5.1. Statistical Quasi-Equilibrium Models
 Emanuel et al.  presented arguments for the so-called statistical quasi-equilibrium model of interactions between moist convection and the large-scale flow. This model appeared in its original and simplest form in Emanuel's  paper on intraseasonal oscillations. In that paper the temperature profile of the atmosphere is assumed to be that of a moist adiabat with saturated moist entropy equal to that of the moist entropy in the boundary layer. The boundary layer entropy is determined by budget considerations for entropy over the entire troposphere, since convection is assumed to redistribute entropy vertically in an infinitely short time. Contributors to the entropy budget are surface entropy fluxes, radiative losses, and zonal advection by the mean zonal wind. On an equatorial beta plane with mean surface easterlies, this model produces an unstable, eastward-moving equatorial mode as well as a number of off-equatorial modes. We will focus on the equatorial mode. Both the growth rate and the propagation speed of this mode depend primarily on the horizontal wavenumber and the strength of the mean easterlies. Examination of Emanuel's  governing equations show that his modes reduce to neutral Matsuno  modes with zero equivalent depth (or zero effective stratification) in the absence of mean zonal flow and radiative damping.
 Emanuel et al.  expanded the reasoning in Emanuel  in two fundamental ways. First, a lag of order an hour or two between convective forcing and response was introduced. This by itself damps modes. Second, convection was assumed to provide an additional negative tendency on the boundary layer entropy in comparison to Emanuel . Conceptually, this was related to the transport of low entropy air into the boundary layer by convective downdrafts. As demonstrated below, this corresponds to introducing a positive GMS; the original Emanuel  model exhibits zero GMS.
 In order to expose the fundamental nature of the Neelin-Yu  model, a highly simplified and incomplete version of the original model is presented here. We begin with the two-dimensional, linearized, Boussinesq, non-rotating governing equations for perturbations on an atmosphere at rest, written largely in the notation of Fuchs and Raymond :
where u and w are the horizontal and vertical velocities, p is the kinematic pressure, b is the buoyancy, equal to the acceleration of gravity times the fractional potential temperature anomaly, and q and e are the mixing ratio and moist entropy perturbations scaled to buoyancy units [Fuchs and Raymond 2007]. The quantities ΓB, ΓQ, and ΓE are the ambient vertical gradients of potential temperature, mixing ratio, and moist entropy, also scaled to buoyancy units. Source terms for buoyancy, mixing ratio, and moist entropy are given by SB, SQ, and SE.
 The moist entropy equation is redundant (but still useful) since e = b + q, ΓE = ΓB + ΓQ, and SE = SB + SQ. We ignore surface fluxes and radiation in this limited treatment, so that
where h is the height of the tropopause. The limitation to two-dimensionality eliminates Rossby wave-like modes from the analysis and limits consideration to convectively coupled gravity waves, which have the same zonal structure as equatorial Kelvin waves. As the latter receive most of the attention in Neelin and Yu , this is not a major limitation.
 Equations (5.1)-(5.3) can be combined into a single equation relating vertical velocity w and buoyancy b,
leaving only the buoyancy to be determined. The simplified Betts-Miller cumulus parameterization consists of the assumptions
where μ–1 is an adjustment time constant equal to about 2 h. The cumulus parameterization thus has the net effect of relaxing the moisture perturbation toward zero and the buoyancy perturbation toward b0, which is taken to be independent of height. The parameter b0 is chosen so that the condition (5.7) on the entropy source term is satisfied. This condition can be expressed
Taking the partial derivative of this with respect to time and substituting (5.6) results in
 This integro-differential equation is difficult to solve analytically. However, Neelin and Yu's assumption for the value of μ makes it much larger in magnitude than the frequency of any plausible large-scale disturbance. If we take the limit of very large μ, then this equation reduces to
which is more tractable. In particular, the solution for vertical velocity w takes the form
where we assume that ΓB is constant and can therefore be extracted from the integral. The parabolic form of this solution in z is not precisely a first baroclinic normal mode, but it is very close to it. The resulting equation is
which we recognize as the one-dimensional wave equation with constant phase speed c given by
Since the speed of adiabatic gravity waves with vertical half-wavelength h (i. e., the fundamental baroclinic mode) is ca = hΓB1/2/π, this can be written
If ΓM ≈ 0.1, we see that c ≈ ca/3, or ≈ 16 m s–1 for a typical fundamental baroclinic mode gravity wave speed of ca = 50 m s–1.
 Thus, in the limit of very large μ, the fundamental convectively coupled mode of the Neelin-Yu system is a neutral, non-dispersive traveling wave. Neelin and Yu  show, in line with the reasoning of Emanuel et al. , that the effect of finite μ is to damp slightly these waves without significantly affecting their phase speed. Furthermore, these waves can be destabilized if wave-induced surface heat exchange (WISHE; Yano and Emanuel 1991] becomes active in an environment with non-zero mean zonal surface flow. This solution reduces to the (trivial) case of no WISHE in Emanuel  when ΓM = 0.
 Modes with vertical structure different than that given by (5.18) are not admitted as solutions to (5.17). Neelin and Yu  show that such solutions exist in the case of non-zero μ, but are very strongly damped.
 The most interesting result is that the square of the convectively coupled gravity wave propagation speed (or the equivalent depth) is proportional to the GMS. Furthermore, according to Emanuel et al. , positive GMS arises from the existence of convective downdrafts which carry low moist entropy air from the free troposphere into the boundary layer. This is in accord with the picture of GMS presented in Figure 6, since downdraft mass fluxes reduce the updraft mass flux at low to middle levels, thus causing the maximum vertical mass flux to occur at higher levels.
 Neelin and Yu  and Yu and Neelin  also found stationary deep convective modes. These are modes which don't propagate except by advection and whose essential dynamics consists just of the interaction of convection and moisture in a single column. They satisfy the condition expressed in Neelin and Yu's equation (5.9h), which can be expressed in our terms as
where d is the boundary layer thickness. The second term on the left side of this equation is equivalent to their boundary layer term, given their bulk boundary layer model. Comparison with (5.13) indicates that (5.22) is equivalent to ΓM = 0. In our simplified representation these modes thus correspond to their convectively coupled modes with zero GMS. However, given that Neelin and Yu  assume an additional degree of freedom in the form of an explicit boundary layer, these modes appear to have an independent existence in their model.
 Though not considered by Neelin and Yu , their mathematical model admits solutions with negative GMS. Making ΓM < 0 in (5.19) results in pairs of stationary modes which respectively amplify and decay with time. However, in the context of Emanuel et al. , negative GMS doesn't make physical sense, since it requires convective downdrafts to increase rather than decrease the moist entropy (or moist static energy) of the atmospheric boundary layer. Below we consider an extension to their model in which negative GMS occurs in a physically plausible manner.
5.2. Negative GMS and the ITCZ
 Back and Bretherton  showed, using reanalysis data, that the east Pacific intertropical convergence zone (ITCZ) exhibits negative ΓV (vertical advection) and positive ΓH (horizontal advection) with an indeterminate sum ΓR. Thus, ITCZ convection has negative GMS in the classical sense of Neelin and Held  and Yu et al. , but the resulting source of moist static energy is removed by horizontal advection. As noted above, this is a situation not envisioned by Neelin and Yu , where the target relative humidity is held constant with height. However, since Neelin's Quasi-equilibrium Tropical Circulation Model (QTCM; Neelin and Zeng, 2000; Zeng et al. 2000] has more freedom in the choice of target relative humidity profile, negative GMS values can in principle occur there.
 Sobel and Neelin  developed a hybrid model which includes an explicit prognostic boundary layer of fixed depth coupled to a free troposphere model similar to that in the original QTCM. The convergence occurring in the boundary layer produces divergence distributed uniformly through the depth of the free troposphere with the associated upward motion profile supplementing that produced by normal QTCM mechanisms. The advection of potential temperature and moisture associated with this upward motion enhances convection produced by the QTCM vertical velocity profile. Due to the “bottom-heavy” structure of this boundary layer-driven ascent, the resulting GMS is highly negative. However, the GMS associated with the normal free tropospheric vertical velocity profile is positive. The net GMS is an average of the two weighted by the relative amplitudes of the two modes of ascent. This hybrid model is thus a way to incorporate the observed behavior of ITCZs into the QTCM.
 Crudely, one can think of the boundary layer mode as representing the shallow meridional circulation described by Zhang et al. . This mode imports entropy where there is surface convergence, which can occur in the absence of deep convection when the horizontal variations in sea surface temperature favor it by the Lindzen-Nigam  mechanism. Sobel and Neelin  argued that since the shallow Lindzen-Nigam flow can exist independently of the presence or absence of deep convection, the entropy (or in their case, moist static energy) imported by that flow can be considered as an external forcing on the local entropy budget, similar to the entropy forcing Fs – R in (2.1). In the absence of other processes, the presence and intensity of the deep baroclinic flow, which is strongly coupled to deep convection, then balances that import plus Fs – R through its own gross moist stability. Such an argument is not completely clean because the surface convergence and shallow entropy import are modulated by the pressure gradients due to the deep baroclinic flow so that the two are not truly independent. Nonetheless the recent analysis of Back and Bretherton [2009a, b] also seems to support this view.
 In the Sobel and Neelin  model, which is zonally symmetric, a horizontal diffusion of moisture (and thus entropy or moist static energy) had to be introduced in order to reduce the ITCZ intensity to observed levels. This was done out of expediency, but it was then argued that perhaps the diffusion might be standing in for transient non-axisymmetric disturbances, such as easterly waves, which might be exporting moisture from the ITCZ to drier adjacent regions. In the model, the dominant balance in the moist static energy budget is between export by this diffusion and net import by the resolved, zonally symmetric flow (including the strong import by the boundary layer mode and much weaker export by the baroclinic mode). The recent diagnostic work of Peters et al.  supports this picture, with the moist static energy transport by transients out of the east Pacific ITCZ being both significant and diffusive in character. The resulting picture is very different from the early Neelin-Held concept of deep convective regions. In that picture, the net entropy forcing by surface fluxes minus radiative cooling was balanced by export due to the mean divergent circulation, with a positive gross moist stability. In the revised picture of Sobel and Neelin  transient eddy fluxes play an important role in balancing the net source (at least for the east Pacific ITCZ), while the mean circulation may either weakly export or even import entropy, adding to rather than balancing the net source.
5.3. Moisture and Kelvin Modes
 The statistical quasi-equilibrium models discussed above have the characteristic that vertical structure inconsistent with their assumed first baroclinic mode structure is rapidly damped. Thus, disturbances for which such alternative vertical structure is an intrinsic characteristic cannot be represented by such models. Examples of disturbances of this type include easterly waves [Reed and Recker, 1971; Reed et al., 1977; Cho and Jenkins, 1987], convectively coupled equatorial Kelvin waves [Straub and Kiladis, 2002; Tulich et al., 2007], and possibly the Madden-Julian oscillation [Kiladis et al., 2005]. Thus, Emanuel et al.'s  elegant idea for determining the vertical structure of tropical disturbances may be over-simplified.
 An alternative class of convectively coupled wave models was originated by Mapes  and expanded upon by Majda and Shefter [2001a,b], Majda et al. , Khouider and Majda [2006, 2007, 2008], Kuang [2008a,b], Andersen and Kuang , etc. In such models convection is controlled by convective inhibition, or alternatively convective available potential energy concentrated in the lower troposphere, which essentially amounts to the same thing. These models produce unstable waves which exhibit vertical structure incorporating both first and second baroclinic modes. They typically predict wave speeds in reasonable agreement with the observed propagation speeds of equatorial, convectively coupled Kelvin waves.
 A different type of mode was predicted by the simplified models of Sobel et al.  and Fuchs and Raymond [2002, 2005]. In these models the precipitation is controlled by the saturation fraction of the troposphere, as described in Section 4.2. In the case of no surface mean wind or cloud-radiation interactions, stationary unstable modes with relatively little scale selection develop when the GMS is negative. If surface easterlies are imposed, WISHE causes eastward propagation, with longer wavelength modes propagating more rapidly. With meridional moisture gradients, these modes become unstable for weakly positive GMS. Cloud-radiation interactions can destabilize these modes in the presence of weakly positive GMS as well. The modes predicted by these models were denoted “moisture waves” by Sobel et al.  and “moisture modes” by Fuchs and Raymond , since moisture anomalies play an important dynamical role in their development.
 The simplicity of the moisture mode allows its essence to be captured in weak temperature gradient cloud resolving model simulations such as those of Raymond and Zeng  and Raymond and Sessions . This is because the interaction between neighboring air columns is via the rapid buoyancy equilibration and the exchange of moisture between columns implied by the weak temperature gradient approximation. In fact, it is fair to say that the moisture mode is what remains after the weak temperature gradient approximation is made, just as only balanced modes survive the quasi-geostrophic approximation [Sobel et al. 2001].
 Fuchs and Raymond [2002, 2005] assumed a first baroclinic mode vertical structure for both the heating and the vertical velocity structure of moisture modes. Fuchs and Raymond  relaxed this condition, keeping the sinusoidal heating with a half-wavelength equal to the depth of the tropopause, but implementing a vertically resolved dynamical model including an upper radiation boundary condition. The results are similar to those of Fuchs and Raymond  for moisture modes, though they differ greatly for convectively coupled gravity modes.
 Raymond and Fuchs  extended the model of Fuchs and Raymond  by including an additional control on convection from convective inhibition, which we describe here. Starting from (5.1)-(5.4) and assuming time and zonal dependencies of the form exp[i(kx – ωt)] yields
where m = kΓB1/2/ω. The convective heating takes the first baroclinic mode form
where m0 = π/h, h being the height of the tropopause, and B is a constant determined below by the convective closure.
 As with our simplified discussion of the Neelin-Yu  model, cloud-radiation interactions and WISHE are ignored. The convective closure condition is
where α–1 is a time constant of order one day and q(z) is the vertical profile of scaled mixing ratio perturbation as before. The first term on the right side of (5.27) represents the effects of perturbations in precipitable water and is responsible for producing moisture modes, whereas the second term results in convectively coupled gravity waves, with the constant μCIN governing the strength of the convective inhibition effect. In this term es is the scaled surface moist entropy perturbation and is set to zero in the absence of WISHE. The quantity et is the perturbation saturation moist entropy at the top of the planetary boundary layer (PBL), the height of which is indicated by the dimensionless parameter D, the ratio of PBL to tropopause height h. Since et is a function only of the buoyancy perturbation b and pressure, et = λtb(D), where b(D) is b at the PBL top, with λt ≈ 3.5.
 Using q(z) = e(z) – b(z) as well as (5.7), (5.13), (5.24), and (5.25) allows (5.27) to be rewritten as
where ΓM is the GMS as defined by (5.13), κ = hΓB1/2k/(πa) is the dimensionless wavenumber, and where Φ = m0/m = πω/(khΓB1/2) is the phase speed ω/k of the disturbance divided by hΓB1/2/π, the phase speed of free, fundamental baroclinic mode gravity waves.
 The first term on the right side of (5.29) is the inhomogeneous part of the solution to (5.23), whereas the second term is the homogeneous part. Since physically the homogeneous solution represents a free gravity wave in the troposphere with dimensionless phase speed Φ, it is this part of the solution which sets the phase speed of the disturbance. It also has a vertical structure significantly different from that of the fundamental baroclinic mode.
 Combination of (5.28), (5.29), and (5.30) results in a rather complex dispersion relation for Φ,
which can only be solved numerically. However, if we ignore convective inhibition by setting μCIN = 0 and keep leading order terms in ∣Φ∣ << 1, it is easily shown that the dispersion relation reduces to
i. e., the mode is stationary with a growth rate proportional to minus the GMS. Thus, negative GMS results in moisture mode instability as one might expect.
 If on the other hand, we suppress moisture modes by setting ΓM = 0 and again retain only leading order terms in ∣Φ∣, then the simplified, but still implicit dispersion relation
holds. If Φ is approximately real, reflecting propagation with a modest growth rate, then the exponential term must be negative imaginary, implying that
where n is a positive integer. (The value n = 0 formally satisfies (5.35), but violates the condition that ∣Φ∣ ≪ 1.) For n = 1 the phase speed is 20 m s–1 if the free, fundamental mode gravity wave speed is 50 m s–1. This is quite close to the observed propagation speed of convectively coupled equatorial Kelvin waves. Furthermore, unlike the Neelin-Yu  and similar statistical quasi-equilibrium solutions, it depends only weakly on the GMS (see Raymond and Fuchs, 2009] and it exhibits a vertical structure similar to that seen in observed Kelvin waves.
 The complex vertical structure of temperature and wind in this model is produced by a fundamental baroclinic mode heating profile, in contrast to the results of Mapes  etc. This is not to say that the pattern of convective heating assumed by these authors does not exist in real Kelvin waves – only that it is not strictly necessary to produce the observed structure of the waves.
 When the full dispersion relation is analyzed numerically, both moisture modes and convectively coupled gravity waves emerge, with only modest modifications to the results discussed above.
5.4. GMS and the MJO
 There are many theories for the Madden-Julian oscillation (MJO; see Zhang, 2005 for a comprehensive review). Nevertheless, most global atmospheric numerical models have failed to provide a realistic representation of this phenomenon [Slingo, 1996; Lin, 2006].
 Raymond and Fuchs  demonstrated robust MJO-like disturbances in an equatorial beta plane model with no land, but with realistic sea surface temperatures and a toy convective parameterization prone to the production of negative GMS in strong convection. Figure 9 shows a snapshot of precipitation, surface wind, saturation fraction, and NGMS in an active MJO phase in the eastern Indian Ocean and western Pacific for a perpetual August simulation. Notice how the convection organizes itself into a line with a long near-equatorial component somewhat like a monsoon trough or an ITCZ. Meridional oscillations in the line suggest the development of largescale dynamical instability. Further examination shows that some of the oscillations break off to form vortices which resemble the outer circulations of a tropical cyclone.
Figure 9. Snapshot of convection in the eastern Indian Ocean and western Pacific during an active MJO phase in the beta plane numerical model of Raymond and Fuchs . (a) Surface wind, precipitation rate (shading) and saturation fraction (contours). (b) NGMS, with the heavy line indicating a zero value, light shading positive values, and dark shading negative values.
Download figure to PowerPoint
 An interesting feature of this simulation is that the existence of strong convection within these lines coincides with the development of negative NGMS. This invites comparison with the results discussed in Section 5.2 of Back and Bretherton  and Sobel and Neelin , who infer the existence of negative GMS in the east Pacific ITCZ and suggest that it is a consequence of the boundary layer flow produced by strong sea surface temperature gradients in this region. However, the negative GMS in the simulation shown in Figure 9 occurs in convective lines over weak sea surface temperature gradients where boundary layer forcing produced by these gradients is unlikely to be as important as it is in the east Pacific.
 As Figure 10 shows, for precipitation rates between 5 mm d–1 and 10 mm d–1 the convective mass flux profile is characteristic of cumulus congestus clouds with a maximum in vertical mass flux near 800 hPa, which is well below the level of minimum moist entropy, resulting in negative GMS (see Figure 6). However for stronger precipitation, Figure 10 shows that the level of maximum vertical mass flux ascends to near 550 hPa. Negative GMS results from the development in this case of a moist entropy profile with a minimum at even higher levels.
Figure 10. Vertical profiles of (a) specific moist entropy and (b) vertical mass flux, averaged over regions in Figure 9 with precipitation rates in the ranges 5 – 10 mm d–1 (thin lines) and 15 – 25 mm d–1 (thick lines).
Download figure to PowerPoint
 Whether the manner in which this model produces negative GMS actually occurs in nature remains to be determined. However, the existence of negative GMS, however produced, strongly suggests that the moisture mode instability discussed in Section 5.3 is active in this model. This conclusion is supported by the strong correlation between precipitation and saturation fraction illustrated in Figure 9 as well as the tendency of the disturbances to move rather slowly. In this model at least, the MJO appears to be a moisture mode which depends on the transient occurrence of negative GMS for its existence.