• Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with large-scale environment: Part I. J. Atmos. Sci., 31 , 674701.

    • Search Google Scholar
    • Export Citation
  • Brown, R. G., and C. D. Zhang, 1997: Variability of midtropospheric moisture and its effect on cloud-top height distribution during TOGA COARE. J. Atmos. Sci., 54 , 27602774.

    • Search Google Scholar
    • Export Citation
  • Derbyshire, S. H., I. Beau, P. Bechtold, J. Y. Grandpeix, J. M. Piriou, J. L. Redelsperger, and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity. Quart. J. Roy. Meteor. Soc., 130 , 30553079.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1987: An air–sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci., 44 , 23242340.

  • Emanuel, K. A., 1993: The effect of convective response time on WISHE modes. J. Atmos. Sci., 50 , 17631775.

  • Emanuel, K. A., 1995: The behavior of a simple hurricane model using a convective scheme based on subcloud-layer entropy equilibrium. J. Atmos. Sci., 52 , 39603968.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., J. D. Neelin, and C. S. Bretherton, 1994: On large-scale circulations in convecting atmospheres. Quart. J. Roy. Meteor. Soc., 120 , 11111143.

    • Search Google Scholar
    • Export Citation
  • Haertel, P. T., and G. N. Kiladis, 2004: Dynamics of 2-day equatorial waves. J. Atmos. Sci., 61 , 27072721.

  • Khairoutdinov, M., and D. Randall, 2006: High-resolution simulation of shallow-to-deep convection transition over land. J. Atmos. Sci., 63 , 34213436.

    • Search Google Scholar
    • Export Citation
  • Khouider, B., and A. J. Majda, 2006a: A simple multicloud parameterization for convectively coupled tropical waves. Part I: Linear analysis. J. Atmos. Sci., 63 , 13081323.

    • Search Google Scholar
    • Export Citation
  • Khouider, B., and A. J. Majda, 2006b: Multicloud convective parametrizations with crude vertical structure. Theor. Comput. Fluid Dyn., 20 , 351375.

    • Search Google Scholar
    • Export Citation
  • Kuang, Z., 2008: Modeling the interaction between cumulus convection and linear waves in a limited-domain cloud system–resolving model. J. Atmos. Sci., 65 , 576591.

    • Search Google Scholar
    • Export Citation
  • Kuang, Z., and C. Bretherton, 2006: A mass-flux scheme view of a high-resolution simulation of a transition from shallow to deep cumulus convection. J. Atmos. Sci., 63 , 18951909.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31 , 156179.

  • Majda, A. J., and M. G. Shefter, 2001: Models for stratiform instability and convectively coupled waves. J. Atmos. Sci., 58 , 15671584.

    • Search Google Scholar
    • Export Citation
  • Majda, A. J., B. Khouider, G. Kiladis, K. H. Straub, and M. G. Shefter, 2004: A model for convectively coupled tropical waves: Nonlinearity, rotation, and comparison with observations. J. Atmos. Sci., 61 , 21882205.

    • Search Google Scholar
    • Export Citation
  • Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57 , 15151535.

    • Search Google Scholar
    • Export Citation
  • Mapes, B. E., S. Tulich, J. Lin, and P. Zuidema, 2006: The mesoscale convection life cycle: Building block or prototype for large-scale tropical waves? Dyn. Atmos. Oceans, 42 , 329.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and J. Y. Yu, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part I: Analytical theory. J. Atmos. Sci., 51 , 18761894.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., I. M. Held, and K. H. Cook, 1987: Evaporation-wind feedback and low-frequency variability in the tropical atmosphere. J. Atmos. Sci., 44 , 23412348.

    • Search Google Scholar
    • Export Citation
  • Parsons, D. B., K. Yoneyama, and J. L. Redelsperger, 2000: The evolution of the tropical western Pacific atmosphere-ocean system following the arrival of a dry intrusion. Quart. J. Roy. Meteor. Soc., 126 , 517548.

    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., 1995: Regulation of moist convection over the west Pacific warm pool. J. Atmos. Sci., 52 , 39453959.

  • Raymond, D. J., 2000: Thermodynamic control of tropical rainfall. Quart. J. Roy. Meteor. Soc., 126 , 889898.

  • Redelsperger, J. L., D. B. Parsons, and F. Guichard, 2002: Recovery processes and factors limiting cloud-top height following the arrival of a dry intrusion observed during TOGA COARE. J. Atmos. Sci., 59 , 24382457.

    • Search Google Scholar
    • Export Citation
  • Ridout, J. A., 2002: Sensitivity of tropical Pacific convection to dry layers at mid- to upper levels: Simulation and parameterization tests. J. Atmos. Sci., 59 , 33623381.

    • Search Google Scholar
    • Export Citation
  • Roca, R., J. P. Lafore, C. Piriou, and J. L. Redelsperger, 2005: Extratropical dry-air intrusions into the West African monsoon midtroposphere: An important factor for the convective activity over the Sahel. J. Atmos. Sci., 62 , 390407.

    • Search Google Scholar
    • Export Citation
  • Sherwood, S. C., 1999: Convective precursors and predictability in the tropical western Pacific. Mon. Wea. Rev., 127 , 29772991.

  • Straub, K. H., and G. N. Kiladis, 2002: Observations of a convectively coupled Kelvin wave in the eastern Pacific ITCZ. J. Atmos. Sci., 59 , 3053.

    • Search Google Scholar
    • Export Citation
  • Takemi, T., O. Hirayama, and C. H. Liu, 2004: Factors responsible for the vertical development of tropical oceanic cumulus convection. Geophys. Res. Lett., 31 .L11109, doi:10.1029/2004GL020225.

    • Search Google Scholar
    • Export Citation
  • Tulich, S. N., D. A. Randall, and B. E. Mapes, 2007: Vertical-mode and cloud decomposition of large-scale convectively coupled gravity waves in a two-dimensional cloud-resolving model. J. Atmos. Sci., 64 , 12101229.

    • Search Google Scholar
    • Export Citation
  • Wang, B., 1988: Dynamics of tropical low-frequency waves—An analysis of the moist Kelvin wave. J. Atmos. Sci., 45 , 20512065.

  • Webster, P. J., and R. Lukas, 1992: TOGA COARE: The Coupled Ocean–Atmosphere Response Experiment. Bull. Amer. Meteor. Soc., 73 , 13771416.

    • Search Google Scholar
    • Export Citation
  • Wheeler, M., G. N. Kiladis, and P. J. Webster, 2000: Large-scale dynamical fields associated with convectively coupled equatorial waves. J. Atmos. Sci., 57 , 613640.

    • Search Google Scholar
    • Export Citation
  • Yanai, M., S. Esbensen, and J. H. Chu, 1973: Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos. Sci., 30 , 611627.

    • Search Google Scholar
    • Export Citation
  • Yu, J-Y., and J. D. Neelin, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part II: Numerical results. J. Atmos. Sci., 51 , 18951914.

    • Search Google Scholar
    • Export Citation
  • Zehnder, J. A., 2001: A comparison of convergence and surface-flux-based convective parameterizations with applications to tropical cyclogenesis. J. Atmos. Sci., 58 , 283301.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    (left) Phase speed and (right) growth rate as functions of wavenumber from the linearized version of the full model described in section 2c (with two-way wave equations), using normative parameter values. Modes with positive growth rates are highlighted with circles in the phase speed diagram. The phase speeds are symmetric about 0.

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    Physical patterns of the eigenmodes of the linearized full model described in section 2c (with two-way wave equations) for an eastward-propagating wave with a wavelength of 8640 km. Normative values are used for all parameters: (a) T1 (solid), T2 (dashed), and q (dotted) as functions of x (zonal distance); (b) J1 (solid) and J2 (dashed) as functions of x; (c) zonal and height pattern of the combined temperature anomaly, with a contour interval of 0.5 K; (d) zonal and height pattern of the combined convective heating anomaly, with a contour interval (CI) = 2 K day−1. In both (c) and (d), negative contours are dashed and the zero contour is omitted.

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    Phase lag between J1 and −J2 as a function of wavenumber for the linearized full model described in section 2c with normative parameter values.

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    Maximum linear growth rates for the linearized full model described in section 2c with individual parameters varied and the other parameters kept at their normative values.

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    Same as Fig. 1 but for the simplified version described in section 3a.

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    Same as Fig. 2 but for the simplified version described in section 3a and a CI of 0.5 K day−1 in (d).

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    Same as Fig. 4 but for the system described in section 3a.

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    Same as Fig. 6 but for the limiting case I ( f = 1) described in section 3b(1), with b2 = 0, γ0 = 0, ε = 0, τJ = 0, and m2 = 0.

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    Same as Fig. 8 but for the limiting case II ( f = 0) described in section 3b(2), with b2 = 0, γ0 = 0, ε = 0, τJ = 0, and m2 = 0.

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    Growth rates as a function of wavenumber for the limiting case II described in section 3b(2), with m2 = τJ = ε = 0 (thick solid), τJ = ε = 0 (thin solid), m2 = ε = 0 (dotted), ε = 0 (diamond symbols), and none of m2, τJ, ε is zero (circles). When a parameter is not zero, it takes its normative value.

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    A schematic of the moisture-stratiform instability, illustrated for an eastward-propagating wave viewed in a reference frame that follows the wave. All fields shown are anomalies. We start with (a) temperature and vertical velocity (arrows) anomalies associated with the wave. The large-scale lifting cools the lower troposphere as part of the wave signal. (b) This induces a positive deep convection anomaly, which cools the subcloud layer to maintain quasi equilibrium with the large-scale flow. (c) The deep convection anomaly also makes the midtroposphere more humid. (d) An anomalously moist midtroposphere allows convection to reach higher, while an anomalously dry one makes convection lower. This produces a convective heating anomaly pattern that is in phase with the original temperature anomaly and causes instability.

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    Growth rates as a function of wavenumber and f with b2 = 0, γ0 = 0, ε = 0, τJ = 0, and m2 = 0.

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    (a) Linear growth rate of the unstable mode as a function of wavenumber and r0 for the system described by Eq. (32) with f = 0.5. (b) Same as (a) but with the effect of a 2-h adjustment time to QE included.

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    Same as Fig. 8, but for the system described by Eq. (32), with f = 0.5 and γ0 = −0.2.

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    Fig. A1. (a) Contributions to convective drying of the midtroposphere (∂qmid/∂t)conv (solid) by −d1J1 (dashed) and −d2J2 (dotted) based on a linear regression. (b) Contributions to advective moistening of the midtroposphere (∂qmid/∂t)adv (solid) by a1w1 (dashed) and a2w2 (dotted) based on a linear regression. (c) Total tendencies of the midtroposphere humidity ∂qmid/∂t (solid) and (a1d1)J1d2J2 (dashed). (d) A linear regression of convective tendencies of boundary moist static energy ∂hb/∂t (solid) against J1 and J2. (e) The second-mode heating (solid) and rq(1.5T1qmid) (dashed) with rq = 0.7. (f) A linear regression of hb against lower-tropospheric temperature Tlow.

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A Moisture-Stratiform Instability for Convectively Coupled Waves

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  • 1 Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
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Abstract

A simple model of two vertical modes is constructed and analyzed to reveal the basic instability mechanisms of convectively coupled waves. The main novelty of this model is a convective parameterization based on the quasi-equilibrium concept and simplified for a model of two vertical modes. It hypothesizes 1) the approximate invariance of the difference between saturation moist static energy in the lower half of the troposphere and moist static energy in the subcloud layer, regardless of free troposphere humidity, and 2) that variations in the depth of convection are determined by moisture-deficit variations in the midtroposphere. Physical arguments for such a treatment are presented. For realistic model parameters chosen based on cloud system resolving model simulations (CSRMs) of an earlier study, the model produces unstable waves at wavelengths and with structures that compare well with the CSRM simulations and observations.

A moisture–stratiform instability and a direct–stratiform instability are identified as the main instability mechanisms in the model. The former relies on the effect of midtroposphere humidity on the depth of convection. The latter relies on the climatological mean convective heating profile being top heavy, and it is identified to be the same as the stratiform instability mechanism proposed by B. E. Mapes. The moisture–stratiform instability appears to be the main instability mechanism for the convectively coupled wave development in the CSRM simulations. The finite response time of convection has a damping effect on the waves that is stronger at high wavenumbers. The net moistening effect of the second-mode convective heating also damps the waves, but more strongly at low wavenumbers. These effects help to shape the growth rate curve so that the most unstable waves are of a few thousand kilometers in scale.

Corresponding author address: Zhiming Kuang, Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138. Email: kuang@fas.harvard.edu

Abstract

A simple model of two vertical modes is constructed and analyzed to reveal the basic instability mechanisms of convectively coupled waves. The main novelty of this model is a convective parameterization based on the quasi-equilibrium concept and simplified for a model of two vertical modes. It hypothesizes 1) the approximate invariance of the difference between saturation moist static energy in the lower half of the troposphere and moist static energy in the subcloud layer, regardless of free troposphere humidity, and 2) that variations in the depth of convection are determined by moisture-deficit variations in the midtroposphere. Physical arguments for such a treatment are presented. For realistic model parameters chosen based on cloud system resolving model simulations (CSRMs) of an earlier study, the model produces unstable waves at wavelengths and with structures that compare well with the CSRM simulations and observations.

A moisture–stratiform instability and a direct–stratiform instability are identified as the main instability mechanisms in the model. The former relies on the effect of midtroposphere humidity on the depth of convection. The latter relies on the climatological mean convective heating profile being top heavy, and it is identified to be the same as the stratiform instability mechanism proposed by B. E. Mapes. The moisture–stratiform instability appears to be the main instability mechanism for the convectively coupled wave development in the CSRM simulations. The finite response time of convection has a damping effect on the waves that is stronger at high wavenumbers. The net moistening effect of the second-mode convective heating also damps the waves, but more strongly at low wavenumbers. These effects help to shape the growth rate curve so that the most unstable waves are of a few thousand kilometers in scale.

Corresponding author address: Zhiming Kuang, Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138. Email: kuang@fas.harvard.edu

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

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
i1520-0469-65-3-834-e1
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 (Gi),
i1520-0469-65-3-834-e2
and obtain
i1520-0469-65-3-834-e3
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 cj. When the buoyancy frequency, the square of which is N2 = g[(d lnT/dz) + (g/cpT)], is constant, the vertical modes are
i1520-0469-65-3-834-e4
where HT 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 hb and an equation for the midtropospheric humidity qmid. The equation for hb follows prior studies (e.g., M00; MS01; KM06):
i1520-0469-65-3-834-e5
where E is the tendency from surface heat flux anomalies, and b1 and b2 represent the reduction of hb per unit J1 and J2, respectively. In actuality, large-scale vertical advection has a smaller but nonnegligible effect on hb, 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 qmid can be written as
i1520-0469-65-3-834-e6
where a1 and a2 represent the effective moisture stratification for the two modes, and d1 and d2 represent the convective drying effects on qmid 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 qmid 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, w1J1. Furthermore, because qmid is located around the nodal point of w2, the effect of large-scale advection by w2 on qmid (i.e., a2) is small. We may therefore simplify Eq. (6) to
i1520-0469-65-3-834-e7
where m1a1d1 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 Tj, hb, and q are expressed in temperature unit (1 K) so that the scale for Jj and wj 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
i1520-0469-65-3-834-e8
and an integrated lower-tropospheric heating anomaly L as
i1520-0469-65-3-834-e9
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
i1520-0469-65-3-834-e10
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
i1520-0469-65-3-834-e11
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
i1520-0469-65-3-834-e12
where req is the ratio anomaly that is in statistical equilibrium with its large-scale environment [Eq. (10)] and τr is the adjustment time for r to approach that equilibrium. We consider τr 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 τr 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 τr 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
i1520-0469-65-3-834-e13
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
i1520-0469-65-3-834-e14
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 Leq:
i1520-0469-65-3-834-e15
where
i1520-0469-65-3-834-e16
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, Leq 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 qmid to simplify the notation. Taking into consideration the finite adjustment time to achieve QE, denoted as τL, we have
i1520-0469-65-3-834-e17
We expect τL 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
i1520-0469-65-3-834-e18
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/HT), j = 1, 2 as the vertical structures with HT = 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 rq 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; rq 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.

3. The basic instability mechanisms

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,
i1520-0469-65-3-834-e19
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:
i1520-0469-65-3-834-e20
where
i1520-0469-65-3-834-e21
and include the finite adjustment time to achieve QE as
i1520-0469-65-3-834-e22
where
i1520-0469-65-3-834-e23
Here, we have expressed J1,eq in terms of ∂T/∂t, although one can also write it in a form similar to Eq. (15). We shall take τJ = τL. 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 + rqq+, the difference in relaxing J1 instead of L lies in the ∂q+/∂t term. Because τL 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 hb 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
i1520-0469-65-3-834-e24
Physically, this represents a case in which convective mass flux is dominated by undiluted updrafts. Here, the subcloud-layer moist static energy hb is changed only by J1 (as b2 = 0) and is in equilibrium with T1 alone (as f = 1). Equation (24) simply states that J1,eq is that required to keep ∂hb/∂t the same as FT1/∂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 τJ = 0 (i.e., in SQE), so that
i1520-0469-65-3-834-e25
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, τJ = 0, and m2 = 0; other parameters take their normative values) are shown in Fig. 8. The q field in this case is simply −J2/γq. 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 τJ 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 hb 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
i1520-0469-65-3-834-e26
that is, J1 is that required to keep ∂hb/∂t the same as FT2/∂t. First consider a system in SQE,
i1520-0469-65-3-834-e27
i1520-0469-65-3-834-e28
where J2 = −γqq (as γ0 = 0). Assuming solutions of the form exp[i(kx − ωt)], we obtain the dispersion relationship
i1520-0469-65-3-834-e29
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 + im1γqF/b1; that is, the phase speed is c2 and the growth rate is m1γqF/b1 at all wavelengths. The growth rate is proportional to m1γq, which measures how strongly the depth of convection depends on q (the factor γq) 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, 1 ≡ (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/γq.

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 m2γq 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 −m2γqq 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 + im1γqF/b1). Plugging this into (29) and neglecting the second-order terms of δω, we have
i1520-0469-65-3-834-e30
Therefore, this damping effect becomes significant for wavenumbers lower than (m2 + m1F/b1)λq/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 m2γq. For general parameter choices, an effective m2, 2 ≡ (b1m2b2m1/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
i1520-0469-65-3-834-e31
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 τJ included (dotted line) and with the effects of both m2 and τJ included (diamond symbol). The mathematical reason for the different effects of m2 and τJ 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 = −γqq, 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).

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 J1 is more strongly tied to ∂T/∂t (controlled by shallow or deep CAPE) while the second baroclinic heating J2 is more strongly tied to moisture (moisture control). This allows J2 to be more in phase with T2 when J1 is controlled by T2, but constrains J1 to be largely in quadrature with T1 when it is controlled by T1. Although we have taken b2 = 0 and γ0 = 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 m1 or γq is zero), and the moistening effect of J2 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 m2 = 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 J2. 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 c2c*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 γ0 < 0, γq = 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 γq = 0. To simplify the discussion, we will also take b2 = 0, ε = 0, and τJ = 0, although these are not required. The system is now reduced to
i1520-0469-65-3-834-e32
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 (τJ = 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 rq = 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:
i1520-0469-65-3-834-e33
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 γq = 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 τJ = 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 γq = 0, one may wish to include the effect of undiluted CAPE and replace Eq. (20) with
i1520-0469-65-3-834-e34
which changes the second equation in (32) to
i1520-0469-65-3-834-e35
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.

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 m2) 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.

REFERENCES

  • Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with large-scale environment: Part I. J. Atmos. Sci., 31 , 674701.

    • Search Google Scholar
    • Export Citation
  • Brown, R. G., and C. D. Zhang, 1997: Variability of midtropospheric moisture and its effect on cloud-top height distribution during TOGA COARE. J. Atmos. Sci., 54 , 27602774.

    • Search Google Scholar
    • Export Citation
  • Derbyshire, S. H., I. Beau, P. Bechtold, J. Y. Grandpeix, J. M. Piriou, J. L. Redelsperger, and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity. Quart. J. Roy. Meteor. Soc., 130 , 30553079.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1987: An air–sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci., 44 , 23242340.

  • Emanuel, K. A., 1993: The effect of convective response time on WISHE modes. J. Atmos. Sci., 50 , 17631775.

  • Emanuel, K. A., 1995: The behavior of a simple hurricane model using a convective scheme based on subcloud-layer entropy equilibrium. J. Atmos. Sci., 52 , 39603968.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., J. D. Neelin, and C. S. Bretherton, 1994: On large-scale circulations in convecting atmospheres. Quart. J. Roy. Meteor. Soc., 120 , 11111143.

    • Search Google Scholar
    • Export Citation
  • Haertel, P. T., and G. N. Kiladis, 2004: Dynamics of 2-day equatorial waves. J. Atmos. Sci., 61 , 27072721.

  • Khairoutdinov, M., and D. Randall, 2006: High-resolution simulation of shallow-to-deep convection transition over land. J. Atmos. Sci., 63 , 34213436.

    • Search Google Scholar
    • Export Citation
  • Khouider, B., and A. J. Majda, 2006a: A simple multicloud parameterization for convectively coupled tropical waves. Part I: Linear analysis. J. Atmos. Sci., 63 , 13081323.

    • Search Google Scholar
    • Export Citation
  • Khouider, B., and A. J. Majda, 2006b: Multicloud convective parametrizations with crude vertical structure. Theor. Comput. Fluid Dyn., 20 , 351375.

    • Search Google Scholar
    • Export Citation
  • Kuang, Z., 2008: Modeling the interaction between cumulus convection and linear waves in a limited-domain cloud system–resolving model. J. Atmos. Sci., 65 , 576591.

    • Search Google Scholar
    • Export Citation
  • Kuang, Z., and C. Bretherton, 2006: A mass-flux scheme view of a high-resolution simulation of a transition from shallow to deep cumulus convection. J. Atmos. Sci., 63 , 18951909.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31 , 156179.

  • Majda, A. J., and M. G. Shefter, 2001: Models for stratiform instability and convectively coupled waves. J. Atmos. Sci., 58 , 15671584.

    • Search Google Scholar
    • Export Citation
  • Majda, A. J., B. Khouider, G. Kiladis, K. H. Straub, and M. G. Shefter, 2004: A model for convectively coupled tropical waves: Nonlinearity, rotation, and comparison with observations. J. Atmos. Sci., 61 , 21882205.

    • Search Google Scholar
    • Export Citation
  • Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57 , 15151535.

    • Search Google Scholar
    • Export Citation
  • Mapes, B. E., S. Tulich, J. Lin, and P. Zuidema, 2006: The mesoscale convection life cycle: Building block or prototype for large-scale tropical waves? Dyn. Atmos. Oceans, 42 , 329.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and J. Y. Yu, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part I: Analytical theory. J. Atmos. Sci., 51 , 18761894.

    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., I. M. Held, and K. H. Cook, 1987: Evaporation-wind feedback and low-frequency variability in the tropical atmosphere. J. Atmos. Sci., 44 , 23412348.

    • Search Google Scholar
    • Export Citation
  • Parsons, D. B., K. Yoneyama, and J. L. Redelsperger, 2000: The evolution of the tropical western Pacific atmosphere-ocean system following the arrival of a dry intrusion. Quart. J. Roy. Meteor. Soc., 126 , 517548.

    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., 1995: Regulation of moist convection over the west Pacific warm pool. J. Atmos. Sci., 52 , 39453959.

  • Raymond, D. J., 2000: Thermodynamic control of tropical rainfall. Quart. J. Roy. Meteor. Soc., 126 , 889898.

  • Redelsperger, J. L., D. B. Parsons, and F. Guichard, 2002: Recovery processes and factors limiting cloud-top height following the arrival of a dry intrusion observed during TOGA COARE. J. Atmos. Sci., 59 , 24382457.

    • Search Google Scholar
    • Export Citation
  • Ridout, J. A., 2002: Sensitivity of tropical Pacific convection to dry layers at mid- to upper levels: Simulation and parameterization tests. J. Atmos. Sci., 59 , 33623381.

    • Search Google Scholar
    • Export Citation
  • Roca, R., J. P. Lafore, C. Piriou, and J. L. Redelsperger, 2005: Extratropical dry-air intrusions into the West African monsoon midtroposphere: An important factor for the convective activity over the Sahel. J. Atmos. Sci., 62 , 390407.

    • Search Google Scholar
    • Export Citation
  • Sherwood, S. C., 1999: Convective precursors and predictability in the tropical western Pacific. Mon. Wea. Rev., 127 , 29772991.

  • Straub, K. H., and G. N. Kiladis, 2002: Observations of a convectively coupled Kelvin wave in the eastern Pacific ITCZ. J. Atmos. Sci., 59 , 3053.

    • Search Google Scholar
    • Export Citation
  • Takemi, T., O. Hirayama, and C. H. Liu, 2004: Factors responsible for the vertical development of tropical oceanic cumulus convection. Geophys. Res. Lett., 31 .L11109, doi:10.1029/2004GL020225.

    • Search Google Scholar
    • Export Citation
  • Tulich, S. N., D. A. Randall, and B. E. Mapes, 2007: Vertical-mode and cloud decomposition of large-scale convectively coupled gravity waves in a two-dimensional cloud-resolving model. J. Atmos. Sci., 64 , 12101229.

    • Search Google Scholar
    • Export Citation
  • Wang, B., 1988: Dynamics of tropical low-frequency waves—An analysis of the moist Kelvin wave. J. Atmos. Sci., 45 , 20512065.

  • Webster, P. J., and R. Lukas, 1992: TOGA COARE: The Coupled Ocean–Atmosphere Response Experiment. Bull. Amer. Meteor. Soc., 73 , 13771416.

    • Search Google Scholar
    • Export Citation
  • Wheeler, M., G. N. Kiladis, and P. J. Webster, 2000: Large-scale dynamical fields associated with convectively coupled equatorial waves. J. Atmos. Sci., 57 , 613640.

    • Search Google Scholar
    • Export Citation
  • Yanai, M., S. Esbensen, and J. H. Chu, 1973: Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos. Sci., 30 , 611627.

    • Search Google Scholar
    • Export Citation
  • Yu, J-Y., and J. D. Neelin, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part II: Numerical results. J. Atmos. Sci., 51 , 18951914.

    • Search Google Scholar
    • Export Citation
  • Zehnder, J. A., 2001: A comparison of convergence and surface-flux-based convective parameterizations with applications to tropical cyclogenesis. J. Atmos. Sci., 58 , 283301.

    • Search Google Scholar
    • Export Citation

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 J1, J2, w1, w2, T1, and T2. 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 qmid 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 (Q2) of qmid against the two convective heating modes J1 and J2. 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 J1 and J2 to convective moistening are shown in red and green, respectively. Their sum reproduces Q2 almost perfectly on the scale plotted and thus is omitted. Convective drying (negative Q2) is dominated by the first heating mode (d1 = 1.3). The second heating mode has a moistening effect (d2 = −1.1). This represents a moistening (drying) effect in the midtroposphere by congestus (stratiform) convection. Figure A1b shows the regression of the qmid tendency due to vertical advection against w1 and w2, which yields a1 = 1.6 and a2 = 0.0. The effect of the second mode is small because the midtroposphere is around the nodal point of w2. Similar results are obtained by applying the vertical structures of w1 and w2 on the background moisture stratification and integrating from 400 to 600 hPa. The actual time derivative of qmid 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: w1 is well correlated with J1 (correlation 0.98) and is only slightly larger (by ∼3%) than J1. Note that a larger adiabatic cooling (w1) than convective heating J1 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 a2, it appears reasonable to neglect moisture advection by w2 and combine convective drying and vertical moisture advection effects of the first mode, as in Eq. (7). This simplified treatment reproduces the total qmid tendencies reasonably well (Fig. A1c). While the estimates are subject to many uncertainties, that m1 = a1d1 is positive (i.e., the net effect of deep convective heating, J1, is to make the midtroposphere more humid) is a robust result.

We compute ∂hb/∂t as the mass weighted averages of moist static energy tendencies over the depth of the subcloud layer. Figure A1d shows a regression of ∂hb/∂t by convection against J1 and J2 [Eq. (5) with E = 0]. Contributions from large-scale vertical advection in the subcloud layer are smaller and neglected. This yields b1 = 1.0 and b2 = 2.3. A positive b2 implies that convective heating in the lower troposphere is more effective at reducing hb 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 r0 = 1 and rq = 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 hb variations as air parcels rise through the lower troposphere.

APPENDIX B

Connections between Eqs. (3) and (19)

Let us first add a Newtonian cooling ε in the thermodynamic equations in Eq. (3). Assuming a wave solution of the form exp(iωtikx) and diagonalizing the system, we have
i1520-0469-65-3-834-eb1
Eliminating wj leads to
i1520-0469-65-3-834-eb2
For waves with periods substantially shorter than 2π/ε (∼60 days with our choice of ε), we have approximately
i1520-0469-65-3-834-eb3
Therefore, Eq. (3) is connected to Eq. (19) with
i1520-0469-65-3-834-eb4
Equation (B4) indicates that convective heating is more effective in forcing temperature variations in the one-way wave equations. This is particularly true for the first vertical mode. For waves with a phase speed of 25 m s−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 τJ is similar to that with the simplifying parameter choices.

When f = 1, Eq. (23) becomes
i1520-0469-65-3-834-ec1
We have continued to neglect T1’s contribution in q+. At SQE, we have
i1520-0469-65-3-834-ec2
where
i1520-0469-65-3-834-ec3
Here T2 is forced by modes of Eq. (C2) and does not feed back onto T1 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.
When f = 0, we have
i1520-0469-65-3-834-ec4
Therefore
i1520-0469-65-3-834-ec5
where
i1520-0469-65-3-834-ec6
Here T1 is forced by the modes of Eq. (C5) and does not feed back on T2 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, K1 and K2. Using our normative parameter values and allowing for a reasonable range, K1 is close to 0.4 and K2 is close to −1. Therefore, the f = 1 case with general parameter choices may be viewed as having a negative effective γq.

Fig. 1.
Fig. 1.

(left) Phase speed and (right) growth rate as functions of wavenumber from the linearized version of the full model described in section 2c (with two-way wave equations), using normative parameter values. Modes with positive growth rates are highlighted with circles in the phase speed diagram. The phase speeds are symmetric about 0.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 2.
Fig. 2.

Physical patterns of the eigenmodes of the linearized full model described in section 2c (with two-way wave equations) for an eastward-propagating wave with a wavelength of 8640 km. Normative values are used for all parameters: (a) T1 (solid), T2 (dashed), and q (dotted) as functions of x (zonal distance); (b) J1 (solid) and J2 (dashed) as functions of x; (c) zonal and height pattern of the combined temperature anomaly, with a contour interval of 0.5 K; (d) zonal and height pattern of the combined convective heating anomaly, with a contour interval (CI) = 2 K day−1. In both (c) and (d), negative contours are dashed and the zero contour is omitted.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 3.
Fig. 3.

Phase lag between J1 and −J2 as a function of wavenumber for the linearized full model described in section 2c with normative parameter values.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 4.
Fig. 4.

Maximum linear growth rates for the linearized full model described in section 2c with individual parameters varied and the other parameters kept at their normative values.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 5.
Fig. 5.

Same as Fig. 1 but for the simplified version described in section 3a.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 6.
Fig. 6.

Same as Fig. 2 but for the simplified version described in section 3a and a CI of 0.5 K day−1 in (d).

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 7.
Fig. 7.

Same as Fig. 4 but for the system described in section 3a.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 8.
Fig. 8.

Same as Fig. 6 but for the limiting case I ( f = 1) described in section 3b(1), with b2 = 0, γ0 = 0, ε = 0, τJ = 0, and m2 = 0.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 9.
Fig. 9.

Same as Fig. 8 but for the limiting case II ( f = 0) described in section 3b(2), with b2 = 0, γ0 = 0, ε = 0, τJ = 0, and m2 = 0.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 10.
Fig. 10.

Growth rates as a function of wavenumber for the limiting case II described in section 3b(2), with m2 = τJ = ε = 0 (thick solid), τJ = ε = 0 (thin solid), m2 = ε = 0 (dotted), ε = 0 (diamond symbols), and none of m2, τJ, ε is zero (circles). When a parameter is not zero, it takes its normative value.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 11.
Fig. 11.

A schematic of the moisture-stratiform instability, illustrated for an eastward-propagating wave viewed in a reference frame that follows the wave. All fields shown are anomalies. We start with (a) temperature and vertical velocity (arrows) anomalies associated with the wave. The large-scale lifting cools the lower troposphere as part of the wave signal. (b) This induces a positive deep convection anomaly, which cools the subcloud layer to maintain quasi equilibrium with the large-scale flow. (c) The deep convection anomaly also makes the midtroposphere more humid. (d) An anomalously moist midtroposphere allows convection to reach higher, while an anomalously dry one makes convection lower. This produces a convective heating anomaly pattern that is in phase with the original temperature anomaly and causes instability.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 12.
Fig. 12.

Growth rates as a function of wavenumber and f with b2 = 0, γ0 = 0, ε = 0, τJ = 0, and m2 = 0.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 13.
Fig. 13.

(a) Linear growth rate of the unstable mode as a function of wavenumber and r0 for the system described by Eq. (32) with f = 0.5. (b) Same as (a) but with the effect of a 2-h adjustment time to QE included.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Fig. 14.
Fig. 14.

Same as Fig. 8, but for the system described by Eq. (32), with f = 0.5 and γ0 = −0.2.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

i1520-0469-65-3-834-fa01

Fig. A1. (a) Contributions to convective drying of the midtroposphere (∂qmid/∂t)conv (solid) by −d1J1 (dashed) and −d2J2 (dotted) based on a linear regression. (b) Contributions to advective moistening of the midtroposphere (∂qmid/∂t)adv (solid) by a1w1 (dashed) and a2w2 (dotted) based on a linear regression. (c) Total tendencies of the midtroposphere humidity ∂qmid/∂t (solid) and (a1d1)J1d2J2 (dashed). (d) A linear regression of convective tendencies of boundary moist static energy ∂hb/∂t (solid) against J1 and J2. (e) The second-mode heating (solid) and rq(1.5T1qmid) (dashed) with rq = 0.7. (f) A linear regression of hb against lower-tropospheric temperature Tlow.

Citation: Journal of the Atmospheric Sciences 65, 3; 10.1175/2007JAS2444.1

Table 1.

A summary of parameters used in the simple model.

Table 1.

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.

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