## 1. Introduction

There is a long history of constructing models with gross vertical structures to capture the basic dynamics of convectively coupled tropical waves (e.g., Lindzen 1974; Emanuel 1987; Neelin et al. 1987; Wang 1988; Mapes 2000, hereafter M00; Majda and Shefter 2001, hereafter MS01; Khouider and Majda 2006a, hereafter KM06). Earlier models emphasized a first baroclinic structure (or mode) that is of one sign over the full depth of the free troposphere (e.g., Lindzen 1974; Emanuel 1987; Neelin et al. 1987; Wang 1988). More recent observations of these waves revealed a significant second baroclinic component in their vertical temperature structures (e.g., Wheeler et al. 2000; Straub and Kiladis 2002; Haertel and Kiladis 2004). Such observations alone do not contradict models based only on the first baroclinic mode; it is possible that the first baroclinic mode captures the basic dynamics and that the second-mode temperature structure is simply a by-product. The more convincing evidence for the inadequacy of the first baroclinic-mode models is that they do not yield instability without external destabilization mechanisms (Emanuel et al. 1994), which is inconsistent with the results of the cloud system resolving model (CSRM) simulations (e.g., Tulich et al. 2007; Kuang 2008, hereafter K08). M00 proposed the first instability model that contains both the first and the second baroclinic modes, and he identified a stratiform instability for the wave–convection coupling. The behavior of such models has been analyzed in some detail (M00; MS01; Majda et al. 2004). In these models, modulation of convection by the second-mode temperature anomaly is emphasized and effects of free troposphere moisture variations are ignored. Results from K08, however, indicate that moisture is an essential component for allowing convectively coupled waves to develop. Moisture was included in the two vertical-mode models of KM06 as a major component of the system, along with a third cloud type, congestus (in addition to the deep convective and stratiform clouds in M00). Linear stability analyses of that model indicated that moisture plays a major role in destabilizing the system (KM06; Khouider and Majda 2006b). However, the actual instability mechanism was not clearly identified, which is considered here to be the identification of the processes by which an initial perturbation amplifies, as in, for example, Emanuel et al. (1994) and M00. For the purpose of understanding the basic instability mechanisms, it also seems useful to develop treatments of convection that are conceptually simpler than those used in KM06.

This study continues the effort to construct models of convectively coupled waves with crude vertical structures. Our emphasis will be on conceptually simple convective parameterizations and on revealing the model’s basic instability mechanisms. The model formulation is presented in section 2, along with results from linear analyses. The model parameters are selected based on CSRM simulations of convectively coupled waves described in K07 so that the model resides in realistic parameter regimes (see appendix A). The model is then simplified further and limiting cases are considered to reveal its basic instability mechanisms (section 3). This is followed by a summary and discussion (section 4) and three appendixes.

## 2. Formulation and linear analyses of the simple model

Like earlier models (e.g., M00; MS01; KM06), the present model has two components: the first describes the response to convective heating and the second describes the convective parameterization.

### a. Response to convective heating

*k*with a background state of no motion and eliminate pressure and horizontal winds. This gives

*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).

*G*),

_{i}*c*. When the buoyancy frequency, the square of which is

_{j}*N*

^{2}=

*g*[(

*d*ln

*/*T

*dz*) + (

*g*/

*c*

_{p}

*)], is constant, the vertical modes are*T

*H*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 (

_{T}*J*

_{2}). 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.

*h*

_{b}and an equation for the midtropospheric humidity

*q*

_{mid}. The equation for

*h*follows prior studies (e.g., M00; MS01; KM06):

_{b}*E*is the tendency from surface heat flux anomalies, and

*b*

_{1}and

*b*

_{2}represent the reduction of

*h*per unit

_{b}*J*

_{1}and

*J*

_{2}, respectively. In actuality, large-scale vertical advection has a smaller but nonnegligible effect on

*h*, 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

_{b}*E*= 0 to eliminate any surface heat flux feedback, which, as shown in K07, does not change the basic characteristics of the waves.

*q*

_{mid}can be written as

*a*and

_{1}*a*represent the effective moisture stratification for the two modes, and

_{2}*d*

_{1}and

*d*

_{2}represent the convective drying effects on

*q*

_{mid}per unit

*J*

_{1}and

*J*

_{2}. We have neglected horizontal advection of moisture. The parameters

*a*

_{1}and

*a*

_{2}can be derived given the background moisture stratification and the vertical structure of

*w*

_{1}and

*w*

_{2}. 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

*d*

_{1}and

*d*

_{2}. 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*〉.

*w*

_{1}∼

*J*

_{1}. Furthermore, because

*q*

_{mid}is located around the nodal point of

*w*

_{2}, the effect of large-scale advection by

*w*

_{2}on

*q*

_{mid}(i.e.,

*a*

_{2}) is small. We may therefore simplify Eq. (6) to

*m*

_{1}≈

*a*

_{1}−

*d*

_{1}and

*m*

_{2}≈ −

*d*

_{2}are the moistening effects per unit

*J*

_{1}and

*J*

_{2}. 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 *c*_{1} is 1 and *c*_{2} is set to ½. We use 1 day as the time scale, so the length scale is 4320 km, and *T _{j}*,

*h*, and

_{b}*q*are expressed in temperature unit (1 K) so that the scale for

*J*and

_{j}*w*is 1 K day

_{j}^{−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 *J*_{1} and *J*_{2}, described below.

### b. Convective parameterizations

*U*(scaled by the depth of the troposphere) as

*L*as

*U*

_{0}=

*r*

_{0}

*L*

_{0}, 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

*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

*r*

_{eq}is the ratio anomaly that is in statistical equilibrium with its large-scale environment [Eq. (10)] and

*τ*is the adjustment time for

_{r}*r*to approach that equilibrium. We consider

*τ*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

_{r}*τ*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

_{r}*τ*is not the relaxation time for moisture anomalies, which is on the order of a day and will be discussed in section 3.

_{r}*h** is the saturation moist static energy. We then rewrite Eq. (13) as

*F*= ∂

*h*/∂T*. The factors

*f*and (1 −

*f*) are the relative weights of

*T*

_{1}and

*T*

_{2}. 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.

*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}:

*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

*τ*to be on the order of a few hours—the time for a few turnovers of shallow cumulus convection. From

_{L}*L*and the

*U/L*ratio,

*J*

_{1}and

*J*

_{2}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

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 *T*_{1} 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*= 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

_{T}*J*

_{1}and −

*J*

_{2}, compared to Fig. 2d. Figure 3 shows the phase lag between

*J*

_{1}and −

*J*

_{2}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 *b*_{1} < 0.6, *b*_{2} > 3.3, *d*_{1} < 0.9, *d*_{2} > −0.6, *a*_{1} > 1.6, or *a*_{2} < −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*will be large and

_{q}*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.

## 3. The basic instability mechanisms

### a. A slightly simplified version of the model

*J*

_{1}and

*J*

_{2}:

*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*. Physically, it is perhaps more natural to apply the finite adjustment time on lower-tropospheric heating

_{L}*L*instead of

*J*

_{1}, as

*L*is more locally determined by the subcloud layer and the lower troposphere. However, since

*J*

_{1}= (1 +

*r*

_{0})

*L*+

*r*

_{q}q^{+}, the difference in relaxing

*J*

_{1}instead of

*L*lies in the ∂

*q*term. Because

^{+}/∂t*τ*is considerably shorter than the time scale for

_{L}*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 *J*_{1} and *J*_{2} 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 *J*_{1} and *J*_{2} 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 *m*_{1} and *m*_{2} are shown instead of *a*_{1}, *a*_{2}, *d*_{1}, and *d*_{2} 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 *b*_{1} < 0.6, *b*_{2} > 10/3, *m*_{1} > 0.5, or *m*_{2} < 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 *m*_{1} > 0. Moistening of the midtroposphere by deep convection is clearly seen in K07. With *m*_{1} > 0, the basic behavior of the system is preserved without including the contribution of *T*_{1} in *q*^{+}, so we will take *q*^{+} = *q*. The contribution of *T*_{1} in *q*^{+} has an important stabilizing role when *m*_{1} < 0. This regime, however, is not the subject of this paper and will not be discussed further.

### b. The regime with γ_{0} ≥ 0 and a moisture-stratiform instability

We first consider the model behavior with *γ*_{0} ≥ 0 (i.e., *r*_{0} ≤ 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 *b*_{2} = 0 and *γ*_{0} = 0. Physically, *b*_{2} = 0 means that convection in the upper and lower troposphere has the same effect on *h _{b}* per unit heating, and

*γ*

_{0}= 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

*γ*

_{0}≥ 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

*f*= 1 so that Eq. (23) reduces to

*h*is changed only by

_{b}*J*

_{1}(as

*b*

_{2}= 0) and is in equilibrium with

*T*

_{1}alone (as

*f*= 1). Equation (24) simply states that

*J*

_{1,eq}is that required to keep ∂

*h*/∂

_{b}*t*the same as

*F*∂

*T*

_{1}/∂

*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*= 0 (i.e., in SQE), so that

_{J}*c**

_{1}=

*c*

_{1}/(1 +

*F/b*

_{1}) because of the reduced effective static stability. The first vertical mode also forces a response in

*T*

_{2}and

*q*(in a one-way interaction) through the effect of

*J*

_{1}on

*J*

_{2}, both directly [Eq. (20)] and indirectly through

*J*

_{1}’s effect on moisture, and resonance occurs when

*c*

_{2}=

*c**

_{1}. The direct effect vanishes with

*γ*

_{0}= 0, but does not vanish with more general parameter choices. The temperature and heating structures for a simple case (

*b*

_{2}= 0,

*γ*

_{0}= 0, ε = 0,

*τ*

_{J}= 0, and

*m*

_{2}= 0; other parameters take their normative values) are shown in Fig. 8. The

*q*field in this case is simply −

*J*

_{2}/

*γ*. The phase speed

_{q}*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 *J*_{1} to lag *T*_{1} 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

*f*= 0, corresponding to an atmosphere in which heavily entraining parcels dominate the convective mass flux. Here, the subcloud layer

*h*is in equilibrium with

_{b}*T*

_{2}alone, and there is no more dependence on

*T*

_{1}by the other variables. Again, we take

*b*

_{2}= 0 and

*γ*

_{0}= 0 for simplicity. In this case, Eq. (23) reduces to

*J*

_{1}is that required to keep ∂

*h*/∂

_{b}*t*the same as

*F*∂

*T*

_{2}/∂

*t*. First consider a system in SQE,

*J*

_{2}= −

*γ*(as

_{q}q*γ*

_{0}= 0). Assuming solutions of the form exp[

*i*(

*kx − ωt*)], we obtain the dispersion relationship

*T*

_{2}and q. The effect of

*T*

_{2}on

*q*[through its effect on

*J*

_{1}and, in turn, the moistening effect of

*J*

_{1}on

*q*, as expressed in Eq. (28)] coupled with the effect of

*q*on

*T*

_{2}[through its effect on

*J*

_{2}and, in turn, the heating effect of

*J*

_{2}on T

_{2}, as expressed in Eq. (27)] give rise to an instability. This is best seen with

*m*

_{2}= 0. Putting aside the less interesting solution

*ω*= 0, we have

*q*= −(

*m*

_{1}

*F/b*

_{1})

*T*

_{2}, and the heating

*J*

_{2}is exactly in phase with

*T*

_{2}and

*ω*=

*c*

_{2}

*k*+

*im*

_{1}

*γ*

_{q}F/b_{1}; that is, the phase speed is

*c*

_{2}and the growth rate is

*m*

_{1}

*γ*

_{q}F/b_{1}at all wavelengths. The growth rate is proportional to

*m*

_{1}

*γ*

_{q}, which measures how strongly the depth of convection depends on

*q*(the factor

*γ*) and how strongly moisture depends on

_{q}*J*

_{1}(the factor

*m*

_{1}). It is also proportional to

*F/b*

_{1}, which measures how strongly

*J*

_{1}depends on ∂

*T*

_{2}/∂

*t*. While we have chosen

*b*

_{2}= 0 and

*γ*

_{0}= 0 here, this picture holds for more general parameter choices as well (see appendix C), except for the strict dependence on

*m*

_{1}, which is specific to the choice of

*b*

_{2}= 0,

*γ*

_{0}= 0. For more general parameter choices, a modified

*m*

_{1},

*m̂*

_{1}≡ (

*m*

_{1}+

*m*

_{2}

*γ*

_{0}/1 +

*b*

_{2}

*γ*

_{0}/

*b*

_{1}), should be used (see appendix C).

In this limiting case, *T*_{1} is forced by *J*_{1} but does not feed back onto *J*_{1}. 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 −*J*_{2}/*γ*_{q}.

A positive *m*_{2}, 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 (*J*_{2} > 0); the combined effect of vertical advection and precipitation associated with the second-mode heating moistens the atmosphere (i.e., *m*_{2}*J*_{2} > 0), reducing the dry anomaly. That *m*_{2} 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 *m*_{2}*γ _{q}* defines a relaxation time scale for moisture anomalies when the ∂

*T*

_{2}/∂

*t*term in Eq. (28) vanishes. For our normative parameter choices, the relaxation time scale is about a day and a half.

*T*

_{2}/∂

*t*term in Eq. (28), moisture is coupled to

*T*

_{2}. In this case, a relaxation time scale for moisture alone is not defined, and the −

*m*

_{2}

*γ*term acts to reduce the growth rate of the unstable mode. This effect is stronger at lower frequencies; that is, the moistening effect of

_{q}q*J*

_{2}acts to preferentially damp low wavenumbers. More quantitatively, consider small departures in

*ω*from its value for

*m*

_{2}= 0 (i.e.,

*ω*=

*δω*+

*c*

_{2}

*k*+

*im*

_{1}

*γ*

_{q}F/b_{1}). Plugging this into (29) and neglecting the second-order terms of

*δ*

*ω*, we have

*m*

_{2}+

*m*

_{1}

*F*/

*b*

_{1})

*λ*/

_{q}*c*

_{2}. With the normative parameter values, this corresponds to a wavelength of 9000 km. The effect of

*m*

_{2}on the growth rate is shown in Fig. 10 (thin solid line). With

*m*

_{2}< 0, Eq. (28) contains an unstable moisture mode; without the ∂

*T*

_{2}/∂

*t*term, it has zero phase speed and a growth rate of

*m*

_{2}

*γ*. For general parameter choices, an effective

_{q}*m*

_{2},

*m̂*

_{2}≡ (

*b*

_{1}

*m*

_{2}−

*b*

_{2}

*m*

_{1}/

*b*

_{1}+

*b*

_{2}

*γ*

_{0}), should be used; the effective

*m*

_{2}becomes negative when

*b*

_{1}< 0.6,

*b*

_{2}> 10/3,

*m*

_{1}> 0.5, or

*m*

_{2}< 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)

*m*

_{2}. These modes are similar to the standing modes found in KM06 (their Fig. 4a).

*J*

_{2}out of phase with

*T*

_{2}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*included (dotted line) and with the effects of both

_{J}*m*

_{2}and

*τ*

_{J}included (diamond symbol). The mathematical reason for the different effects of

*m*

_{2}and

*τ*is simply that the latter involves ∂/∂

_{J}*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 (*m*_{2} = 0) and no dissipation (ε = 0), and the propagation of a second vertical-mode temperature anomaly *T*_{2}. The *T*_{2} anomaly modulates deep (or first baroclinic) convective heating *J*_{1} by perturbing the statistical equilibrium between the lower troposphere and the subcloud layer. The result is a *J*_{1} field that lags *T*_{2} by 90° in phase. This changes the moisture field, which lags *J*_{1} by another 90° and is therefore 180° out of phase with *T*_{2}. As *J*_{2} = −*γ _{q}q*, it is in phase with

*T*

_{2}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

*J*

_{2}(i.e., both stratiform and shallow or congestus convection). Building upon this basic instability, we now add the moistening effect of

*J*

_{2}, 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).

#### 3) Basic difference between the two limiting cases

The following steps, as shown in Fig. 11, are the same in both limiting cases: (b) → (c) → (d) → (a). The main difference between the two is that in the limiting case II with *f* = 0, the second baroclinic temperature anomaly controls the first baroclinic heating, and the feedback loop illustrated in Fig. 11 is complete, giving rise to the moisture-stratiform instability; in the limiting case I with *f* = 1, the first baroclinic temperature anomaly controls the first baroclinic heating, the feedback loop is not complete, and all waves are stable. The basic reason for the different behavior of the two vertical modes is that the first baroclinic heating *J*_{1} is more strongly tied to ∂*T*/∂*t* (controlled by shallow or deep CAPE) while the second baroclinic heating *J*_{2} is more strongly tied to moisture (moisture control). This allows *J*_{2} to be more in phase with *T*_{2} when *J*_{1} is controlled by *T*_{2}, but constrains *J*_{1} to be largely in quadrature with *T*_{1} when it is controlled by *T*_{1}. Although we have taken *b*_{2} = 0 and *γ*_{0} = 0 for conceptual simplicity, the same conclusion can be drawn with more general parameter choices, as discussed in appendix C.

#### 4) Intermediate cases: 0 < *f* < 1

We are unaware of general mathematical results that relate intermediate cases to the two limiting cases in a simple and physically meaningful way. We have therefore examined the model behavior for intermediate values of *f* empirically. The moisture–stratiform instability remains the basic instability (all waves are stable when either *m*_{1} or *γ _{q}* is zero), and the moistening effect of

*J*

_{2}and the MCD effect continue to shape the growth rate curve by reducing the growth rates, more strongly at low and high wavenumbers, respectively. However, even with

*m*

_{2}= 0, as

*f*increases, the instability is reduced more strongly at low wavenumbers (Fig. 12), indicating the presence of other stabilization effects at low wavenumbers in addition to the moistening effect of

*J*

_{2}. There is enhanced instability near

*f*∼ 0.2 particularly at higher wavenumbers. This is associated with the resonance effect present in the limiting case I (when

*c*

_{2}∼

*c**

_{1}). The larger resonance effects at higher wavenumbers can be understood in mathematical terms. However, given the highly idealized nature of the present model and the strong MCD damping effect at high wavenumbers, the relevance of such resonance to the real atmosphere is not clear.

### c. The case of γ_{0} < 0, γ_{q} = 0, and the stratiform instability of Mapes (2000)

_{0}> 1 (or

*γ*

_{0}< 0) that behaves differently from that with

*r*

_{0}≤ 1 (or

*γ*

_{0}≥ 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

*b*

_{2}= 0, ε = 0, and

*τ*

_{J}= 0, although these are not required. The system is now reduced to

*r*

_{0}> 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*= 2 h) is included (Fig. 13b).

_{J}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 *γ*_{0} < 0 and that the growth rate is proportional to wavenumber. Therefore, this mechanism depends on positive stratiform heating (negative *J*_{2}) being tied to positive deep convective heating (*J*_{1}) directly through a negative *γ*_{0} instead of indirectly through *J*_{1}’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 *γ*_{0} = −0.2 is shown in Fig. 14. With *f* = 0.5, the phase and amplitude relationship between *T*_{1} and *T*_{2} is such that *J*_{1}, which is proportional to ∂(*T*_{1} + *T*_{2})/∂*t*, is roughly in opposite phase to *T*_{2}. This sets up a feedback loop that is the same as that of M00: when *T*_{2} is negative (cold below and warm above), deep convection is enhanced; with *γ*_{0} < 0, a negative *J*_{2} (cooling below and heating above) is tied to enhanced deep convection (positive *J*_{1}) and amplifies the *T*_{2} 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 *r*_{0} = 1.5 (i.e., *γ*_{0} = −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 *T*_{2} and *T*_{1} on *J*_{1} are similar to their roles in the CAPE calculation in M00 and MS01, but here *J*_{1} 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 ∂*T*_{2}/∂*t* and ∂*T*_{1}/∂*t* in determining *J*_{1}, 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 *T*_{2} 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 *T*_{1} and *T*_{2} in the CAPE calculation in M00 imply an *f* value of 0.8, although the role of *T*_{2} 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.

*J*

_{2}to be in opposite phase to

*J*

_{1}, so there is no tilted heating structure as in observations. The tilt can be introduced with a time lag between

*J*

_{2}and

*J*

_{1}:

^{−1}. For a wavelength of 864 km, for example, and a 3-h time lag,

*J*

_{2}lags −

*J*

_{1}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 (

*J*

_{2}lags −

*J*

_{1}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 (

*γ*

_{0}< 0 and

*γ*

_{q}= 0) to produce the significant tilt seen in the observed heating field of large-scale waves. While we have taken

*b*

_{2}= 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).

*γ*

_{q}= 0, one may wish to include the effect of undiluted CAPE and replace Eq. (20) with

*T*

_{2}, 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), *J*_{2} lags −*J*_{1} 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.

## 4. Summary and discussion

In this paper, we have developed a toy model of convectively coupled waves. Its main new feature is a conceptually simple treatment of convection based on the quasi-equilibrium concept, simplified for a model of crude vertical structure. For convection in the lower troposphere we neglect the effect of entrainment; for convection reaching the upper troposphere, we emphasize the effect of entrainment and thus the impact of the environmental moisture deficit. For realistic model parameters based on the results of CSRM simulations of K07, the toy model produces unstable waves at wavelengths and with structures that compare reasonably well with the CSRM results.

It is of interest to contrast the present treatment with that used by M00, which is an influential model of convectively coupled waves. M00 introduced his model by raising the question, “If CIN is in equilibrium with a statistically ubiquitous population of small entraining cumuli, then how can it be a significant factor inhibiting deep convective cells, which should presumably suffer less entrainment due to their larger size?” This led to his separate treatments for shallow and deep convection, where the role of shallow convection is a simple damping effect on the second-mode temperature anomaly, and the effect of inhibition and triggering is emphasized through the effect of convective inhibition (CIN) and subcloud-layer kinetic energy (or triggering energy) on the deep convective mass flux. This separate treatment, however, does not resolve the inconsistency raised by M00, who recognized it as a conceptual deficiency, as shallow and deep convections are obviously interrelated. This inconsistency is absent in our treatment, where CIN is indeed not a significant factor in inhibiting deep convective cells; instead, midtroposphere moisture deficit is the main factor. We also eliminate triggering and inhibition from the conceptual picture and consider convection to be in quasi equilibrium with the large-scale flow. Triggering and inhibition do occur; however, they reflect more of a view on individual storm scale instead of that on a large scale. On a large scale, we maintain that a quasi-equilibrium view is an adequate conceptual simplification.

We further analyzed the basic instability mechanisms of this model. We identified a moisture-stratiform instability, illustrated in Fig. 11, which arises from the effect of a midtropospheric humidity deficit on the depth of convection. We found that the net moistening effect of the second-mode convective heating and the finite time to approach QE both act to reduce the growth rates, preferentially at low and high wavenumbers, respectively. These damping mechanisms help to select wavelengths of a few thousand kilometers as the fastest growing. An earlier study (KM06) also found that moisture plays an important role in the instability seen in their model. However, the instability mechanism, that is, the mechanism by which an initial perturbation gets amplified, was not as clearly identified. Moreover, KM06 concluded that second baroclinic-mode low-level moisture convergence plays a major role in the generation of the basic instability, whereas here this effect (included in *m*_{2}) is found to damp the waves, as discussed in section 3b(2).

When the background convective heating profile is stronger in the upper troposphere than in the lower troposphere, the model contains an additional instability mechanism. This is named the direct-stratiform instability and is identified to be the same as the stratiform instability of M00. The direct-stratiform instability mechanism, however, is inconsistent with the importance of moisture advection and the substantial tilt in the convective heating structure (especially at low wavenumbers) seen in the CSRM simulations of K07. The moisture-stratiform instability, on the other hand, is consistent with the CSRM simulation results and is suggested as the main instability mechanism in these simulations—and likely in the actual atmosphere under TOGA COARE conditions as well.

## Acknowledgments

The author thanks three anonymous reviewers for their helpful comments and suggestions, as well as Brian Mapes, Masahiro Sugiyama, Chris Walker, Joe Andersen, and, in particular, Dave Raymond for their helpful comments and suggestions on an earlier version of this paper. This work was supported partly by the Modeling, Analysis and Prediction (MAP) program in the NASA Earth Science Division.

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## APPENDIX A

### Parameter Estimation

In this appendix, we try to obtain rough estimates of the parameters used in the simple model. The CSRM simulations of K07 will be used to guide the estimates. In those simulations, a CSRM is coupled to linear gravity wave dynamics and convectively coupled waves spontaneously develop. The reader is referred to K07 for details about the simulations. We will use the first 20 days (i.e., the initial growth period) of the case with a lid at 14 km, constant surface fluxes, and a wavelength of 10 000 km. A longer wavelength (and wave period) is preferable because convection and the large-scale wave can be expected to be closer to statistical equilibrium and the effect of the finite response time of convection, which would complicate the interpretation, is smaller.

We first construct CSRM counterparts of the simple model variables. Vertical-mode decomposition (with a lid at 14 km) is used to obtain *J*_{1}, *J*_{2}, *w*_{1}, *w*_{2}, *T*_{1}, and *T*_{2}. To be consistent with Eq. (4), the modes are normalized so that their absolute values average to 1. The subcloud layer is defined to be between the surface and 930 hPa. The variable *q*_{mid} is chosen to be that averaged over 400–600 hPa. It is important to emphasize that the simple model is a gross simplification of the CSRM simulations (and the actual atmosphere). Therefore, viewing the CSRM results in the simple model framework is approximate and uncertainties in the resulting estimates are far greater than those implied by the goodness of the fit. We intend to use these estimates only as educated guesses of plausible values.

With the above cautionary words in mind, we show in Fig. A1a the regression of convective moistening (*Q*_{2}) of *q*_{mid} against the two convective heating modes *J*_{1} and *J*_{2}. In the CSRM simulations, the large-scale advective tendencies are explicitly calculated and convective tendencies such as convective moistening and heating are computed as residuals, similar to Yanai et al. (1973), for example. Contributions from *J*_{1} and *J*_{2} to convective moistening are shown in red and green, respectively. Their sum reproduces *Q*_{2} almost perfectly on the scale plotted and thus is omitted. Convective drying (negative *Q*_{2}) is dominated by the first heating mode (*d*_{1} = 1.3). The second heating mode has a moistening effect (*d*_{2} = −1.1). This represents a moistening (drying) effect in the midtroposphere by congestus (stratiform) convection. Figure A1b shows the regression of the *q*_{mid} tendency due to vertical advection against *w*_{1} and *w*_{2}, which yields *a*_{1} = 1.6 and *a*_{2} = 0.0. The effect of the second mode is small because the midtroposphere is around the nodal point of *w*_{2}. Similar results are obtained by applying the vertical structures of *w*_{1} and *w*_{2} on the background moisture stratification and integrating from 400 to 600 hPa. The actual time derivative of *q*_{mid} is the sum of convective drying and vertical moisture advection and is substantially smaller because these two effects tend to cancel each other out. In particular, there is a large compensation between adiabatic cooling and convective heating associated with the first mode: w_{1} is well correlated with *J*_{1} (correlation 0.98) and is only slightly larger (by ∼3%) than *J*_{1}. Note that a larger adiabatic cooling (*w*_{1}) than convective heating *J*_{1} is consistent with the notion of a positive gross moist stability for the deep convective heating mode (e.g., Emanuel et al. 1994). Given the smallness of *a*_{2}, it appears reasonable to neglect moisture advection by *w*_{2} and combine convective drying and vertical moisture advection effects of the first mode, as in Eq. (7). This simplified treatment reproduces the total *q*_{mid} tendencies reasonably well (Fig. A1c). While the estimates are subject to many uncertainties, that *m*_{1} = *a*_{1} − *d*_{1} is positive (i.e., the net effect of deep convective heating, *J*_{1}, is to make the midtroposphere more humid) is a robust result.

We compute ∂*h _{b}*/∂

*t*as the mass weighted averages of moist static energy tendencies over the depth of the subcloud layer. Figure A1d shows a regression of ∂

*h*/∂

_{b}*t*by convection against

*J*

_{1}and

*J*

_{2}[Eq. (5) with E = 0]. Contributions from large-scale vertical advection in the subcloud layer are smaller and neglected. This yields

*b*

_{1}= 1.0 and

*b*

_{2}= 2.3. A positive

*b*

_{2}implies that convective heating in the lower troposphere is more effective at reducing

*h*

_{b}than that in the upper troposphere on a per-unit heating basis. This is perhaps somewhat counterintuitive, but it is quite clear in the CSRM simulations because boundary layer cooling and drying peaks before the maximum first-mode heating; that is, it is shifted toward the congestus phase (Fig. A1d). Some indication of this is also seen in the 2-day wave study of Haertel and Kiladis (2004).

We now try to constrain the parameters in Eq. (10). This formulation on the control of the height of convection is very approximate, so uncertainties in the estimates are large. To reflect these uncertainties, we simply chose *r*_{0} = 1 and *r*_{q} = 0.7. Figure A1e indicates this is a plausible choice. A better fit can be achieved through a regression but would convey a false sense of accuracy and is deemed more misleading than informative.

In Fig. A1f, we plot the regression of subcloud layer moist static energy to the temperature averaged over the lower troposphere (930–500 hPa), assuming that the wave period is sufficiently long so that strict (or close to strict) statistical equilibrium is satisfied [Eq. (13)]. This gives *F* ≈ 4. This is larger than ∂*s*/∂T* at, say, 3 km (∼285 K, ∼700 hPa), which is ∼3. A possible reason for this is that factors such as entrainment may have diluted the *h _{b}* variations as air parcels rise through the lower troposphere.

## APPENDIX B

### Connections between Eqs. (3) and (19)

*iωt*−

*ikx*) and diagonalizing the system, we have

*w*leads to

_{j}*π*/ε (∼60 days with our choice of ε), we have approximately

^{−1}, the first vertical-mode heating is 3 times as effective and the second vertical-mode heating is 2 times as effective in the one-way wave equations compared to the two-way wave equations.

## APPENDIX C

### Limiting Cases with General Parameter Choices

In this appendix, we consider the limiting cases with general parameter choices to extend the results with the simplifying parameter choices presented in sections 3b and c. We shall consider the system to be in SQE. The effect of *τ _{J}* is similar to that with the simplifying parameter choices.

*f*= 1, Eq. (23) becomes

*T*

_{1}’s contribution in

*q*

^{+}. At SQE, we have

*T*

_{2}is forced by modes of Eq. (C2) and does not feed back onto

*T*

_{1}and

*q*. This is the same as the limiting case discussed in section 3b(1) but with the parameters modified and a moisture equation added.

*f*= 0, we have

*T*

_{1}is forced by the modes of Eq. (C5) and does not feed back on

*T*

_{2}and

*q*. This is the same as the limiting case II discussed in section 3b(2) but with the modified parameters. Comparing Eqs. (C5) and (C2), we see that the two systems have the same form. In terms of linear stability, the main difference is in the coefficients in front of

*q*in the temperature equation,

*K*

_{1}and

*K*

_{2}. Using our normative parameter values and allowing for a reasonable range,

*K*

_{1}is close to 0.4 and K

_{2}is close to −1. Therefore, the

*f*= 1 case with general parameter choices may be viewed as having a negative effective

*γ*

_{q}.

A summary of parameters used in the simple model.

^{1}

The *q* is expressed in kelvins by dividing the associated latent energy by the specific heat. The same applies to saturation humidity *q** below.