a. The Lagrangian parcel adjustment scheme

In order to obtain a closed system for convectively driven large-scale dynamics, it is necessary to use a convective parameterization that defines feedbacks from convective activity to the large-scale dynamics and thermodynamics. The specific form of this relationship determines the models discussed in this paper.

In the LPA scheme, the convective updrafts adjust dynamically to positive feedbacks from the fluctuations of the CAPE and to negative feedbacks from the kinetic energy of convective updrafts. The prognostic equation associated with the LPA scheme, formulated in YM²E and YM², has the form

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In this expression, B = θ_eb − θ^*_em is the convective buoyancy, where saturated moist potential temperature, θ^*_em, is related to the dry potential temperature by

θ^*_emγθ.

The term (c_pΓ_mB)/(θ₀) in (2.12) expresses buoyancy force due to the fluctuations in CAPE, and (w²_c)/(2H) stands for kinetic energy of updrafts. The Heaviside factor is utilized here to emphasize that convection can occur only in the positive vertical direction, with warm and moist air evaporating from the ocean surface and rising through the boundary layer.

In the LPA parametrization, the prognostic variables involve the large-scale wind, v_H; the dry potential temperature, θ; the vertical velocity of convective updrafts, w_c; the moist equivalent potential temperature in the boundary layer, θ_eb; and the averaged moist potential temperature, 〈θ_e〉_z, or equivalently, θ_em. Downdrafts, both environmental and convective, are determined diagnostically, using conservation of mass in (2.2) and the relation in (2.8).

b. Instantaneous adjustment to CAPE fluctuations

The LPA equation in (2.12) expresses dynamic adjustment of convective updrafts to the variations in CAPE. When CAPE fluctuations are entirely balanced by the kinetic energy of updrafts, the left-hand side of (2.12) vanishes and no dynamics for the small-scale vertical velocity occurs. Thus, in the ICAPE scheme the equation in (2.12) is replaced by the diagnostic relation

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Here, the Heaviside factor is utilized because only positive CAPE fluctuations can drive convective updrafts. Otherwise, all other equations in the ICAPE model are exactly equivalent to their analogs in the LPA scheme.

c. Quasi-equilibrium scheme

Following YM²E and YM², no buoyancy fluctuations are permitted and B = 0, based on the original instantaneous statistical QE idea of Arakawa and Schubert (1974). This restriction of zero buoyancy yields that

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By using (2.15) and combining the equations for θ and θ_eb, one easily obtains a nonlinear equation for the intensity of convective updrafts in terms of the large-scale winds and remaining thermodynamic variables:

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The number of prognostic variables needed to describe the dynamics based on the QE scheme is further decreased, as compared with the LPA and ICAPE schemes. Only the large-scale winds, v_H; the dry potential temperature, θ; and the quantity 〈θ_e〉_z are obtained prognostically. The moist equivalent potential temperature θ_eb is determined trivially from the CAPE conservation condition, θ_eb = γθ, while w_c is found from (2.16). Environmental descent and convective downdrafts are related to the other variables through the same diagnostic relations as in the LPA and ICAPE schemes.

a. Radiative–convective equilibrium and the linear system

In radiative–convective equilibrium (RCE), forcing generated by fluctuations in the SST is exactly balanced by radiative cooling from the top of the troposphere through time-independent convective updrafts and downdrafts. Since σ_c is typically a small number, the approximation (1 − σ_c) ≈ 1 in (2.4) will be henceforth used for simplicity. The following radiative–convective equilibrium conditions, valid for both the LPA and ICAPE schemes, are

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The relations above define equilibrium variables denoted by bars. The remaining variables, such as w_e and w_d, can be recovered by using diagnostic relations (2.2) and (2.8). A key point is that with the turbulent drag in (2.11), an RCE state can be defined in a meaningful way even if the barotropic mean wind is zero.

Henceforth it will be convenient to use nondimensional variables that are naturally set by the RCE state. In particular, the following velocity scale for large-scale dynamics is utilized, , which coincides with the propagation speed of dry Kelvin wave. The equatorial Rossby radius sets the length scale, L = 1500 km. These scales yield the timescale, T = L/V ≈ ⅓ day. Note that with this choice of scales the equatorial parameter β has a unit nondimensional size. A natural scale for temperature fluctuations is given by the moist saturated equivalent potential temperature in the boundary layer, θ^*_eb. Finally, the value of equilibrium convective updrafts,

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sets the scale for fluctuations in w_c.

The linearization procedure for the three parameterizations, LPA, ICAPE, and QE, about the RCE is straightforward. The resulting linear systems for all three parameterizations are reported below in the nondimensional units described earlier. In these equations, advection by the barotropic mean wind has been removed by using a standard Galilean shift, x′ = x − ut. Here in section 3, the effects of rotation are also neglected by assuming a quasi-one-dimensional structure in the zonal direction for all the variables, or equivalently, by ignoring y variations. Under these assumptions the linearized equations around the RCE state become

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with the constants

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For simplicity, the primes in linearized equations will be discarded throughout the rest of this section. It is easy to verify that large-scale horizontal wind and dry temperature in the equations in (3.3) evolve according to the linear damped shallow-water equations, coupled with moist thermodynamic processes through the convective updrafts, in the last term of the second equation in (3.3).

In the following sections various properties of linear eigenmodes in the form, U = Ce^{i(kx−ω(k)t) + d(k)t}, will be discussed and compared for the linearized systems that arise for several convective parameterizations considered in this paper. In this analysis, for each wavenumber, k (or wavelength, 40 000/k km), real and imaginary parts of a corresponding eigenvalue are treated separately using the following terminology. The properly scaled real part, d(k), is naturally called the growth rate, while the imaginary part, ω(k), is referred to as the phase. The scaled ratio ω(k)/k and derivative dω(k)/dk are called the generalized phase velocity (GPhV) and generalized group velocity (GGrV). Naturally, GPhV and GGrV coincide with regular phase and group velocity only if the underlying eigenvalue is purely imaginary and has no growth associated with it.

b. Waves in the LPA system

The system of linearized equations for LPA in (3.3) involves five prognostic equations and, therefore, at each wavenumber it has five complex eigenvalues. The meaning of all five branches of eigenvalues will be illustrated for the standard set of physical constants from Table 1, along with easterly mean wind, u = −2 m s⁻¹, and turbulent velocity fluctuations with intensity, u₀ = 2 m s⁻¹. Figure 1 presents the GPhV diagram for this case. Since the GPhV diagram is essentially symmetric about ω(k) = 0, it suffices to present only its upper part, where ω(k)/k > 0. Among the five branches, there are two eastward waves, shown in Fig. 1, with ω(k) > 0, two westward waves with ω(k) < 0, and one wave that is essentially standing, with ω(k) ≈ 0, for all wavenumbers k.

In order to give a physical interpretation of the various waves, it is important to analyze the eigenvector structure corresponding to each of the eigenvalues. An economical and illustrative way of representing eigenvector structure is given by using bar diagrams (such as depicted in Figs. 2 and 4). Here, each of the variables is represented by a single bar on a diagram, weighted according to the magnitude of the corresponding (typically complex valued) nondimensional component of the eigenvector. Each magnitude is then multiplied by its appropriate dimensional scale, and the overall set of bars is normalized so that it has a total unit weight. Such a representation allows one to compare relative intensities of various components in an eigenmode.

For the LPA scheme, one pair of eastward and westward propagating waves is significantly faster than the second pair of propagating waves, for all wavenumbers reported in Fig. 1. For example, the generalized phase velocity of the slower pair of waves at a supercluster scale, L = 2000 km, is approximately 12 m s⁻¹, while the faster waves with the same wavenumber propagate with an enormous GPhV of approximately 700 m s⁻¹. Note that the generalized phase velocity of the fast waves is large even at the small scales, with the smallest value of 50 m s⁻¹, which is achieved in the limit of large k. The speed of dry gravity wave, 50 m s⁻¹, is also shown in this figure as a benchmark.

The bar diagrams in Figs. 2a,b show that the faster modes are essentially dominated by small-scale convective gusts, expressed by the variable w_c, coupled to variations in the boundary layer equivalent potential temperature, θ_eb, in a wide range of scales, illustrated with a relatively large scale, L = 5000 km, and a smaller scale of L = 200 km. On the other hand, the slower waves, illustrated in Figs. 2c,d, carry strong fluctuations of the large-scale wind, u, coupled with the dry potential temperature fluctuations through a mechanism similar to dry gravity waves. Convective coupling of these waves, reflected by the large amplitudes of fluctuations in w_c, θ_eb, and 〈θ_e〉_z in Figs. 2c,d, causes a significant decrease in phase velocity from 50 m s⁻¹, the value for dry gravity waves, to approximately 12 m s⁻¹. These points indicate that one may identify the slower pair of waves as the eastward and westward moist gravity waves, while the faster pair of waves can be called the superfast waves, motivated by their extremely large phase velocities. The remaining standing wave corresponds to the almost conservation of 〈θ_e〉_z discussed at the beginning of section 2, and is dominated by the fluctuations of 〈θ_e〉_z; this wave will be referred to as the 〈θ_e〉_z mode.

It is worthwhile to analyze the full physical picture associated with eigenvectors of (3.3) for all waves involved in this system. The intensity of updraft convective activity in the ICAPE parametrization is diagnostically linked to the large-scale flow (which is entirely two-dimensional in the zonal and vertical directions) and, therefore, must be in phase with the ascending branch of the large-scale circulation. This effect is indeed observed for moist gravity waves and the 〈θ_e〉_z mode (not graphed here due to the space limitations). For the superfast waves, though, the opposite occurs, where convective updrafts are out of phase with the ascending branch of the large-scale circulation. This fact, along with the superfast phase velocities and the eigenstructure from Figs. 2a,b dominated by vertical gusts, demonstrates that the superfast modes are, in fact, nonphysical parasitic waves, generated as artifacts of the LPA convective parameterization.

Up to this point the discussion involved only the wave properties of normal modes based on the imaginary parts of eigenvalues. On the other side, the stability properties, associated with growth rates, select the modes that will most probably play an important role in the fully nonlinear regime of dynamics. For the current example it is observed that in response to easterly mean wind, the eastward moist gravity wave (the slower eastward branch in Fig. 1) has instability in the wide range of wavelengths, 150 km < L < 20 000 km. The peak of instability is located near L = 1000 km, with the maximum growth rate of approximately 1.2 day⁻¹. The growth rate diagram is represented by a long-dashed curve in Fig. 3a. All other modes in this example are damped at all wavenumbers with various damping rates, except for the 〈θ_e〉_z mode, which is essentially neutral. Clearly, due to the absence of rotation or any other symmetry-breaking effects apart from the direction of mean wind in (3.3), west and east are entirely equivalent here. Therefore, the response to westerly mean wind is exactly symmetric, with the same instability of a westward moist gravity wave instead of an eastward moist gravity wave. Following common practice, easterly mean winds will be given a preference here since they naturally mimic circulation associated with a Walker cell observed in the Pacific.

Finally, it is important to point out that the superfast waves can themselves create weak linear instabilities. For example, in the regime involving the set of parameters from Table 1 completed with u₀ = 0.1 m s⁻¹ and u = −0.1 m s⁻¹, the superfast waves are unstable at the short wavelengths, L < 300 km. This situation can be distinctly problematic for numerical models operating in the nonlinear regime, since the fast moving modes without physical significance can rapidly propagate and contaminate the solutions through nonlinear interactions. Therefore, it is desirable to design convective parameterizations that do not have superfast or other physically meaningless and potentially harmful modes.

The linearized analysis for the LPA scheme from YM² only examined special solutions involving the linearized moist gravity waves; the more detailed analysis of the complete set of all normal-mode solutions developed here reveals the existence of such nonphysical parasitic waves for the LPA scheme. The nonlinear numerical solutions with the LPA scheme presented in YM²E are extremely noisy. It is interesting to speculate that one nonphysical contribution to this noise involves the parasitic waves documented here. This is an interesting topic for future investigation.

c. Waves in the ICAPE system

In the ICAPE system dynamic adjustment to CAPE fluctuations is replaced by instantaneous adjustment. Mathematically, the dimension of the linearized system and the number of normal modes in (3.3) are each reduced by one. Among the four normal modes in the ICAPE system, three can be identified as analogs of waves in the LPA system. Indeed, two of the modes are almost entirely equivalent to eastward and westward moist gravity waves, both in stability and wave properties, as illustrated in Fig. 3 for the eastward moist gravity waves. Figure 4 depicts the bar diagram for the eigenvector structure of the moist gravity wave in the ICAPE system at sample wavelengths 5000 and 200 km. It is easy to see that the eigenvectors are virtually indistinguishable from those depicted for the LPA-generated moist gravity waves in Figs. 2c,d. The ICAPE-generated 〈θ_e〉_z mode, not displayed here, is also entirely similar to its LPA-generated analog.

The remaining normal mode generated by the ICAPE scheme, however, has no counterpart in the LPA system. It is virtually a standing convective adjustment mode, with enormous damping rates at all wavenumbers, d(k) ≈ −1000 day⁻¹. At all wavenumbers, the eigenstructure of this mode has essentially all of its amplitude proportional to w_c; thus, this mode represents rapid damping of CAPE perturbations. This single mode replaces the two parasitic superfast waves that are observed in the LPA scheme. The authors note that the superfast waves arise due to the presence of a large constant,

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in the nondimensional version of the dynamic CAPE adjustment equation in (2.12) that sets a short timescale for spurious waves. In the ICAPE scheme this mechanism is shut off, which removes the unwanted superfast modes and replaces them with a superdamped standing wave, while keeping all physically relevant modes intact. The superdamped mode is associated with rapid adjustment to equilibrium of CAPE perturbations. This indicates that the ICAPE scheme has some features similar to CAPE adjustment schemes (Neelin and Yu 1994).

Moist gravity waves in the ICAPE system have stability properties that are almost entirely equivalent to those of moist gravity waves of the LPA system, as follows from the direct comparison of solid and long-dashed curves in Fig. 3a. In particular, the maximum growth rate in both schemes is approximately equal to 1.2 day⁻¹. This rate corresponds to a moist gravity wave with wavelength 1000 km. The band of unstable waves for both schemes covers wavelengths in the range 150 km < L < 20 000 km. The maximum growth rate, the wavelength of the most unstable mode, and the range of instability all form useful diagnostics that will be utilized later in section 4c to address stability properties of the ICAPE system with rotation.

d. Waves in ICAPE scheme without convective downdrafts and in the QE scheme

An interesting question raised earlier in section 2c concerns the importance of small-scale convective downdrafts for the stability properties of convectively coupled models. This issue will be addressed within the simple framework of the ICAPE scheme, where convective downdrafts are switched off by setting ϵ_p = 1 in the equation for θ_eb in (3.3). The dotted curve in Fig. 3a represents the growth rates associated with the ICAPE scheme without downdrafts. A comparison with the solid curve in Fig. 3a shows that instability is much more strongly pronounced without downdrafts, with the growth without bound at smaller wavelengths. This confirms the intuition that convective downdrafts tend to stabilize convectively coupled dynamics. In fact, the scheme without downdrafts is dominated by the effects of environmental convergence and, thus, effectively becomes a wave–conditional instability of the second kind (CISK) parameterization (Kuo 1974; Lindzen 1974; Emanuel 1994), where such catastrophic small-scale instabilities are prominent.

The generalized phase speeds for the ICAPE moist gravity waves without convective downdrafts are represented by the dotted curve in Fig. 3b. It is evident from this figure that at the largest planetary scales, the convective downdrafts have very little influence on the generalized phase speed. For scales shorter than 5000 km this situation changes dramatically. There, the absence of convective downdrafts slows down the waves significantly. For example, at a typical scale of 1000 km the generalized phase speeds of the ICAPE system with and without convective downdrafts are 12 and 3 m s⁻¹, correspondingly, while at a scale, L = 100 km, the discrepancy is even stronger, with phase velocities of 11.5 and 0.8 m s⁻¹.

Next, a brief discussion is given for wave and stability properties of the scheme based on the QE assumption in (2.15). This model involves only three prognostic equations. A simple analysis shows that the three normalmodes include eastward and westward moist gravity waves, extremely similar in their wave properties and eigenvector structure to moist gravity waves in the LPA and ICAPE systems, and the 〈θ_e〉_z mode. The growth rates for eastward moist gravity waves are represented by the dot–dashed line in Fig. 3a. Deviations of the QE growth rates from the ICAPE and LPA growth rates at large scales are relatively weak, while at small scales the picture changes qualitatively. Contrasting with the LPA and ICAPE schemes, where growth rates decrease with k and become negative at a finite wavelength, growth rates in the QE model quickly saturate with k at a positive value close to the maximum growth rate in the ICAPE model and remain at this value for all small scales.

A comparison of the solid and dot–dashed curves in Fig. 3b shows that the wave properties of QE moist gravity waves are remarkably similar to those of ICAPE waves; this is also true for the 〈θ_e〉_z mode. Eigenvector structures of both QE-generated moist gravity waves and the 〈θ_e〉_z mode are indistinguishable from their counterparts for the ICAPE-generated waves in an extremely wide range of scales, with at least 50 km < L < 40 000 km.

e. Summary of properties for ICAPE, LPA, and QE schemes

The results presented in the previous three sections demonstrate that ICAPE, LPA, and QE schemes present very similar wave and stability properties at large scales, as concerns physically relevant moist gravity waves and the 〈θ_e〉_z mode. The behavior at small scales depends on the details of the particular convective parameterization. Of the three schemes, the authors argue that the ICAPE-based model, developed in section 2b, is preferable for numerical simulations in the fully nonlinear regime for the following reasons. The short-wave instability for moist gravity waves is suppressed in the ICAPE scheme, which is not true for the QE scheme, and the ICAPE scheme does not possess the parasitic nonphysical super-fast waves that are present in the LPA scheme. Therefore, the subsequent discussions will be focused on various additional properties for the model based on the ICAPE parameterization, with only occasional references to the other parameterizations.

a. Meridional truncated equations and ICAPE linearization

Here, a reduced set of equations is derived based on truncation of the convectively coupled system in the meridional direction, and the subsequent linearization of these equations near the radiative–convective equilibrium. Formally, the truncation step precedes linearization and will be reported first. However, it can be shown that truncation and linearization can be performed in the reverse order, which is simpler for the actual computations.

For shallow-water dynamics on the equatorial β plane the following Riemann invariants (Gill 1982) are utilized,

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With these variables, the nondimensional equations for wind, u, and dry potential temperature, θ, can be rewritten in terms of the Riemann invariants, q, r, and meridional velocity, υ, as follows:

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where N₁(U), N₂(U), and N₃(U) are the nonlinear advection and drag terms; and U = (q, r, υ, w_c, θ_eb, 〈θ_e〉_z) is the solution vector. The remaining equations are transformed in a similar way and are not reported here for brevity.

In the dry case, the linear part of (4.1) can be decomposed into a sequence of fully decoupled wave systems, based on expansion in terms of the parabolic cylinder functions, D_n:

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Convective feedbacks to the linearized dry dynamics are explicitly represented by the terms with w_c in the first two equations in (4.1). These feedbacks generate nontrivial coupling among wave systems associated with different meridional numbers, n.

The purpose of the meridional truncation procedure is to retain the most observationally prominent waves and to discard the rest of the less important meridionally higher-order waves. For the simplest truncated system, only a few waves are retained that correspond to the four lowest meridional modes, with the following variables: 1) q₀, Kelvin wave; 2) q₁, υ₀, mixed Rossby–gravity waves; 3) q₂, υ₁, r₀, lowest-order Rossby and gravity waves, symmetric about the equator; and 4) q₃, υ₂, r₁, lowest-order Rossby and gravity waves, antisymmetric about the equator. To obtain a closed set of convectively coupled equations, the lowest symmetric and antisymmetric components in the meridional expansion in (4.2) of the remaining variables, w_c,0, w_c,1, θ_eb,0, θ_eb,1, 〈θ_e〉_z,0, and 〈θ_e〉_z,1, are added to the set of large-scale variables above. To complete the truncation procedure, the moist coupling terms and nonlinear terms in each of the equations are projected onto the meridional mode that corresponds to a particular wave system. Since the parabolic cylinder functions form an orthonormal basis, the projection operator has the following simple form:

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In the second step, the truncated equations are linearized near the radiative–convective equilibrium state. Also, the diagnostic relation in (2.13) based on the ICAPE assumption is utilized to eliminate w_c and to reduce the set of prognostic equations. After some algebra, the following linear system arises:

Symmetric modes

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Antisymmetric modes

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The constants Q₁, . . . , Q₁₃ depend on various physical parameters and are listed in the appendix. Remarkably enough, the variables that describe the modes symmetric in y (symmetric variables) are found only in the first six equations of the system in (4.3), while the antisymmetric variables are found only in the last seven equations. Therefore, the symmetric and antisymmetric parts of (4.3) decouple completely in the linearized system. This decoupling, of course, is destroyed in the fully nonlinear truncated system so these wave fields can interact through nonlinearity. The analysis of the linear system above utilizes eigenmodes in the form U(x, t) = Ce^{i(kx−ω(k)t)+d(k)t}, with the phase ω(k) and normalized growth rate, d(k), similar to section 3.

b. Classification of waves

The linear system in (4.3) describes the dynamics of dry variables coupled with moist exchanges of energy through convection, restricted to the first few meridional modes. Here, a discussion of wave properties of this system is given. The convectively coupled waves are also compared to the waves that arise in the dry system.

At large scales, errors introduced by truncation and convective parameterization have the least influence on the properties of the convectively coupled system. Therefore, a classification of moist waves will be based on the comparison of normal modes for (4.3) and the dry modes obtained by the same meridional truncation at the large scales (Gill 1982). Figure 5a presents the phases of the normal modes for (4.3), while Fig. 5b shows the phases of the corresponding dry waves, that is, dry Kelvin mode, dry mixed Rossby–gravity (MRG), and the lowest-order symmetric and antisymmetric dry Rossby and gravity waves. The values of physical parameters in this example are taken from Table 1, with u₀ = 2 m s⁻¹, u = −2 m s⁻¹, and σ_c = 0.01. Remarkably enough, all dry equatorial waves have analogs in the system described by (4.3). This analogy is fully confirmed by the comparison of the underlying wave structures and flow patterns of moist and dry normal modes and provides quantitative support for the idea that the equatorial waves observed by Wheeler and Kiladis (1999) are moist convectively coupled waves. Based on this similarity, it is natural to extend the names of the dry modes to their moist counterparts and to introduce a moist Kelvin wave, a moist MRG wave, etc. The direct comparison of Figs. 5a and 5b shows that (generalized) phase velocities of the moist modes are slower than phase velocities of their dry counterparts. For example, at the wavelength L = 10 000 km, a moist Kelvin wave and a moist MRG propagate with phase velocities that are 65% and 55% slower than phase velocities of dry Kelvin and dry MRG waves. A similar deceleration is observed for symmetric and antisymmetric Rossby waves, as well as symmetric and antisymmetric westward gravity modes.

In addition to the moist modes that are classified according to their dry analogs, the 13 × 13 system in (4.3) generates four more normal modes. All these extra modes are essentially standing waves that are represented by four nearly indistinguishable curves near ω(k) = 0 on the phase diagram in Fig. 5a. A more detailed analysis of growth rates and wave structure reveals that these modes are the symmetric and antisymmetric analogs of the 〈θ_e〉_z mode and the superdamped convective adjustment mode discussed in section 3c. A simple check shows that the generalized phase speeds of the moist modes are fairly independent of the strength and direction of the mean flow, which means that the classification of moist waves given above can be extended for any mean flow.

At scales much shorter than the equatorial Rossby deformation radius, the effects of rotation are expected to play no significant role in the dynamics. Therefore, one expects that some of the normal modes with rotation become similar to the normal modes that arise in the absence of rotation, in the limit of large wavenumbers. This similarity is indeed observed and illustrated in Table 2. In the left column of this table various moist modes are listed, according to their large-scale classification given above. Apart from the moist Kelvin and moist MRG waves, all other modes include a symmetric and antisymmetric variety. In the right column the analogs of moist normal modes are reported based on comparisons at small scales. In particular, simple calculations show that the phase speed of the moist Kelvin and the eastward branch (k > 0) of moist MRG wave decrease significantly with k, so that both modes become essentially equivalent to the eastward moist gravity wave generated by the system without rotation. Both symmetric and antisymmetric westward gravity waves converge at small scales to the moist westward gravity wave of the system without rotation. Interestingly enough, symmetric and antisymmetric moist eastward gravity waves at small scales carry essentially no moisture and convection fluctuations, and both travel at the speed of a dry gravity wave, 50 m s⁻¹. The moist Rossby wave and the westward branch (k < 0) of moist MRG waves become essentially standing waves at small scales and are dominated by meridional flow with very weak small-scale convection. Finally, both symmetric and antisymmetric 〈θ_e〉_z modes and superdamped modes converge to their analogs from the system without rotation.

c. Easterly mean wind and the transition to instability

Here the discussion is concentrated on the stability properties of normal modes for the system in (4.3) and the structure of unstable modes. Rotation breaks the east–west symmetry, and for simplicity only the system driven by easterly mean wind is considered here.

For compatibility with other parts of this paper, the standard set of parameters is adopted from Table 1, along with u₀ = 2 m s⁻¹, σ_c = 0.01, and an easterly mean flow of u = −2 m s⁻¹. Simple stability analysis reveals that an easterly mean wind triggers instability in the following eastward waves: 1) the moist Kelvin wave, and 2) the eastward branch of moist MRG wave. To estimate how the effects of rotation modify stability properties, the same stability diagnostics are utilized for these two modes as in the absence of rotation at the end of section 3c. The maximum growth rates remain almost unchanged by rotation, with 1.2 and 1.1 day⁻¹ for the moist Kelvin and the eastward branch of moist MRG waves, as compared with 1.2 day⁻¹ for the eastward gravity wave without rotation. The strongest instability tends to occur at slightly shorter wavelengths—compare approximately 800 and 550 km for the moist Kelvin and moist MRG waves to approximately 1000 km without rotation. The range of instability for the moist Kelvin waves includes the wavelengths 125 km < L < 40 000 km and is almost equivalent to the range of unstable modes without rotation, while the moist MRG waves are unstable in a slightly narrower range, 150 km < L < 10 000 km.

In Figs. 6 and 7 the actual flow patterns are presented that are associated with moist Kelvin and moist mixed Rossby–gravity waves, at the wavelength 700 km, near the peak of maximum growth for both waves. At this wavelength, moist Kelvin and moist MRG waves propagate with the generalized phase speeds of 12.1 and 12.6 m s⁻¹, which are significantly slower than 50 and 50.6 m s⁻¹, the phase speeds of dry Kelvin and dry MRG waves at this scale. Convective coupling introduces a weak meridional circulation to the flow pattern of the moist Kelvin wave. Strong zonal equatorial convergence at the bottom of the troposphere is augmented by weak meridional convergence, and it produces the ascending branch of a Walker-like circulation. The intensity of convective updrafts sets the spatial distribution of the strength of convective heating in this model, in accordance with the spatial structure of the baroclinic heating mode. Figure 6 clearly indicates that regions of strong convective heating, represented by lighter shading, are positively correlated with the ascending branch of large-scale motion.

The flow pattern for the moist MRG wave presented in Fig. 7 is almost equivalent to its dry analog. Here, the regions of strong zonal convergence are located away from the equator and are accompanied by a weak meridional cross flow. Meridional circulation in this case is strong enough to generate a flow in the (y, z) plane similar to the Hadley cell pattern, clearly identifiable on the diagram. Positive convective heating is again in phase with the ascending branch of the vertical circulation associated with this wave.

Next, the transition to instability is studied here. The nonzero barotropic mean wind triggers the wind-induced surface heat exchange (WISHE) (Emanuel 1987;Yano and Emanuel 1991) mechanism that drives instabilities of the system in (4.3) through the forcing term,

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projected on the appropriate meridional components in the equations for θ_eb,0 and θ_eb,1 in (4.3). On the other hand, a simple check shows that in the absence of mean wind the WISHE mechanism is suppressed, and the linear system in (4.3) is stable. The mean wind in this model crudely represents the effects of midlatitude dynamics and, in general, does not have to be constant. Therefore, sensitivity to the strength of the mean wind can be utilized as a simplified model for studying the dependence of stability of the tropical wave system on the midlatitude dynamics.

The second important parameter that is intrinsically nonconstant and that significantly affects stability properties is the intensity of turbulent drag in the boundary layer, parameterized by u₀, according to (2.11). The combined effect of barotropic mean wind and turbulent drag on stability is illustrated in Fig. 8. Here we show the boundaries separating stable and unstable regimes for moist Kelvin and moist MRG waves of the system in (4.3), in the parameter space of |u| and u₀. The two curves are almost identical. The curve associated with the moist Kelvin wave is located just above the curve for the moist MRG wave, which corresponds to slightly stronger instability for the moist Kelvin wave. Also, transitions to instability for moist Kelvin waves (left branch of the curve) occur at slightly smaller values of |u|, while reverse transitions to stability occur at larger values of |u|. However, the moist MRG waves have nearly zero growth rates when the moist Kelvin waves become critical; thus, any transition to instability in the convectively coupled system involves both of these waves with the possibility of their nonlinear interaction. Table 3 lists various wave properties at criticality for both the Kelvin and moist MRG wave, for the values of u₀ marked by diamonds and asterisks in Fig. 8. It is easy to see that both the phase and group velocities are fairly independent of the wavelength. Thus, both the moist Kelvin and moist MRG waves tend to propagate together in the transition to instability.