a. Response to convective heating

Similar to K07, we start with the linear inviscid anelastic 2D primitive equations for a horizontal wavenumber k with a background state of no motion and eliminate pressure and horizontal winds. This gives

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where ε is the mechanical damping coefficient, taken to be a constant, J is convective heating, and all other symbols assume their usual meteorological meaning. The overbar denotes the background mean variables, and prime denotes deviations from the mean. Despite the various assumptions and simplifications, systems such as Eq. (1) captures well the basic wind and temperature distributions of convectively coupled waves given the convective heating and cooling. This was shown for the 2-day waves (Haertel and Kiladis 2004) and is true for the present case as well (not shown).

We then assume rigid plate boundary conditions at the surface and at the top of the troposphere and expand the forcing and the solution in terms of the vertical eigenmodes (G_i),

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and obtain

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Without forcing and damping, the solutions to Eq. (3) for each vertical mode are two neutral waves propagating in opposite directions with a dry wave speed of c_j. When the buoyancy frequency, the square of which is N ² = g[(d lnT/dz) + (g/c_pT)], is constant, the vertical modes are

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where H_T is the height of the troposphere. The modes are normalized so that their absolute values average to unity over the depth of the troposphere. We have retained only the first two vertical modes with the goal of constructing a minimal model to reveal the basic instability mechanisms. Note that congestus and stratiform heating are treated here as opposite phases of the same mode (J₂). This simpler treatment represents the observed and CSRM-simulated heating structures very well (Haertel and Kiladis 2004; K07).

It is useful to remind ourselves of the empirical nature of the two-mode model: the two vertical modes are chosen not because they are mathematically the first two eigenmodes of Eq. (1) with the rigid plate boundary conditions but because the empirical evidence shows that the basic vertical structure of the waves can be captured with these two modes (e.g., Haertel and Kiladis 2004; K07). Indeed, it is based on this empirical evidence, and that a radiation upper boundary condition does not have a major effect on the wave characteristics (K07), that the rigid plate boundary conditions were then chosen, allowing Eq. (1) to be conveniently decomposed into vertical modes that resemble the vertical modes seen empirically. Therefore, the present model does not address why these particular vertical structures or modes dominate; the answer requires a model that allows vertical modes to be selected naturally.

Equation (3) is augmented by an equation for the subcloud layer moist static energy h_b and an equation for the midtropospheric humidity q_mid. The equation for h_b follows prior studies (e.g., M00; MS01; KM06):

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where E is the tendency from surface heat flux anomalies, and b₁ and b₂ represent the reduction of h_b per unit J₁ and J₂, respectively. In actuality, large-scale vertical advection has a smaller but nonnegligible effect on h_b, although including them does not appear to change the basic behaviors of the model, so they are left out for simplicity. In this paper, we further set E = 0 to eliminate any surface heat flux feedback, which, as shown in K07, does not change the basic characteristics of the waves.

The equation for q_mid can be written as

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where a₁ and a₂ represent the effective moisture stratification for the two modes, and d₁ and d₂ represent the convective drying effects on q_mid per unit J₁ and J₂. We have neglected horizontal advection of moisture. The parameters a₁ and a₂ can be derived given the background moisture stratification and the vertical structure of w₁ and w₂. In KM06, vertically averaged free troposphere humidity 〈q〉 was used in Eq. (6) so that column moist static energy conservation can be used to constrain d₁ and d₂. However, the main purpose of including an equation for moisture is to include the role of tropospheric humidity on convection, as discussed in more detail in section 2b. While conveniently constrained by column moist static energy conservation, 〈q〉 is not necessarily the most relevant quantity for this purpose, even though it could serve as a reasonable approximation. In this paper, we will use the midtropospheric humidity q_mid instead and forgo the convenience of using 〈q〉.

As discussed further in appendix A and noted in many previous studies (e.g., Haertel and Kiladis 2004), there is substantial compensation between adiabatic cooling and convective heating associated with the first mode; that is, w₁ ∼ J₁. Furthermore, because q_mid is located around the nodal point of w₂, the effect of large-scale advection by w₂ on q_mid (i.e., a₂) is small. We may therefore simplify Eq. (6) to

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where m₁ ≈ a₁ − d₁ and m₂ ≈ −d₂ are the moistening effects per unit J₁ and J₂. We have again verified that this simplification does not modify the basic behaviors of the model discussed in this paper.

Equations (3), (5), and (7) describe the atmosphere’s response to convective heating. We nondimensionalize the above equations using the first dry baroclinic gravity wave speed (50 m s⁻¹) as the velocity scale so that c₁ is 1 and c₂ is set to ½. We use 1 day as the time scale, so the length scale is 4320 km, and T_j, h_b, and q are expressed in temperature unit (1 K) so that the scale for J_j and w_j is 1 K day⁻¹.¹

The dry dynamics are similar to those used in previous studies (e.g., M00; MS01; KM06). The main new feature of this model is its convective parameterizations that determine J₁ and J₂, described below.

b. Convective parameterizations

First, we define an integrated upper-tropospheric heating anomaly U (scaled by the depth of the troposphere) as

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and an integrated lower-tropospheric heating anomaly L as

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and we assume that in statistical equilibrium, the ratio of the total upper-tropospheric heating (mean plus anomaly) to the total lower-tropospheric heating is related to the anomalous moisture deficit (relative to saturation) in the midtroposphere by

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where the subscript 0 denotes background mean values, U₀ = r₀L₀, and a negative q⁺ indicates an anomalous moisture deficit. In the midtroposphere (say, 500 hPa and 270 K), upon expressing the saturation humidity q* in kelvins, we have ∂q*/∂T ∼ 1. Further taking into account the fact that the first mode is near its peak value (π/2) in the midtroposphere, we have

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Equation (10) states that for the same amount of convection in the lower troposphere, there is less convection in the upper troposphere when the midtroposphere is dry. The notion that a dry midtroposphere limits the depth of tropical convection is well supported by observations, numerical simulations, and theoretical reasoning (Brown and Zhang 1997; Sherwood 1999; Parsons et al. 2000; Redelsperger et al. 2002; Ridout 2002; Derbyshire et al. 2004; Takemi et al. 2004; Roca et al. 2005; Kuang and Bretherton 2006). One plausible interpretation (e.g., Brown and Zhang 1997; Derbyshire et al. 2004; Kuang and Bretherton 2006) is that all else being equal, with a dry midtroposphere, convection does not reach as high because entrainment of drier environmental air by the rising air parcels leads to more evaporative cooling, and hence negative buoyancy. More detailed studies, however, are needed to place this interpretation on a firmer footing. Moisture deficit, or saturation deficit, has been used before as a control on convection (e.g., Emanuel 1995; Raymond 2000; Zehnder 2001; KM06), but in terms of precipitation or precipitation efficiency instead of the height of convection.

We shall consider the adjustment of the U/L ratio (denoted as r) to moisture deficit to be instantaneous. One could take into consideration the finite response time of r so that

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where r_eq is the ratio anomaly that is in statistical equilibrium with its large-scale environment [Eq. (10)] and τ_r is the adjustment time for r to approach that equilibrium. We consider τ_r to be of the same order as the time for convective updrafts to rise from the lower troposphere to the upper troposphere (hours or less). For τ_r values in this range, inclusion of this adjustment process does not change the basic behavior of the model, so we will leave it out in this paper. Note that τ_r is not the relaxation time for moisture anomalies, which is on the order of a day and will be discussed in section 3.

Second, we consider that saturation moist static energy averaged over a layer above the cloud base is in quasi-statistical equilibrium (QE) with the subcloud-layer moist static energy. When the equilibrium is achieved instantaneously, we have

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The bracket on the right-hand side denotes averaging over a layer above the cloud base. The variable h* is the saturation moist static energy. We then rewrite Eq. (13) as

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where F = ∂h*/∂T. The factors f and (1 − f ) are the relative weights of T₁ and T₂. We shall interpret f as measuring the importance of undiluted parcels in the total convective mass flux. An f close to 1 implies that the convective mass flux is dominated by undiluted parcels, and Eq. (13) holds with 〈h*〉 taken as an average over the whole troposphere. In this case, the approximate invariance of convective available potential energy (CAPE) is effectively used as a simplification for QE over the whole depth of the troposphere, as in Emanuel et al. (1994), for example. An f close to 0 implies that the convective mass flux is dominated by heavily entraining parcels, and Eq. (13) holds with 〈h*〉 averaged over a shallow layer above the cloud base. In this case, within the two-vertical-mode framework, Eq. (14) effectively assumes convective inhibition (CIN) to be approximately invariant. This is known as the boundary layer quasi equilibrium (BLQ; Emanuel 1995; Raymond 1995). Our normative value for f is 0.5, where 〈h*〉 may be viewed as an average over the lower half of the troposphere. To emphasize the instantaneous adjustment, we will refer to Eq. (14) [or Eq. (13)] as strict quasi equilibrium (SQE), following Emanuel et al. (1994), where the word strict simply means that the adjustment is instantaneous.

For a given U/L ratio, we plug Eqs. (3) and (5) into Eq. (14) and make use of Eqs. (8), (9), and (10) to solve for the lower-tropospheric heating in SQE, denoted as L_eq:

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where

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Equation (15) is a straightforward restatement of the SQE condition of Eq. (14), and the various terms have clear physical meanings; provided that A and B are positive, uplifting in the lower troposphere or a moist midtroposphere increases convective heating in the lower troposphere in SQE. Additionally, L_eq can be expressed in terms of ∂T/∂t. This form will be used in section 3. In Eq. (16) and for the rest of the paper, we omit the subscript in q_mid to simplify the notation. Taking into consideration the finite adjustment time to achieve QE, denoted as τ_L, we have

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We expect τ_L to be on the order of a few hours—the time for a few turnovers of shallow cumulus convection. From L and the U/L ratio, J₁ and J₂ can easily be determined, completing our convective parameterization.

The present convective parameterization falls within the QE framework first introduced by Arakawa and Schubert (1974), which states that convection should be in a state of statistical equilibrium with the large-scale flow. In some previous simple models of the interaction between large-scale circulation and deep convection (e.g., Emanuel et al. 1994), the approximate invariance of CAPE is used as a simplification for SQE over the whole depth of the troposphere (i.e., f = 1). As noted earlier, this emphasizes the role of undiluted parcels in deep convective mass flux, since undiluted parcels are used in the CAPE computation. Recent evidence, however, indicates that this is not a good simplification of SQE, at least in cases such as convectively coupled waves. High-resolution numerical studies show that undiluted parcels do not make a significant contribution to the overall convective mass flux (Khairoutdinov and Randall 2006; Kuang and Bretherton 2006). This is corroborated by the observed and simulated sensitivity of convection to tropospheric moisture [see the discussion in relation to Eq. (10)].

In the present treatment, we are using the invariance of a shallow CAPE as a simplification of SQE over the lower half of the troposphere [with f = ½ in Eq. (14)]. The shallow CAPE measures the integrated buoyancy for undiluted parcels only up to the midtroposphere. This is a simplification for the present simple model with only two vertical modes in the free troposphere. It by no means suggests that cloud parcels do not experience entrainment in the lower troposphere. However, the cumulative effect of entrainment is smaller in the lower troposphere because of the shorter distance traveled by the cloud parcels and their smaller moist static energy difference from the environment (for the regions that we are concerned with, the lower troposphere is taken to always be sufficiently moist so that its moist static energy is close to that of the cloud parcels). Therefore, neglecting the effect of entrainment and using Eq. (14) is a reasonable simplification for SQE over the lower half of the troposphere. We then use Eq. (10) to explicitly include the effect of entrainment on the convective mass flux that can reach from the lower to the upper half of the troposphere. In Eq. (10), we have neglected the role of the traditional CAPE (defined for undiluted parcels over the whole depth of the free troposphere). This reflects the view that the mass flux reaching the upper troposphere is in significantly diluted updrafts and that the midtropospheric moisture deficit is the dominating factor that controls the depth of convection (through entrainment). The CAPE anomalies can become important when they are sufficiently negative so that the background CAPE is substantially consumed. This is a nonlinearity that can limit the growth of the waves. Further discussions of nonlinearity, however, will not be presented in this paper.

c. Linearized equations and normative parameter values

The system is linearized by replacing Eq. (10) with

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The linearized model thus consists of the prognostic Eqs. (3), (6), and (17), and the auxiliary Eqs. (8), (9), (15), and (18).

Observed and simulated data from K07 and Haertel and Kiladis (2004), for example, may be used to estimate the parameters by viewing such data in the framework of the present model. This is discussed in appendix A. Table 1 lists the normative parameter values used in the paper based on these estimates. It is important to stress that because of the highly simplified nature of the present model, viewing the observations or the CSRM results in its framework is very approximate and the parameter estimates are intended only as educated guesses of plausible values.

d. Linear analysis

Results from a linear analysis of this system are shown in Fig. 1. Waves with wavelengths from 1500 to 40 000 km are unstable, with a maximum growth rate of about 0.13 day⁻¹ at 5000 km. The unstable waves have phase speeds around 20 m s⁻¹, slightly slower than the dry wave speed of the second vertical mode. The eigenvector at a wavelength of 8640 km is expressed in physical space in Figs. 2a,b. The eigenvector is scaled so that T₁ is a sine function with an amplitude of 1. We have further reconstructed the vertical structure of temperature and convective heating (Figs. 2c,d) so that it is visually more direct to compare with observations. We have used sin(jπz/H_T), j = 1, 2 as the vertical structures with H_T = 14 km. These figures show that the linear model yields instability at wavelengths and with structures that compare well with the CSRM simulations and observations. Note that convective heating here is dominated by the first vertical mode and has a significant tilt, consistent with the observations and simulations (Haertel and Kiladis 2004; K07). In contrast, in the model of KM06, heating in the upper troposphere is substantially stronger than that in the lower troposphere (their Fig. 6). There is also a tendency for this to be true in MS01 as the wavelength increases to beyond 2000 km (their Fig. 4). The substantially stronger heating in the upper troposphere indicates a larger contribution from the second-mode heating and a more in-phase relation between J₁ and −J₂, compared to Fig. 2d. Figure 3 shows the phase lag between J₁ and −J₂ as a function of wavenumber. Substantial phase lag is seen at all wavenumbers. Over a wide range of wavelengths (1500–10 000 km), the phase lag is between ∼85° and ∼65°. This is consistent with the CSRM simulation results from K07.

We have repeated the linear analysis with the parameters perturbed around their normative values one at a time. The parameter dependence of the maximum linear growth rate is shown in Fig. 4. While not shown, the phase speeds of the most unstable modes are between 10 and 25 m s⁻¹, except for b₁ < 0.6, b₂ > 3.3, d₁ < 0.9, d₂ > −0.6, a₁ > 1.6, or a₂ < −0.4, where the phase speeds of the most unstable modes are a few meters per second or less. While it is useful to know the parameter sensitivities of the model, for the purpose of revealing the basic instability mechanisms, it is more informative to consider certain limiting cases, as discussed in section 3.

It is useful to note that while r_q and f are separate parameters in the model and are varied independently in Fig. 4, they are related to the conceptual picture of whether upper-troposphere convective mass flux is dominated by nearly undiluted or significantly diluted updrafts. When significantly diluted updrafts dominate, the midtroposphere humidity has an important effect on the depth of convection; r_q will be large and f will be small (a smaller depth of the atmosphere can be assumed to be in QE regardless of the environment humidity). Thus in principle one should vary these parameters together to be conceptually consistent, although this is not done here.

a. A slightly simplified version of the model

To reveal its basic instability mechanisms, we make a few simplifications to the model described in section 2. First, we replace the two-way wave equations in Eq. (3) with one-way wave equations that have a Newtonian cooling coefficient ε,

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and use Eq. (7) instead of Eq. (6) to evolve midtropospheric humidity. Replacing the two-way wave equations by one-way wave equations has some quantitative effects, as discussed in appendix B. For example, heating drives temperature anomalies more effectively in the one-way wave equations, and this effect is stronger for the first vertical mode. However, these changes do not alter the basic behavior of the model. It is also convenient to rewrite Eq. (18) in terms of J₁ and J₂:

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where

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and include the finite adjustment time to achieve QE as

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where

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Here, we have expressed J_1,eq in terms of ∂T/∂t, although one can also write it in a form similar to Eq. (15). We shall take τ_J = τ_L. Physically, it is perhaps more natural to apply the finite adjustment time on lower-tropospheric heating L instead of J₁, as L is more locally determined by the subcloud layer and the lower troposphere. However, since J₁ = (1 + r₀)L + r_qq⁺, the difference in relaxing J₁ instead of L lies in the ∂q⁺/∂t term. Because τ_L is considerably shorter than the time scale for q⁺ to vary (on the order of the wave period), Eq. (22) has the same basic effect as does Eq. (17).

The phase speeds and the linear growth rates for this simplified version are shown in Fig. 5, and the structures for a wavelength of 8640 km are shown in Fig. 6. The higher wavenumbers are more stable compared to Fig. 1, and the amplitudes of J₁ and J₂ are smaller compared to those in Fig. 2. The latter is because heating is more effective in driving temperature anomalies in one-way equations (appendix B). When this difference is accounted for following discussion in appendix B, the amplitudes of J₁ and J₂ become similar to those in Fig. 2. We have also repeated the linear analysis with the parameters perturbed around their normative values one at a time for this simplified version of the model. The parameter dependence of the maximum linear growth rate is shown in Fig. 7 and is similar to that in Fig. 4 in terms of its basic pattern. The parameter dependence on m₁ and m₂ are shown instead of a₁, a₂, d₁, and d₂ because Eq. (7) is used. The dependence on ε is also similar to that in Fig. 4 (not shown). Similar to the model in section 2, all unstable waves have phase speeds between 10 and 25 m s⁻¹, except for b₁ < 0.6, b₂ > 10/3, m₁ > 0.5, or m₂ < 0.6, where the phase speeds of the most unstable modes are a few meters per second or less. This dependence is explained in section 3b(2) and appendix C. The similarity in the basic parameter dependence indicates that the simplified system captures the basic behavior of the model described in section 2c.

In this paper, we focus on regimes with m₁ > 0. Moistening of the midtroposphere by deep convection is clearly seen in K07. With m₁ > 0, the basic behavior of the system is preserved without including the contribution of T₁ in q⁺, so we will take q⁺ = q. The contribution of T₁ in q⁺ has an important stabilizing role when m₁ < 0. This regime, however, is not the subject of this paper and will not be discussed further.

b. The regime with γ₀ ≥ 0 and a moisture-stratiform instability

We first consider the model behavior with γ₀ ≥ 0 (i.e., r₀ ≤ 1), which corresponds to an atmosphere where climatologically speaking the convective heating in the lower troposphere is greater than or equal to that in the upper troposphere. We shall try to identify the model’s basic instabilities by considering limiting cases. To simplify the discussion, we take b₂ = 0 and γ₀ = 0. Physically, b₂ = 0 means that convection in the upper and lower troposphere has the same effect on h_b per unit heating, and γ₀ = 0 means that the background convective heating is of the same strength as in the upper and lower troposphere. These are not required but help to simplify our discussion. The results are largely representative of general cases with γ₀ ≥ 0, and nonrepresentative results will be pointed out along the way. Limiting cases with general parameter choices can be reduced to the same form by redefining the parameters as discussed in appendix C.

1) Limiting case I: f = 1

Let us first consider the limiting case with f = 1 so that Eq. (23) reduces to

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Physically, this represents a case in which convective mass flux is dominated by undiluted updrafts. Here, the subcloud-layer moist static energy h_b is changed only by J₁ (as b₂ = 0) and is in equilibrium with T₁ alone (as f = 1). Equation (24) simply states that J_1,eq is that required to keep ∂h_b/∂t the same as F∂T₁/∂t. In this case, the first mode (temperature and heating) is no longer affected by the second mode or moisture. This reduces the system to the first baroclinic-mode model discussed in previous studies (Emanuel 1987; Neelin et al. 1987). Let us start with ε = 0 (no Newtonian cooling) and τ_J = 0 (i.e., in SQE), so that

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The convectively coupled first vertical mode has a zero growth rate and a phase speed of c*₁ = c₁/(1 + F/b₁) because of the reduced effective static stability. The first vertical mode also forces a response in T₂ and q (in a one-way interaction) through the effect of J₁ on J₂, both directly [Eq. (20)] and indirectly through J₁’s effect on moisture, and resonance occurs when c₂ = c*₁. The direct effect vanishes with γ₀ = 0, but does not vanish with more general parameter choices. The temperature and heating structures for a simple case (b₂ = 0, γ₀ = 0, ε = 0, τ_J = 0, and m₂ = 0; other parameters take their normative values) are shown in Fig. 8. The q field in this case is simply −J₂/γ_q. The phase speed c*₁ is 10 m s⁻¹, and the wave structures in many aspects resemble the observed patterns.

So far, the growth rate is 0 for all wavenumbers. Introducing a finite τ_J causes heating J₁ to lag T₁ by more than π/2 in phase, which gives rise to a damping effect that is stronger at high wavenumbers. This has been pointed out before and was named moist convective damping (MCD) (Emanuel 1993; Emanuel et al. 1994; Neelin and Yu 1994; Yu and Neelin 1994). A positive ε further damps the waves.

2) Limiting case II: f = 0

Let us now consider f = 0, corresponding to an atmosphere in which heavily entraining parcels dominate the convective mass flux. Here, the subcloud layer h_b is in equilibrium with T₂ alone, and there is no more dependence on T₁ by the other variables. Again, we take b₂ = 0 and γ₀ = 0 for simplicity. In this case, Eq. (23) reduces to

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that is, J₁ is that required to keep ∂h_b/∂t the same as F∂T₂/∂t. First consider a system in SQE,

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where J₂ = −γ_qq (as γ₀ = 0). Assuming solutions of the form exp[i(kx − ωt)], we obtain the dispersion relationship

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Equations (27) and (28) describe a coupled system of T₂ and q. The effect of T₂ on q [through its effect on J₁ and, in turn, the moistening effect of J₁ on q, as expressed in Eq. (28)] coupled with the effect of q on T₂ [through its effect on J₂ and, in turn, the heating effect of J₂ on T₂, as expressed in Eq. (27)] give rise to an instability. This is best seen with m₂ = 0. Putting aside the less interesting solution ω = 0, we have q = −(m₁F/b₁)T₂, and the heating J₂ is exactly in phase with T₂ and ω = c₂k + im₁γ_qF/b₁; that is, the phase speed is c₂ and the growth rate is m₁γ_qF/b₁ at all wavelengths. The growth rate is proportional to m₁γ_q, which measures how strongly the depth of convection depends on q (the factor γ_q) and how strongly moisture depends on J₁ (the factor m₁). It is also proportional to F/b₁, which measures how strongly J₁ depends on ∂T₂/∂t. While we have chosen b₂ = 0 and γ₀ = 0 here, this picture holds for more general parameter choices as well (see appendix C), except for the strict dependence on m₁, which is specific to the choice of b₂ = 0, γ₀ = 0. For more general parameter choices, a modified m₁, m̂₁ ≡ (m₁ + m₂γ₀/1 + b₂γ₀/b₁), should be used (see appendix C).

In this limiting case, T₁ is forced by J₁ but does not feed back onto J₁. The physical structure of the wave with a wavelength of 8640 km from this limiting case in SQE is shown in Fig. 9. The q field in this case is again simply −J₂/γ_q.

A positive m₂, which implies that the net effect of the second-mode heating is to moisten the troposphere, brings a damping effect on q [Eq. (28)]. The physical picture is simple: when, for instance, the troposphere is dry (q < 0), convection is shallower (J₂ > 0); the combined effect of vertical advection and precipitation associated with the second-mode heating moistens the atmosphere (i.e., m₂J₂ > 0), reducing the dry anomaly. That m₂ is positive should be expected based on observations: the effect of second-mode vertical advection on midtroposphere moisture is small; increased shallow convection and less stratiform precipitation reduce the removal of moisture by precipitation. The factor m₂γ_q defines a relaxation time scale for moisture anomalies when the ∂T₂/∂t term in Eq. (28) vanishes. For our normative parameter choices, the relaxation time scale is about a day and a half.

With a nonzero ∂T₂/∂t term in Eq. (28), moisture is coupled to T₂. In this case, a relaxation time scale for moisture alone is not defined, and the −m₂γ_qq term acts to reduce the growth rate of the unstable mode. This effect is stronger at lower frequencies; that is, the moistening effect of J₂ acts to preferentially damp low wavenumbers. More quantitatively, consider small departures in ω from its value for m₂ = 0 (i.e., ω = δω + c₂k + im₁γ_qF/b₁). Plugging this into (29) and neglecting the second-order terms of δω, we have

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Therefore, this damping effect becomes significant for wavenumbers lower than (m₂ + m₁F/b₁)λ_q/c₂. With the normative parameter values, this corresponds to a wavelength of 9000 km. The effect of m₂ on the growth rate is shown in Fig. 10 (thin solid line). With m₂ < 0, Eq. (28) contains an unstable moisture mode; without the ∂T₂/∂t term, it has zero phase speed and a growth rate of m₂γ_q. For general parameter choices, an effective m₂, m̂₂ ≡ (b₁m₂ − b₂m₁/b₁ + b₂γ₀), should be used; the effective m₂ becomes negative when b₁ < 0.6, b₂ > 10/3, m₁ > 0.5, or m₂ < 0.6, which are unstable regimes in Fig. 7 with very small phase speeds. These modes are therefore attributed to a moisture instability described by Eq. (28) with a negative (effective) m₂. These modes are similar to the standing modes found in KM06 (their Fig. 4a).

Departing from SQE by introducing a finite response time, that is, replacing Eq. (28) by

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shifts J₂ out of phase with T₂ and reduces the growth rate. This is similar to the MCD effect discussed in the f = 1 case and is stronger for higher wavenumbers. Figure 10 shows the growth rate with the effect of τ_J included (dotted line) and with the effects of both m₂ and τ_J included (diamond symbol). The mathematical reason for the different effects of m₂ and τ_J is simply that the latter involves ∂/∂t and the former does not.

The above discussion paints the following physical picture for the basic instability in the limiting case II: start with SQE, zero net moistening from the second-mode heating (m₂ = 0) and no dissipation (ε = 0), and the propagation of a second vertical-mode temperature anomaly T₂. The T₂ anomaly modulates deep (or first baroclinic) convective heating J₁ by perturbing the statistical equilibrium between the lower troposphere and the subcloud layer. The result is a J₁ field that lags T₂ by 90° in phase. This changes the moisture field, which lags J₁ by another 90° and is therefore 180° out of phase with T₂. As J₂ = −γ_qq, it is in phase with T₂ and causes growth. This basic instability is illustrated schematically in Fig. 11 and is referred to as the moisture-stratiform instability. While an eastward-propagating wave is chosen for the illustration, the same instability mechanism also operates in westward-propagating or standing waves. In the case of a standing wave, the phase lag will manifest as a lag in time. Note that while we have used the name “stratiform” following M00, it is intended here to represent both phases of J₂ (i.e., both stratiform and shallow or congestus convection). Building upon this basic instability, we now add the moistening effect of J₂, which reduces growth rates more strongly at low wavenumbers; the finite time to approach QE, which reduces growth rates more strongly at high wavenumbers; and the dissipation ε, which damps the waves more or less uniformly in wavenumber. These damping mechanisms shape the otherwise uniform growth rate curve to favor wavelengths of a few thousand kilometers (circles in Fig. 10).

c. The case of γ₀ < 0, γ_q = 0, and the stratiform instability of Mapes (2000)

As seen in Figs. 7d and 4d, there is a branch of unstable waves with r₀ > 1 (or γ₀ < 0) that behaves differently from that with r₀ ≤ 1 (or γ₀ ≥ 0). This represents a case in which the background mean convective heating is stronger in the upper troposphere. Its behavior is best exposed by setting γ_q = 0. To simplify the discussion, we will also take b₂ = 0, ε = 0, and τ_J = 0, although these are not required. The system is now reduced to

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Figure 13a shows that this simple system qualitatively reproduces the r₀ > 1 branch of the unstable modes seen in Figs. 7d and 4d. The large growth rates in Fig. 13a, particularly those at high wavenumbers, are reduced and come to closer agreement with Fig. 7d when an adjustment time to QE (τ_J = 2 h) is included (Fig. 13b).

The dispersion relation for this system is quadratic and can readily be derived. It is easy to show analytically that a necessary condition for instability is γ₀ < 0 and that the growth rate is proportional to wavenumber. Therefore, this mechanism depends on positive stratiform heating (negative J₂) being tied to positive deep convective heating (J₁) directly through a negative γ₀ instead of indirectly through J₁’s effect on q. We shall refer to this as the direct-stratiform instability mechanism to distinguish it from the moisture-stratiform instability mechanism discussed in section 3b.

An example of the structure of the unstable waves from Eq. (32) with f = 0.5 and γ₀ = −0.2 is shown in Fig. 14. With f = 0.5, the phase and amplitude relationship between T₁ and T₂ is such that J₁, which is proportional to ∂(T₁ + T₂)/∂t, is roughly in opposite phase to T₂. This sets up a feedback loop that is the same as that of M00: when T₂ is negative (cold below and warm above), deep convection is enhanced; with γ₀ < 0, a negative J₂ (cooling below and heating above) is tied to enhanced deep convection (positive J₁) and amplifies the T₂ anomaly. Therefore, the basic instability mechanism in this regime is identified to be the same as the stratiform instability of M00. The wave structure is affected by replacing two-way wave equations with one-way wave equations. While not shown, when the two-way wave equations are used with r₀ = 1.5 (i.e., γ₀ = −0.2) and r_q = 0, the temperature structure of the wave captures the salient features of the observed waves quite well.

The formulation in Eq. (32) is indeed similar to that of M00 and MS01, but M00 has an additional prognostic equation for the subgrid-scale triggering energy. The effects of T₂ and T₁ on J₁ are similar to their roles in the CAPE calculation in M00 and MS01, but here J₁ is determined implicitly from ∂T/∂t based on the QE concept instead of from the explicit prognostic approach of M00 and MS01. As discussed in section 2b, the relative importance of ∂T₂/∂t and ∂T₁/∂t in determining J₁, as measured by the parameter f, indicates the depth of the troposphere that is in QE regardless of the environmental humidity and is physically interpreted as the importance of undiluted parcels in the convective mass flux. A similar interpretation can be made for the role of T₂ in the CAPE calculation in MS01, as this includes the effect of entrainment. Values used in MS01 and KM06 imply an f of 0.9. The relative weights of T₁ and T₂ in the CAPE calculation in M00 imply an f value of 0.8, although the role of T₂ on convection is further enhanced by its role in his CIN calculation, which would imply an f value of ∼0.1–0.2. The effective f in M00 therefore varies between 0.1 and 0.8, depending on how strong the CIN control is relative to the CAPE control, and can be close to our normative value.

The system of Eq. (32) constrains J₂ to be in opposite phase to J₁, so there is no tilted heating structure as in observations. The tilt can be introduced with a time lag between J₂ and J₁:

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as done in M00, where a 3-h time lag is used, representing the time lag between the stratiform phase and the convective phase of a mesoscale convective system (MCS). This however provides appreciable tilts only at short wavelengths; a 3-h time lag corresponds only to 270 km with a wave speed of 25 m s⁻¹. For a wavelength of 864 km, for example, and a 3-h time lag, J₂ lags −J₁ by ∼40° and appreciable tilt is indeed seen in the heating structure. However, for longer wavelengths (e.g., the case with a wavelength of 8640 km shown in Fig. 14), the same time lag produces little tilt (J₂ lags −J₁ by ∼8°), as one would expect from Eq. (33). Indications of this behavior are also evident in MS01 (e.g., their Fig. 4). Therefore, it is difficult for the basic instability mechanism in this regime (γ₀ < 0 and γ_q = 0) to produce the significant tilt seen in the observed heating field of large-scale waves. While we have taken b₂ = 0, ε = 0, and τ_J = 0 to simplify the discussion, the basic results remain the same without these simplifications. Equation (33), with an adjustment time of 3 h, also has the effect of further reducing the growth rates at high wavenumbers and selecting waves of synoptic scale (∼2000 km in wavelength) as the fastest growing (not shown).

As discussed in section 2b, we have neglected the role of undiluted CAPE on the depth of convection based on the view that all updrafts experience significant entrainment and that midtropospheric moisture deficit is the main factor affecting the depth of convection. In this section, as we have set γ_q = 0, one may wish to include the effect of undiluted CAPE and replace Eq. (20) with

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which changes the second equation in (32) to

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This adds a simple damping effect on T₂, which is the same as the cumulus congestus damping effect on the second-mode temperature in section 2b(3) of M00.

d. Moisture-stratiform instability versus direct-stratiform instability

From the CSRM simulations of K07, which are idealized simulations based on the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE; Webster and Lukas 1992), there are two pieces of evidence against the direct-stratiform instability being the main instability. First, when vertical advection of moisture is disabled in K07, convectively coupled waves are largely suppressed, but the direct-stratiform instability mechanism is not affected by the removal of vertical moisture advection. Second, substantial tilt in the heating structure is found in K07 across a wide range of wavenumbers. This is also inconsistent with the direct-stratiform instability mechanism. When stratiform heating is tied to deep convective heating with a fixed time lag based on the life cycle of MCSs, the tilt in the heating structure is expected to become increasingly small as wavelength increases. This is noted by Mapes et al. (2006). In contrast, the moisture-stratiform instability requires the vertical advection of moisture and yields substantial tilt in the convective heating structure over a wide range of wavenumbers (Fig. 3). Indeed, in its limiting form, as in Eqs. (27) and (28), J₂ lags −J₁ by a quarter cycle. The moisture-stratiform instability is therefore more in line with the CSRM simulations of K07 and is suggested as the main instability mechanism in these simulations—and likely in the real atmosphere under TOGA COARE conditions as well. Whether and how this might change with the background mean state is a subject of interest and warrants further research.