left to right (see KM08b for details). (top right) Zonal structure of the zonal and meridional divergence terms (div

*y*and div

*x*). The quantities div1 and div2 on top of the panel are the relative contributions of the meridional convergence for the first and second baroclinic velocity components. (bottom) Zonal structure of the potential temperature (shaded) and heating contours at 8 km and the surface horizontal velocity (arrows) of the (left) convectively coupled Kelvin wave compared to the potential temperature and horizontal velocity profiles of the (right) second baroclinic dry Kelvin wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 10 but for the *n* = 1 westward inertio-gravity (two-day) wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 10 but for the *n* = 1 westward inertio-gravity (two-day) wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 10 but for the *n* = 1 westward inertio-gravity (two-day) wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 10 but for the (westward) mixed Rossby–gravity wave in the case of a moderate easterly wind *u*_{0} = 15 m s^{−1}: (top right) vertical structure of the potential temperature (shaded), heating and cooling anomalies (contours), and the *u*–*w* velocity components (arrows) meridionally averaged over the Northern Hemisphere.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 10 but for the (westward) mixed Rossby–gravity wave in the case of a moderate easterly wind *u*_{0} = 15 m s^{−1}: (top right) vertical structure of the potential temperature (shaded), heating and cooling anomalies (contours), and the *u*–*w* velocity components (arrows) meridionally averaged over the Northern Hemisphere.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 10 but for the (westward) mixed Rossby–gravity wave in the case of a moderate easterly wind *u*_{0} = 15 m s^{−1}: (top right) vertical structure of the potential temperature (shaded), heating and cooling anomalies (contours), and the *u*–*w* velocity components (arrows) meridionally averaged over the Northern Hemisphere.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 12 but for the symmetric *n* = 1 Rossby wave in the case of a strong easterly wind *u*_{0} = 20 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 12 but for the symmetric *n* = 1 Rossby wave in the case of a strong easterly wind *u*_{0} = 20 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 12 but for the symmetric *n* = 1 Rossby wave in the case of a strong easterly wind *u*_{0} = 20 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 13 but for the antisymmetric (*n* = 2) Rossby wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 13 but for the antisymmetric (*n* = 2) Rossby wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

As in Fig. 13 but for the antisymmetric (*n* = 2) Rossby wave.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Physical and dynamical structure of the (top) low, (middle) moderate, and (bottom) high moisture modes: (left) bar diagrams and (right) horizontal structure of the vertical velocity contours and the zonal velocity arrows at the top of the troposphere.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Physical and dynamical structure of the (top) low, (middle) moderate, and (bottom) high moisture modes: (left) bar diagrams and (right) horizontal structure of the vertical velocity contours and the zonal velocity arrows at the top of the troposphere.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Physical and dynamical structure of the (top) low, (middle) moderate, and (bottom) high moisture modes: (left) bar diagrams and (right) horizontal structure of the vertical velocity contours and the zonal velocity arrows at the top of the troposphere.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Frequency and growth rate diagrams for the dry system demonstrating the pure shear instability of certain waves for the case of a (top and middle) vertical/baroclinic and (bottom) barotropic/meridional shear background: (top) *u*_{0} = 0, _{1} = 7.5, and _{2} = 0 m s^{−1}; (middle) *u*_{0} = 0, _{1} = 2.5, and _{2} = 2.5 m s^{−1}; (bottom) *u*_{0} = 20 m s^{−1}, _{1} = _{2} = 0.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Frequency and growth rate diagrams for the dry system demonstrating the pure shear instability of certain waves for the case of a (top and middle) vertical/baroclinic and (bottom) barotropic/meridional shear background: (top) *u*_{0} = 0, _{1} = 7.5, and _{2} = 0 m s^{−1}; (middle) *u*_{0} = 0, _{1} = 2.5, and _{2} = 2.5 m s^{−1}; (bottom) *u*_{0} = 20 m s^{−1}, _{1} = _{2} = 0.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Frequency and growth rate diagrams for the dry system demonstrating the pure shear instability of certain waves for the case of a (top and middle) vertical/baroclinic and (bottom) barotropic/meridional shear background: (top) *u*_{0} = 0, _{1} = 7.5, and _{2} = 0 m s^{−1}; (middle) *u*_{0} = 0, _{1} = 2.5, and _{2} = 2.5 m s^{−1}; (bottom) *u*_{0} = 20 m s^{−1}, _{1} = _{2} = 0.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

The structure of the most unstable modes in the dry system. (top) Typical highly oscillatory nontrapped mode due to vertical shear instability (_{1} = 7.5 m s^{−1}, _{2} = 0); second baroclinic (middle) MRG and (bottom) Rossby waves that are unstable for a barotropic shear with equatorial easterly wind with *u*_{0} = 20 m s^{−1}. Notice the deformations of the MRG and Rossby waves due to the meridional shear effect similar to their convectively coupled analogs.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

The structure of the most unstable modes in the dry system. (top) Typical highly oscillatory nontrapped mode due to vertical shear instability (_{1} = 7.5 m s^{−1}, _{2} = 0); second baroclinic (middle) MRG and (bottom) Rossby waves that are unstable for a barotropic shear with equatorial easterly wind with *u*_{0} = 20 m s^{−1}. Notice the deformations of the MRG and Rossby waves due to the meridional shear effect similar to their convectively coupled analogs.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

The structure of the most unstable modes in the dry system. (top) Typical highly oscillatory nontrapped mode due to vertical shear instability (_{1} = 7.5 m s^{−1}, _{2} = 0); second baroclinic (middle) MRG and (bottom) Rossby waves that are unstable for a barotropic shear with equatorial easterly wind with *u*_{0} = 20 m s^{−1}. Notice the deformations of the MRG and Rossby waves due to the meridional shear effect similar to their convectively coupled analogs.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Schematic of the MJO envelope showing a quadruple vortex, indicated by (c) and (a) for cyclonic (c) and anticyclonic (a) gyres, and the implied vertical shears at the equator, characterized by easterlies at the surface and westerlies aloft, in the front (on the right) of the active convection, and by westerlies at the surface surmounted by easterlies aloft, in the back. Convectively coupled waves that are thought to be favored by the prevailing vertical shear background according to the results in Figs. 3 –6 are indicated. Note that Kelvin and *n* = 0 EIG waves are believed to prevail in both the front and back of the MJO active phase owing to the high-level and low-level westerlies, respectively, while higher index EIG and WIG waves are shown only on the front and back due to the prevailing high-level easterlies and westerlies, respectively.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Schematic of the MJO envelope showing a quadruple vortex, indicated by (c) and (a) for cyclonic (c) and anticyclonic (a) gyres, and the implied vertical shears at the equator, characterized by easterlies at the surface and westerlies aloft, in the front (on the right) of the active convection, and by westerlies at the surface surmounted by easterlies aloft, in the back. Convectively coupled waves that are thought to be favored by the prevailing vertical shear background according to the results in Figs. 3 –6 are indicated. Note that Kelvin and *n* = 0 EIG waves are believed to prevail in both the front and back of the MJO active phase owing to the high-level and low-level westerlies, respectively, while higher index EIG and WIG waves are shown only on the front and back due to the prevailing high-level easterlies and westerlies, respectively.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Schematic of the MJO envelope showing a quadruple vortex, indicated by (c) and (a) for cyclonic (c) and anticyclonic (a) gyres, and the implied vertical shears at the equator, characterized by easterlies at the surface and westerlies aloft, in the front (on the right) of the active convection, and by westerlies at the surface surmounted by easterlies aloft, in the back. Convectively coupled waves that are thought to be favored by the prevailing vertical shear background according to the results in Figs. 3 –6 are indicated. Note that Kelvin and *n* = 0 EIG waves are believed to prevail in both the front and back of the MJO active phase owing to the high-level and low-level westerlies, respectively, while higher index EIG and WIG waves are shown only on the front and back due to the prevailing high-level easterlies and westerlies, respectively.

Citation: Journal of the Atmospheric Sciences 67, 9; 10.1175/2010JAS3335.1

Forcing variables of the multicloud parameterization.

Parameters of the multicloud parameterization.