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  • View in gallery

    Schematic describing the vertical structure of a wave that grows from the two instabilities discussed in this study in regions of easterly shear: (a) baroclinic instability and (b) moisture–vortex instability. Anomalous large-scale adiabatic lifting (subsidence) is shown as a upward (downward)-pointing pink arrow while anomalous ascent (descent) associated with enhanced (suppressed) convection is shown as a upward (downward)-pointing teal arrow. In (b) the moisture tendency qualitatively corresponds to the pink vertical arrow. Solid (dashed) contours depict anomalous poleward (equatorward) flow in the lower, mid-, and upper troposphere.

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    Schematic describing the vertical arrangement of variables used in the two-layer moist QG model. The shading indicates the specific humidity in the layer.

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    Schematic describing (a) the phasing between ψT and ψB according to αψ and (b) the phasing between ψB and P′ according to αP. Regions in (a) where the resulting structure is top-heavy (wave signature stronger in the upper troposphere) is shown as purple while the region where the structure is more bottom-heavy is shown as green. Regions in (b) where the convection has an in-phase component with the midtropospheric cyclone (anticyclone) is shown in red (blue).

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    Growth rate (shading) and phase speed (contours) as a function of zonal wavenumber and thermal wind u¯T for the growing wave solution when τc = 0, βq = 0, and m = 0.2. The contour interval is 2.5 m s−1. The dot–dashed line depicts a phase speed of zero, while dashed lines depict negative (westward) phase speeds. The left ordinate shows u¯T and the right ordinate shows βT.

  • View in gallery

    Horizontal map of a zonal wavenumber-8 wave that grows from baroclinic instability, where u¯T=20m s1, βq = 0, τc = 0, and m = 0.2. (top) Upper-tropospheric streamfunction ψ1 and divergence D1. (middle) Midtropospheric streamfunction ψ2 and precipitation anomalies P′. (bottom) Lower-tropospheric streamfunction ψ3 and divergence D3. The contours and shading in both panels are based on an initial anomaly of ψT (ψT0) of 5 × 105 m2 s−1. The contour interval is 2 × 105 m2 s−1. While the meridional structure is not discussed in this study, it is chosen here so that it is much longer than the horizontal structure, effectively making it negligible.

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    As in Fig. 4, but showing the two growing wave solutions that arise when τc = 6 h.

  • View in gallery

    (a) Growth rate (shading) and phase speed (contours) obtained from Eq. (31a) as a function of varying βq and assuming u¯T=0. (b) As in (a), but assuming β = 0. For (a) and (b) τc = 6 h and m = 0.2. (c) As in (a), but showing the approximate solution obtained from Eq. (41a) and τc = 6 min. Units on the ordinate are multiplied by 1011. Note that the scale in the color bar of (c) is reduced by a factor of 26 compared to (a) and (b), largely as a result of the smallness of τc in the approximate solution.

  • View in gallery

    Growth rate as obtained from Eq. (44a) for u¯T-induced moisture–vortex instability (red) and for βq-induced moisture–vortex instability (blue). For the red line uT = −5 m s−1 (βT = 0.3 × 10−11 m−1 s−1) and βq = 0. For the blue line uT = 0 and βq = 0.3 × 10−11 m−1 s−1. For both lines τc = 6 min and m = 0.2.

  • View in gallery

    As in Fig. 5, but showing a zonal wavenumber-15 wave that grows from (a) u¯T-based moisture–vortex instability, where u¯T=7m s1, βq = 0, and (b) βq-based moisture–vortex instability, where u¯T=0 and βq = 0.4 m−1 s−1. In both panels m = 0.2 and τc = 6 h.

  • View in gallery

    Schematic describing how the mean thermal wind u¯T (red arrows) affects the phasing between anomalous lower-tropospheric circulation and anomalous rainfall. A westerly (u¯T>0) thermal wind corresponds to mean lower-tropospheric easterlies when u¯B=0. The lower-tropospheric moisture q3 is shown as a blue–green line, and the vertical velocity associated with convection (ωQ) is shown as the teal vertical arrow.

  • View in gallery

    Growth rate contribution from (a),(b) MVI and (c),(d) the modulation of baroclinic instability when τc > 0, which is written as BCI + BMI − δi for the growing modes obtained from Eq. (44a). (a),(b) The section in Fig. 6a where u¯T>0. (c),(d) The section in Fig. 6a where u¯T<0. In all panels the contribution of baroclinic instability δi to growth is shown as contours. The contour interval is 0.2 day−1.

  • View in gallery

    Growth rates for the unstable mode obtained from Eq. (31a) for βq = 0, m = 0, and (top) u¯T>0 and (bottom) u¯T<0 for τc values of (a) 0.1, (b) 0.5, (c) 1, (d) 2, and (e) 4 h. Growth from baroclinic instability δi is shown as contours for each panel. The panels are arranged as in Fig. 11. The zeroth contour is shown as a dot–dashed line.

  • View in gallery

    As in Fig. 12, but showing αψ as shading and αP as contours defined as in Fig. 3. The angle αψ is defined as a tilt against the shear, with increasing αψ meaning increased tilt against the shear. The angle αP is defined as a westward shift of P′ with respect to ψT.

  • View in gallery

    (top three rows) As in Fig. 5, but showing the growing zonal wavenumber-8 solutions of Eq. (31a) for (a) u¯T=20m s1 and (b) u¯T=20m s1. For both solutions τc = 6 h, m = 0.2, and βq = 0. (bottom) Adiabatic ωa (pink), diabatic (teal), and total (purple dashed) vertical velocity anomaly averaged over 10°–25°N.

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Interactions between Water Vapor, Potential Vorticity, and Vertical Wind Shear in Quasi-Geostrophic Motions: Implications for Rotational Tropical Motion Systems

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  • 1 Department of Climate and Space Science and Engineering, University of Michigan, Ann Arbor, Michigan
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Abstract

A linear two-layer model is used to elucidate the role of prognostic moisture on quasigeostrophic (QG) motions in the presence of a mean thermal wind (u¯T). Solutions to the basic equations reveal two instabilities that can explain the growth of moist QG systems. The well-documented baroclinic instability is characterized by growth at the synoptic scale (horizontal scale of ~1000 km) and systems that grow from this instability tilt against the shear. Moisture–vortex instability—an instability that occurs when moisture and lower-tropospheric vorticity exhibit an in-phase component—exists only when moisture is prognostic. The instability is also strongest at the synoptic scale, but systems that grow from it exhibit a vertically stacked structure. When moisture is prognostic and u¯T is easterly, baroclinic instability exhibits a pronounced weakening while moisture vortex instability is amplified. The strengthening of moisture–vortex instability at the expense of baroclinic instability is due to the baroclinic (u¯T) component of the lower-tropospheric flow. In westward-propagating systems, lower-tropospheric westerlies associated with an easterly u¯T advect anomalous moisture and the associated convection toward the low-level vortex. The advected convection causes the vertical structure of the wave to shift away from one that favors baroclinic instability to one that favors moisture–vortex instability. On the other hand, a westerly u¯T reinforces the phasing between moisture and vorticity necessary for baroclinic instability to occur. Based on these results, it is hypothesized that moisture–vortex instability is an important instability in humid regions of easterly u¯T such as the South Asian and West African monsoons.

Current affiliation: Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JAS-D-20-0205.s1.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher’s Note: This article was revised on 5 March 2021 to correct a typographical error in Table 2 that appeared when originally published.

Corresponding author: Ángel F. Adames, angel.adamescorraliza@wisc.edu

Abstract

A linear two-layer model is used to elucidate the role of prognostic moisture on quasigeostrophic (QG) motions in the presence of a mean thermal wind (u¯T). Solutions to the basic equations reveal two instabilities that can explain the growth of moist QG systems. The well-documented baroclinic instability is characterized by growth at the synoptic scale (horizontal scale of ~1000 km) and systems that grow from this instability tilt against the shear. Moisture–vortex instability—an instability that occurs when moisture and lower-tropospheric vorticity exhibit an in-phase component—exists only when moisture is prognostic. The instability is also strongest at the synoptic scale, but systems that grow from it exhibit a vertically stacked structure. When moisture is prognostic and u¯T is easterly, baroclinic instability exhibits a pronounced weakening while moisture vortex instability is amplified. The strengthening of moisture–vortex instability at the expense of baroclinic instability is due to the baroclinic (u¯T) component of the lower-tropospheric flow. In westward-propagating systems, lower-tropospheric westerlies associated with an easterly u¯T advect anomalous moisture and the associated convection toward the low-level vortex. The advected convection causes the vertical structure of the wave to shift away from one that favors baroclinic instability to one that favors moisture–vortex instability. On the other hand, a westerly u¯T reinforces the phasing between moisture and vorticity necessary for baroclinic instability to occur. Based on these results, it is hypothesized that moisture–vortex instability is an important instability in humid regions of easterly u¯T such as the South Asian and West African monsoons.

Current affiliation: Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JAS-D-20-0205.s1.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher’s Note: This article was revised on 5 March 2021 to correct a typographical error in Table 2 that appeared when originally published.

Corresponding author: Ángel F. Adames, angel.adamescorraliza@wisc.edu

1. Introduction

Large-scale (horizontal scale of ~1000 km) motion systems on Earth’s atmosphere tend to be organized into vortices or waves. Decades of research onto these systems has shown that many of these grow from hydrodynamic instabilities. In the midlatitudes, where the Coriolis force is strong and motions are quasigeostrophic (QG), we observe large-scale Rossby waves and storm systems that grow from baroclinic instability (Eady 1949; Charney 1947; Phillips 1954; Bretherton 1966).

In the tropics, however, the weaker Coriolis force and resulting weak horizontal temperature gradients cause baroclinic instability to be less common. Nonetheless, some tropical motion systems have been hypothesized to grow from this instability. Among them are monsoon low pressure systems (MLPSs) and African easterly waves (AEWs).

The most notable similarity between the regions in which MLPSs and AEWs occur is that they are characterized by strong jets that exhibit strong easterly vertical shear (∂u/∂z < 0) (Burpee 1972; Roja Raman et al. 2009, 2011), although the depth of these jets differs in their respective regions. Because of this shear and the deep convection that is observed in association with these systems, it was thought that a variant of baroclinic instability that is modified by deep convection could explain their growth (Mass 1979; Salvekar et al. 1986; Krishnakumar et al. 1992). Baroclinic instability is characterized by geopotential and horizontal winds that are vertically tilted against the ambient shear (Eady 1949; Vallis 2017) (Fig. 1a).

Fig. 1.
Fig. 1.

Schematic describing the vertical structure of a wave that grows from the two instabilities discussed in this study in regions of easterly shear: (a) baroclinic instability and (b) moisture–vortex instability. Anomalous large-scale adiabatic lifting (subsidence) is shown as a upward (downward)-pointing pink arrow while anomalous ascent (descent) associated with enhanced (suppressed) convection is shown as a upward (downward)-pointing teal arrow. In (b) the moisture tendency qualitatively corresponds to the pink vertical arrow. Solid (dashed) contours depict anomalous poleward (equatorward) flow in the lower, mid-, and upper troposphere.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

Recent research into MLPSs and AEWs has put into question the central role that baroclinic instability plays in these waves. Cohen and Boos (2016) found that MLPSs exhibit a vertical structure that is inconsistent with baroclinic instability. Instead MLPSs exhibit an upright structure, akin to tropical depressions (Yoon and Chen 2005; Hunt et al. 2016; Clark et al. 2020). Similarly, Russell and Aiyyer (2020) found that AEWs exhibit a structure that is more upright than what would be observed if the waves were growing from moist baroclinic instability. Russell et al. (2020) found that organized convection maintains this upright structure, and is likely the main source of instability in AEWs.

Given that the mean state in the regions where both AEWs and MLPSs occur seems favorable for baroclinic instability, it is perplexing that it may not be the main mechanism of growth in these systems. One possibility is that the mean state of the regions where these waves occur has properties that dampen baroclinic instability, favoring a different instability. But what mechanism could lead to this change? A possible answer to this question may be found in the numerical study by Lapeyre and Held (2004). They showed that when the atmosphere became humid, eddies driven by baroclinic instability were replaced by low-level vortices that resembled tropical cyclones. They argued that this occurrence is the result of a correlation between the lower tropospheric moisture and vorticity. Thus, it is possible that water vapor’s interaction with convection and the large-scale circulation may fundamentally change the mechanisms by which tropical disturbances grow in the presence of vertical shear.

The role water vapor plays in modulating the occurrence and organization of tropical deep convection is well documented. For example, a humid free troposphere reduces the dilution that updrafts experience as they ascend (Kuo et al. 2017; Ahmed and Neelin 2018). The central role moisture has in tropical convection has led to the hypothesis that waves may exist where water vapor plays a central role in their dynamics (Sobel et al. 2001).

Some studies have hypothesized that moisture is important in MLPSs (Adames and Ming 2018b,a; Clark et al. 2020). Adames and Ming (2018a, AM18 from here on) hypothesized that the mechanism of growth in MLPSs involves interactions between the circulation, water vapor and convection. By solving a system of shallow water equations with prognostic moisture and a simple convective adjustment scheme, AM18 found that vortices can grow from moisture–vortex instability (Fig. 1b). In this instability the anomalous meridional winds advect moisture, resulting in a moisture tendency. The moisture anomalies lag the moisture tendency and are shifted toward the center of the vortex (Fig. 1b). The enhanced convection that results from the higher moisture content intensifies the vortex through vortex stretching. While AM18 show that MLPSs could potentially grow from moisture–vortex instability, they did not take into account the large vertical wind shear that is usually observed in the South Asian monsoon.

In this study, we will expand upon AM18’s framework by applying it to a two-layer QG model. Such models have been used to understand baroclinic instability (Phillips 1954; Bretherton 1966; Vallis 2017) and to simulate the midlatitude circulation (Pavan and Held 1996; Lapeyre and Held 2004). They have also been used to understand the dynamics of MLPSs and AEWs (Aravequia et al. 1995; Salvekar et al. 1986; Grist et al. 2002). However, the impact that a prognostic moisture equation may have in a linear two-layer QG system has not been studied in detail. We will show that including a prognostic moisture equation significantly modulates the dynamics of waves that propagate in the presence of a thermal wind. Specifically, it weakens baroclinic instability in regions of easterly shear while favoring moisture–vortex instability. The results of this study may explain why baroclinic instability is not the dominant mechanism for growth in MLPSs, and may support recent work on the growth of AEWs.

This study is structured as follows. The next section describes the moist two-layer QG model. Wave solutions to the system of equations are obtained in section 3. Section 4 discusses the wave solutions that arise when moisture is diagnosed from the large-scale wind fields. Section 5 discusses how these solutions change when a prognostic moisture equation is included. Section 6 discusses how baroclinic instability is modified when prognostic moisture is included. A synthesis and discussion of the main findings of this study is offered in section 7. Concluding remarks are given in section 8.

2. Two-layer quasigeostrophic model with prognostic moisture

a. Basic equations

Our analysis is based on a two-layer linear QG model with prognostic moisture. The layout of the model closely follows Holton and Hakim (2012) and is described in Fig. 2. The main variables are shown in Table 1. Constants and their values are shown in Table 2. The lower tropospheric layer is bounded by the 1000 and 500 hPa surfaces. We will assume that all moisture is located within this layer. The upper tropospheric layer is bounded by the 0 and 500 hPa layer. Vertical motion is assumed to be a maximum in the 500 hPa layer, and vanishes at the 1000 and 0 hPa surfaces. It would be more realistic to have an upper boundary at 100 hPa instead of 0 since vertical velocities in the tropics approximately vanish at this layer. However, we keep the top at 0 hPa so that readers can directly compare the results of this study with chapter 7 of Holton and Hakim (2012). Our results are not sensitive to the choice of an upper bound between 100 and 0 hPa. The validity of the QG approximation for AEWs and MLPSs is discussed in section a of the appendix.

Fig. 2.
Fig. 2.

Schematic describing the vertical arrangement of variables used in the two-layer moist QG model. The shading indicates the specific humidity in the layer.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

Table 1.

The main variables and definitions used in this study.

Table 1.
Table 2.

Constants used in this study and their value.

Table 2.

Because the vertical velocity is a maximum in the midtroposphere, it follows that the associated low-level convergence and upper-level divergence result in winds and geopotential field components that exhibit a “first baroclinic” structure; i.e., the fields exhibit a reversed polarity in the lower and upper troposphere. Baroclinic disturbances are characterized by temperature perturbations that are related to the geopotential through hydrostatic balance. We can write this relationship in the two-layer model as
Φ3Φ1ln(p3/p1)=RdT2,
where Rd is the dry gas constant, Φ is the geopotential, and T is the temperature. The subscripts describe the layer of the field following Fig. 2.
The planetary vorticity is linearized with respect to a reference latitude by applying the β-plane approximation:
f=f0+βy,
where f0 and β = df/dy are the planetary vorticity and the meridional planetary vorticity gradient evaluated at a reference latitude, respectively. Since we are interested in motions that occur in the “outer tropics” (10°–30°N/S), we will assume a value of f0 of 4 × 10−5 s−1, a value that corresponds to 16°N.1
We can express many of the field variables in terms of a geostrophic streamfunction ψ. In the QG approximation, ψ is linearly related to Φ:
ψ=Φf01.
The horizontal wind v = ui + υj can be obtained from ψ as follows:
u=ψy,υ=ψx,
where i and j are the zonal and meridional unit vectors. The vertical component of the relative vorticity (ζ) is written as
ζ=h2ψ,
where h2=x2+y2 is the horizontal Laplacian.
We will assume that both convective heating Qc and vertical motion ω exhibit a single vertical structure in which they attain a maximum amplitude at 500 hPa (layer 2), and become zero at the top and bottom boundaries. At the 750 hPa layer, Qc3 and ω3 can be obtained by interpolating between the 0 and 500 hPa layers ω3 = (ω4 + ω2)/2. Since ω vanishes in the top and bottom boundaries, it follows that
ω3=ω22.
The same procedure can be done to obtain ω1 and Qc1. To relate the horizontal divergence D = ∇· v to ω, we invoke mass continuity in each discrete layer. Since ω = 0 at the top and bottom boundaries, we can write the divergence in layers 1 and 3 as
D1=ω2Δp,D3=ω2Δp.
We will decompose the main variables into mean state and perturbation components, denoted by an overbar and a prime, respectively:
ψ(x,y,t)=ψ¯(y)+ψ(x,t).
As in Holton and Hakim (2012), and Vallis (2017), we define ψ¯ as only varying in the meridional direction while ψ′ varies zonally and temporally. Assuming that the mean zonal wind is a constant in space and time, we can express the mean streamfunction as
ψ¯1=u¯1y,
ψ¯3=u¯3y.
The same assumption is made for the lower-tropospheric water vapor (q3) and midtropospheric temperature (T2), i.e.,
q3(x,y,t)=q¯3(y)+q3(x,t),
T2(x,y,t)=T¯2(y)+T2(x,t).
We can combine Eqs. (1), (3), and (9) to obtain the thermal wind equation:
T2y=f0Rdu¯3u¯1ln(p3/p1),
which shows that the mean meridional temperature gradient is constant in space and time. We will test the sensitivity of the wave solutions to the mean thermal wind by varying the amplitude of the meridional temperature gradient from 0 to ±5 K (1000 km)−1. For simplicity, we will assume that the mean meridional moisture gradient is also a constant:
q¯3y=constant.
To test the sensitivity of the wave solutions to the meridional moisture gradient, we will vary its amplitude from 0 to ±1.2 g kg−1 (1000 km)−1. The change in moisture with increasing latitude is much smaller than q¯3, which is on the order of 10 g kg−1.

We are assuming that perturbations are only in the zonal plane since instabilities that involve quasigeostrophic motions tend to prefer the longest meridional scales (see, e.g., Vallis 2017), even when prognostic moisture is included (AM18).

With these approximations and definitions, we can write our upper- and lower-tropospheric vorticity, midtropospheric temperature, and lower-tropospheric moisture equations as
(t+u¯1x)ζ1=υ1βf0D1,
(t+u¯3x)ζ3=υ3βf0D3,
(t+u¯2x)CpT2=υ2CpT¯2yω2Δs¯Δp+Qc2,
(t+u¯3x)Lq3=υ3Lq¯3yω3ΔLq¯ΔpgΔpLP,
where Δs=s¯3s¯1<0 is the vertical change in dry static energy (s = CpT + Φ), ΔLq=L¯q4L¯q2>0 is the vertical change in latent energy between 1000 and 500 hPa, Δp = p3p1, and g is the gravitational acceleration

The system of equations in Eq. (13) are the same equations used to describe baroclinic instability in textbooks (Holton and Hakim 2012; Vallis 2017). They differ in that a linearized moisture equation is included and that convective heating is included in the thermodynamic equation. With the addition of moisture, the two-layer model is reminiscent to those employed by Lapeyre and Held (2004), and Lambaerts et al. (2012), except that this model is linear.

b. Convective parameterization

The definitions given by Eqs. (1)(7) are sufficient to solve the dry two-layer QG model discussed in chapter 7 of Holton and Hakim (2012). However, our study includes deep moist convection and a prognostic moisture equation. The representation of convection and the way it couples Eq. (13d) to Eqs. (13a)(13c) will be key to the main findings of this study.

To facilitate the discussion on the convective parameterization, we will decompose ω2 into an adiabatic and diabatic component, denoted by the subscript a and Q, respectively:
ω2=ωa+ωQ.
Both ωa and ωQ can be diagnosed from the QG omega equation, as was done by Nie and Sobel (2016) and Murthy and Boos (2020). However, we will focus on the thermodynamic equation [Eq. (13c)] in order to maintain the discussion simple. The adiabatic component of the vertical velocity is due to the temperature tendency and temperature advection:
ωaΔpCpΔs¯(T2t+u¯2T2x+υ2T¯2y),
while the diabatic component can be defined as the component that exactly satisfies the weak temperature gradient approximation (WTG) (Sobel et al. 2001):
ωQΔpΔs¯Qc2.
Many previous studies diagnose ωQ in terms of ωa (ωQωa) (Mak 1982; Sanders 1984), leading to the in-phase relationship shown in Fig. 1a. This type of diagnosis is appropriate for extratropical motions, where adiabatic lifting is strong and can create broad regions of stratiform cloudiness. For the tropical disturbances considered here, ωa and ωQ are not in phase (see Fig. 12 in Adames and Ming 2018b). In the tropics, the main source of diabatic heating is deep cumulus convection, whose buoyancy is sensitive to the thermodynamic environment rather than the large-scale ascent (Brown and Zhang 1997; Mapes 2000; Ahmed and Neelin 2018). Adames and Ming (2018b) showed that anomalous precipitation in simulated MLPSs is more closely in phase with anomalous water vapor, rather than large-scale lifting (their Fig. 5). A recent study by Nie et al. (2020) showed that rainfall rates from deep convection (ωQ) are much larger than the rates obtained from large-scale forcing (ωa) in the Asian and African monsoons (their Fig. 2).
Simplified models of the tropics often parameterize the deep convection by employing a convective adjustment scheme such as the Betts–Miller scheme (Betts and Miller 1986). In these schemes, convection adjusts the temperature and moisture back to a reference profile. Clark et al. (2020) showed that MLPSs with realistic structures can be simulated by representing convection with a Betts–Miller scheme. We will use a similar parameterization to that used by Clark et al. (2020), but only include the modulation of convection by water vapor, as in other simplified studies of tropical motion systems (Sobel et al. 2001; Fuchs and Raymond 2005; AM18). Thus, we will express Qc2 as
Qc2Lq3τc,
where τc is the convective moisture adjustment time scale. It is estimated to range between 2 and 12 h (Betts 1986; Neelin and Zeng 2000; Bretherton et al. 2004; Sugiyama 2009). We will use a value of 6 h when applicable. Sensitivity tests using various values of τc yield results similar to those discussed by AM18. The main findings of this study are the same regardless of whether τc is 2 or 12 h.

Equation (17) is similar to Eq. (12) in Ahmed et al. (2020), but without the inclusion of temperature. Ahmed et al. (2020) found that the observed relationship between diabatic heating and the buoyancy of an entraining plume can be written in terms of lower-tropospheric temperature and moisture: Qc2=h4/τb+Lq3/τqCpT3/τt, where τq, τt, and τb are sensitivity time scales akin to τc, and h4 = CpT4 + Lq4 is the boundary layer moist enthalpy. While their relationship is empirical, it summarizes the physical processes that lead to precipitation, that is, deep convection occurs when the atmospheric lapse rate is sufficiently unstable to convection and the troposphere sufficiently humid so that updrafts dilute less. If we ignore temperature and assume that q4 = 2q3, we can show that τc = τqτb/(2τq + τb). While we recognize that the temperature anomalies may play a role in determining the distribution of precipitation in systems like MLPSs and AEWs, focusing on water vapor will yield simpler equations with solutions that are easier to interpret. However, we recognize that the exclusion of temperature’s role in Qc2 may be a limitation of this study.

The anomalous precipitation P can be related to the column-integrated convective heating Qc following Yanai et al. (1973):
LP=ΔpgQc2.
Equation (18) will allow us to combine Eqs. (13c) and (13d) into a single equation for moist enthalpy, as discussed in the following subsection.

It is worth pointing out that the equations describing vertical velocity in this study do not apply the WTG approximation strictly. Rather, by allowing ωa to be nonzero, we are applying what is referred to as a “relaxed” WTG approximation (Raymond and Zeng 2005). While the WTG approximation is generally considered reasonable in MLPSs and AEWs (Adames and Ming 2018b; Hannah and Aiyyer 2017), the relaxed WTG approximation allows for the temperature tendency and the temperature advection to play a role in the dynamics of the waves analyzed here. Their contributions to Eq. (13c) may be small, but they may play an important role in the convective coupling of these systems.

c. Equations for barotropic vorticity and QG potential vorticity

With the approximations and assumptions discussed in sections 2a and 2b, we can reduce the number of variables in Eq. (13) to three. To reduce our four basic equations in Eq. (13) to three, we will eliminate the thermodynamic equation by defining the baroclinic and barotropic streamfunctions, ψT and ψB, respectively, as
ψT=(ψ1ψ3)/2,
ψB=(ψ1ψ3)/2,
and defining the mean thermal (baroclinic) and barotropic zonal winds as
u¯T=(u¯1u¯3)/2,
u¯B=(u¯1+u¯3)/2.
Positive values of u¯T indicate westerly shear and positive ψT indicate positive ψ′ anomalies in the upper troposphere.
Using Eq. (19), we can combine Eqs. (13a)(13c) to obtain an equation for the barotropic vorticity and for the QG potential vorticity (qd). We can also combine Eqs. (13c) and (13d) following AM18 to obtain an equation for the moist enthalpy (CpT + Lq). The resulting equations are written as
D¯hζBDt=υBβu¯TζTx,
D¯hqdDt=υTβυBβTu¯TζBxf0M¯sLP,
D¯hqmDt=υTβqυB(βT+βq)+u¯Tf0τcM¯qPxmf0M¯qLP,
where
D¯hDt=t+u¯Bx
is the horizontal material derivative for a disturbance propagating relative to a mean barotropic flow (u¯B). The dry baroclinic PV anomaly is written as
qd=ζTkd2ψT,
where
kd=f0c
is the inverse of the Rossby radius of deformation and
c=(gRdM¯s2p2Cp)1/2
is the phase speed of free gravity waves.
The moist enthalpy is written in the form of a “moist equivalent” potential vorticity, that is, the enthalpy is written in units of s−1:
qm=f0τcM¯qP+kd2ψT.
Note that the definition of qm used here differs from that in AM18 in the sign of the second term because we have defined ψT to be positive for positive upper-tropospheric ψ′. AM18 defined ψT so that it is positive for positive lower-tropospheric ψ′.
In converting Eq. (13) into Eq. (21) we have defined new variables and constants. The constants M¯q and M¯s are the gross moisture stratification and gross dry stability (Yu et al. 1998; Adames and Kim 2016), respectively defined as
M¯q=LΔq¯Δp2g,
M¯s=Δs¯Δpg.
From Eq. (24) we can see that M¯q and M¯s are the mass-weighted vertical changes in Lq and s, respectively. We also have defined m as the normalized gross moist stability (NGMS):
m=M¯sM¯qM¯s.
Note that we have opted for the use of lowercase m for the NGMS since it is the same definition given by Neelin and Held (1987), except that it is divided by M¯s. The NGMS m can be interpreted as the moist static stability of the column per unit of dry static stability. It has a value of unity in a dry atmosphere, and can be near zero or negative in humid atmospheres. We will use a value of 0.2 in this study, which is close to the NGMS values documented in observations (Sobel et al. 2014; Inoue and Back 2015). The sensitivity of the wave solutions analyzed in this study to variations in m is shown in the online supplementary material.
In Eq. (21) we also define βq as the mean meridional moisture gradient,
βq=2f0Δq¯q¯3y,
and βT as the mean meridional temperature gradient,
βT=u¯Tkd2.
These variables are defined so they are in the same units as β, i.e., m−1 s−1, following AM18.

With the variables and constants in Eq. (21) defined, we will now discuss the processes that lead to the evolution of the three main variables. The evolution of ζB is driven by the advection of planetary vorticity by υB and advection of ζB by u¯T.

The processes that lead to the evolution of qd relative to the mean barotropic flow [rhs in Eq. (21b)] are

  • advection of planetary vorticity by υT,
  • advection of mean temperature by υB,
  • advection of ζB by u¯T,
  • vortex stretching of planetary vorticity by the anomalous divergence in anomalous convection.
The first three processes are those also found in studies of dry baroclinic instability, while the last term is the result of moist convection. It is worth noting that the last rhs term in Eq. (21b) can exist even if moisture is not prognostic. However, when moisture is prognostic, it is this term that is key in the coupling of the moisture anomalies with qd and ζB. The impact that having a prognostic moisture equation without convection is discussed in the supplementary material.

The processes that lead to the evolution of qm relative to the mean barotropic flow [rhs in Eq. (21c)] are

  • advection of mean moisture by υT,
  • advection of mean moist enthalpy by υB,
  • advection of the moisture anomalies by u¯T,
  • vertical advection of moist static energy by the vertical motion in anomalous convection.

Equations (21b) and (21c) contain four terms on the right hand side that mirror one another. However, even though Eqs. (21b) and (21c) are similar, they do not necessarily contribute equally to the evolution of a moist wave.

d. Gross potential vorticity

We can understand the relative contributions of Eqs. (21b) and (21c) to the evolution of a moist wave by following AM18 and combining the equations to obtain the “gross” PV:
qG=(1m)qmmqd.
The evolution of qG is given by
D¯hqGDt=(1m)[υTβqυB(βq+βT)+u¯Tf0τcM¯qPx]m[υTβ+υBβTu¯TζBx].
As discussed by AM18, Eq. (28) describes the relative importance of dry and moist processes to the evolution of a moist wave. The dry processes are multiplied by −m while the moist processes are multiplied by 1 − m. When m → 1, the atmosphere is dry and moist processes play no role in the evolution of the wave. Conversely, when m → 0, moist processes govern the evolution of the wave while dry processes play no role. It is worth noting that the thermal wind (both as u¯T and βT) contributes to both dry and moist processes, indicating that it plays a role in the evolution of the wave regardless of the value of m.

3. Wave solutions

a. Dispersion relation

The system of equations in Eq. (21) can be solved by assuming that they can be described by a wave of the following form:
ψT=ψT0exp(ikxiω*t),
ψB=ψB0exp(ikxiω*t),
P=P0exp(ikxiω*t),
where k is the zonal wavenumber, ω* is the wave frequency, and ψT0, ψB0, and P0 are the initial amplitudes.
Because the only role of u¯B is to advect the wave, it will be convenient to define the wave frequency relative to it:
ω=ω*u¯Bk.
This definition will allow us to simplify the equations that follow without losing any physical insight. With this simplification, we can solve Eq. (21) by substituting the field variables with their solutions in Eq. (29). After substitution, we can find nontrivial solutions to Eq. (21) if the determinant of the coefficients ψT0, ψB0, and P0 is zero. By finding the determinant we obtain a cubic dispersion relation of the form
[miτc(ω+u¯Tk)]Bd+(1m)Bm=0,
where
Bd=ω2k(k2+kd2)+ωβ(kd2+2k2)+β2k+u¯T2k3(kd2k2)
are the terms that define the counterpropagating waves that lead to dry baroclinic instability, and
Bm=ω2kkd2+ω(βkd2+2βqk2)+2ββqku¯Tk3βm
are the terms that define the counterpropagating waves that lead to moist baroclinic instability. Equation (31) is the general form of the dispersion relation for moist QG waves in the two-layer model employed here. It will be used both to elucidate the propagation and growth of these waves and to evaluate approximate solutions that will be discussed in subsequent sections. Additional discussion on Bd and Bm is offered in section b(2) of the appendix.

b. Polarization relations and phase relations

For a given zonal distribution of ψB, the polarization relations for ψT and P′ take the following form:
ψT=β+ωku¯Tk2ψB,
P=iM¯q[(kd2ω+βqk)ψT(βq+βT)kψB]f0[miτc(ω+u¯Tk)].
The polarization relations in Eq. (32) show that the barotropic and baroclinic streamfunctions couple only when u¯T0, consistent with previous studies (Phillips 1954; Wang and Xie 1996). Equation (32b) shows that P′ scales in proportion to M¯q, i.e., P′ is greater in a humid mean state. Both ψT and ψB contribute to P′.
A special case of Eq. (32b) can be obtained when βq = 0, which can be written as
PiM¯qkd2(ωψT+u¯TkψB)f0[miτc(ω+u¯Tk)].
In this case, the relative contribution of ψT and ψB to P′ depends on the magnitude of ω relative to u¯Tk. For u¯Tkω, the polarization relation for P′ simplifies to
P~iM¯qu¯Tkkd2f0[miτc(ω+u¯Tk)]ψB,
which indicates that P′ is governed by the barotropic mode, which governs the advection of mean midtropospheric temperature [Eq. (13c)]. This approximation is accurate in regions where u¯T>10m s1 and for zonal wavenumbers greater than 10. These conditions are reasonable for both MLPSs and AEWs. If we replace P′ by q3 in Eq. (34) we obtain the following approximate proportionality:
q3τcψB,
which illustrates that the moisture anomalies increase in amplitude as τc is increased. When τc = 0, no moisture anomalies exist but P′ can be nonzero, as shown by Eq. (32b). Thus, interactions between the large-scale circulation and moisture require τc > 0.
The phase relation between ψB and ψT will provide key insights into the interpretation of the main instabilities in this study. We can obtain the phase angle between the two fields by rearranging the terms in Eq. (32a) and decomposing ω into its real and imaginary parts ω = ωr + i, which yields the following expression:
tan(αψπ)=ωikβ+ωrksgn(u¯T),
where αψ is defined so that it is zero in regions of easterly shear in which β + ωrk. The sign function (sgn) takes into account the fact that the polarization relation in Eq. (32a) is dependent on the sign of u¯T. A schematic showing how the phasing between ψT and ψB vary as a function of αψ is shown in Fig. 3a.
Fig. 3.
Fig. 3.

Schematic describing (a) the phasing between ψT and ψB according to αψ and (b) the phasing between ψB and P′ according to αP. Regions in (a) where the resulting structure is top-heavy (wave signature stronger in the upper troposphere) is shown as purple while the region where the structure is more bottom-heavy is shown as green. Regions in (b) where the convection has an in-phase component with the midtropospheric cyclone (anticyclone) is shown in red (blue).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

Equation (36) along with Fig. 3a reveal that the larger ωi is relative to ωr, the more tilted the vertical structure of the wave is. This result is clearest when considering the case of β = 0. In this simplified case the angle is just the ratio between ωr and ωi. In the Eady (1949) model of baroclinic instability, ωr = 0 when ωi > 0, which results in αψ = −90°. Neutrally stable waves are vertically stacked and tend to be either top-heavy or bottom-heavy, as discussed by Wang and Xie (1996).

The relationship between P′ and ψB will also elucidate the processes that lead to instability in the moist waves analyzed here. We can obtain an approximate relationship following AM18, rearranging the terms in Eq. (32b) and assuming mτc(ω+u¯Tk). The resulting phase relationship takes the following form:
tanαPmτck(cp+u¯T)sgn(u¯T),
where cp = ωr/k is the phase speed of the wave relative to the barotropic flow, recalling that ωr=ωr*u¯B. Equation (37) is analogous to Eq. (27) in AM18 with the addition of a u¯T term and that the phasing is specifically with respect to ψB. Figure 3b shows the phasing between P′ and ψB. Similarly to AM18, αP = 90° if τc = 0. When τc > 0, a component of P′ will be in phase with ψT, which leads to growth through moisture–vortex instability or decay, depending on the value of αP (Fig. 3b).

4. Solutions for τc = 0

The dispersion relation in Eq. (31a) includes a multitude of terms that involve various processes. As a result, it may be daunting to elucidate the processes that lead to wave propagation and growth without first considering some simplifications. The first simplification worth discussing are the well-documented solutions that arise in an atmosphere where precipitation is diagnosed from large-scale forcing (τc = 0). In such a case, Eq. (31a) simplifies to
mBd+(1m)Bm=0.
The dispersion relation that arises from this case will be denoted with a subscript 0 (i.e., ω0). The full solution to Eq. (38) takes the form
ω0=β(2mk2+kd2)+βqk2(1m)2k(mk2+kd2)±δ,
where
δ={[βkd2βqk2(1m)]24k2(mk2+kd2)2u¯T2k2(kd2mk2)u¯Tk2βq(1m)mk2+kd2}1/2.
The dispersion contains two solutions: a growing mode and a damped mode. The phase speed and growth rate of the growing mode is shown in Fig. 4 for βq = 0 and m = 0.2. Propagation of the wave is westward at the largest scales, although it becomes eastward for zonal wavenumbers greater than 10 when |u¯T|>8m s1. No difference in propagation and growth is seen between easterly u¯T and westerly u¯T. The damped solution mirrors the growing one and is not shown.
Fig. 4.
Fig. 4.

Growth rate (shading) and phase speed (contours) as a function of zonal wavenumber and thermal wind u¯T for the growing wave solution when τc = 0, βq = 0, and m = 0.2. The contour interval is 2.5 m s−1. The dot–dashed line depicts a phase speed of zero, while dashed lines depict negative (westward) phase speeds. The left ordinate shows u¯T and the right ordinate shows βT.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

The terms within the square root in δ are comprised of a combination of terms that are weighted by m or 1 − m. The former terms correspond to the processes that lead to dry baroclinic instability while the latter lead to moist baroclinic instability [see section b(2) of the appendix]. When 0 < m < 1 growth occurs due to a combination of dry and moist baroclinic instabilities. Waves that grow this way are often referred to as diabatic Rossby waves (Parker and Thorpe 1995). We will refer to this instability as baroclinic instability without any distinction, while the growth when m = 1 and m = 0 will be referred to as dry and moist baroclinic instabilities, respectively.

Relative to dry baroclinic instability alone, growth between m = 0 and m = 1 exhibits a larger amplitude due to the contribution from moist processes. This larger amplitude was noted in the diabatic Rossby waves examined by de Vries et al. (2010), and by the balanced moist waves examined by Wetzel et al. (2017). In the case of β = 0 and βq = 0 we find that baroclinic instability occurs when k < kdm−1/2. This result is similar to the dry case [see section b(1) of the appendix], except k is scaled by m−1/2, which indicates that the cutoff for instability occurs at smaller scales (larger k).

Waves that grow from baroclinic instability in easterly shear (u¯T<0) exhibit an eastward tilt with height (Fig. 5). The strong tilt is the result of the large growth rate relative to the propagation speed, as indicated by Eq. (36).

Fig. 5.
Fig. 5.

Horizontal map of a zonal wavenumber-8 wave that grows from baroclinic instability, where u¯T=20m s1, βq = 0, τc = 0, and m = 0.2. (top) Upper-tropospheric streamfunction ψ1 and divergence D1. (middle) Midtropospheric streamfunction ψ2 and precipitation anomalies P′. (bottom) Lower-tropospheric streamfunction ψ3 and divergence D3. The contours and shading in both panels are based on an initial anomaly of ψT (ψT0) of 5 × 105 m2 s−1. The contour interval is 2 × 105 m2 s−1. While the meridional structure is not discussed in this study, it is chosen here so that it is much longer than the horizontal structure, effectively making it negligible.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

5. Solutions for τc > 0

So far, we have discussed the behavior of wave solutions when convection instantaneously responds to large-scale forcing, τc = 0. In this section we will investigate how this behavior changes when moisture is prognostic, i.e., τc > 0.

When τc > 0, Eq. (31a) yields three wave solutions: one that grows when u¯T>0, one that grows when u¯T<0, and a damped mode. The growth rate and phase speeds for the two growing modes is shown in Fig. 6. While the solutions for τc = 0 exhibit symmetry with respect to the direction of u¯T when βq = 0, a nonzero τc breaks this symmetry.2 For τc = 6 h, growth begins near zonal wavenumber 6 for both easterly and westerly u¯T. This is the same zonal wavenumber where baroclinic instability begins to be observed in Fig. 4. However, when τc = 6 h growth is also observed for larger wavenumbers than those seen in Fig. 4. Additionally, while baroclinic instability is seen only when u¯T has a magnitude of 10 m s−1 or greater, growth in easterly shear can be seen for u¯T values of magnitude greater than 4 m s−1. In westerly shear, however, damping is seen between 4 and 10 m s−1. The largest growth rates when τc = 6 h are smaller in amplitude than the largest growth rates when τc = 0.

Fig. 6.
Fig. 6.

As in Fig. 4, but showing the two growing wave solutions that arise when τc = 6 h.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

The growth rates in Fig. 6 are distinct from those Fig. 4, suggesting that these waves may not be growing from baroclinic instability. Instead, they may be growing from an instability that requires a prognostic moisture equation (τc > 0). We can elucidate this instability by obtaining an approximate solution to Eq. (31a). We will assume that, to leading order, the τc = 0 solutions describe the behavior of the wave. With these assumptions we can expand Eq. (31a) into a perturbation series, centered on the τc = 0 solutions and truncating on the second term:
ωω0+εω1.
In Eq. (40), ω0 is the dispersion relation for Eq. (38), ε is a small nondimensional parameter and ω1 is the second-order wave solution. Nondimensionalizing Eq. (31a) reveals that ε = τcf0 (not shown). The approximate solution is robust as long as τcf0m. For m ≃ 0.2 and f0 = 4 × 10−5 s−1, τc would need to be on the order of 8 min for the perturbation expansion to be accurate. We will use a value of 0.1 h (6 min) for our approximate solution. While we recognize that τc is actually on the order of hours (Betts and Miller 1986; Bretherton et al. 2004; Ahmed et al. 2020), this approximate solution is still sufficiently insightful to be worth discussing.
Applying the perturbation series expansion in Eq. (40) to Eq. (31a) leads to the following approximate dispersion relation:
ωω0+iτc(ω0+u¯Tk)ω0M
where
M=BdR
accounts for interactions between moisture and vorticity, and will be referred to as the moisture–vortex interaction term. The denominator term R is defined as
R=ω02k(mk2+kd2)βk[mβ+(1m)βq]u¯T2k3(kd2mk2)+(1m)βqu¯Tk3.
In a dry atmosphere Bd=0 and hence M=0. The moisture–vortex interaction term M can also be written in terms of moist processes by using Eq. (38) and replacing Bd with Bm:
M=(1m)BmmR.
Equation (42) shows that moisture–vortex interactions amplify with decreasing m, consistent with the growth rates obtained by AM18 (see also supplementary material). While Bm0 as m → 0, it does so at a slower pace than (1 − m)/m increases (not shown). The denominator term R also tends to decrease with decreasing m, compensating for the decrease in Bm as m decreases. Thus, interactions between perturbations in moisture and vorticity become more important as the lower troposphere becomes more humid.
The real component of the second term in Eq. (41a) contributes negligibly to the propagation of the wave, as seen when comparing the contours of Figs. 4 and 6a. Because of this, we can simplify Eq. (41a) to elucidate the processes that arise when τc > 0 by decomposing ω0 and M into their real and imaginary components (subscripts r and i, respectively):
ω0=cpk+iδi,
M=Mr+iMi,
where δi is the imaginary component of δ. With this decomposition, we can simplify Eq. (41a) to
ωcpk+i(MVI+BCI+BMI),
where we have defined three different processes that contribute to wave growth or decay. The first imaginary term in Eq. (44a) is
MVI=τck2cp(cp+u¯T)Mr,
which we will refer to as the moisture–vortex instability term due to its similarity to the instability described by AM18. The second term,
BCI=δi(1τcMrδi),
is the growth rate from baroclinic instability. The first term in Eq. (44c) is the baroclinic instability term from Eq. (39), while the second term is a modulation from the addition of prognostic moisture. The third term is written as
BMI=τckδi(2cp+u¯T)Mi,
which will be referred to as the baroclinic moisture–vortex interaction term since it contains contributions from baroclinic instability δi and the parenthesis term is reminiscent to the MVI term.

The three processes defined in Eq. (44a) contain information about how moist waves can grow. We will discuss MVI first since it is a process that occurs independently from baroclinic instability. We will then analyze how the three terms add up to create growth or decay in moist waves.

a. βq-induced moisture–vortex instability

The simplest case of moisture–vortex instability can be obtained when βq ≠ 0 and u¯T=0. In such a case the BCI and BMI terms in Eq. (44a) are zero but MVI can be nonzero. Figures 7a and 7c show the phase speed and growth rate obtained with Eqs. (31a) and (41a), respectively. While the approximate solution from Eq. (41a) exhibits much smaller growth rates than the solution from Eq. (31a) due to the much smaller value of τc, it does qualitatively capture the same pattern. The largest growth is seen when βq > 0 near zonal wavenumber 15. Damping at the largest scales is also seen. It is worth noting that the wavenumber–βq distribution of growth rates for βq-induced moisture–vortex instability is similar to the wavenumber–βT distribution seen for u¯T<0 (βT > 0) in Fig. 6a, hinting that the growth rates are the result of similar instabilities. Growth rates when βq < 0, in contrast, are weaker and are largest near zonal wavenumber 10. No damping is seen at the largest scales.

Fig. 7.
Fig. 7.

(a) Growth rate (shading) and phase speed (contours) obtained from Eq. (31a) as a function of varying βq and assuming u¯T=0. (b) As in (a), but assuming β = 0. For (a) and (b) τc = 6 h and m = 0.2. (c) As in (a), but showing the approximate solution obtained from Eq. (41a) and τc = 6 min. Units on the ordinate are multiplied by 1011. Note that the scale in the color bar of (c) is reduced by a factor of 26 compared to (a) and (b), largely as a result of the smallness of τc in the approximate solution.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

One of the striking features of Fig. 7a is the asymmetry between negative and positive values of βq. This asymmetry is the result of the β effect. When β = 0 (Fig. 7b), no asymmetry is seen between positive and negative values of βq. From inspection of Eq. (44b), we can see that the contribution of β to the asymmetry in Fig. 7a comes from Mr.

b. u¯T-induced moisture–vortex instability

The growth rates shown in Fig. 6 reveal growth beyond the region where baroclinic instability is observed in Fig. 4. Equation (44a) indicates that growth in these regions may be due to moisture–vortex instability induced by a temperature gradient (βT). This hypothesis is based on the fact that MVI is the only growth term in Eq. (44a) that does not require baroclinic instability (positive δi).

The red line in Fig. 8 shows the contribution of Eq. (44b) to the growth of wave when βq = 0 and u¯T=5m s1 (βT = 0.3 × 10−11 m−1 s−1). This value of u¯T is chosen since it can be seen from comparing Figs. 4 and 6 that baroclinic instability does not occur in this region (δi = 0). Comparison with βq-induced moisture vortex reveals many similarities. Both cases exhibit damping at the largest scales and growth that is largest at the synoptic scale. However, some differences are also observed. The growth rate associated with u¯T peaks at larger amplitudes than those associated with βq alone. Additionally, the largest growth rate for the u¯T-only instability is largest at zonal wavenumber 12, whereas it is a maximum at zonal wavenumber 20 for the βq-only moisture vortex instability. Nonetheless, the broad similarities between the two growth rates and the fact that both arise from the MVI term leads us to conclude that the u¯T-based instability examined in Fig. 8 is a type of moisture–vortex instability.

Fig. 8.
Fig. 8.

Growth rate as obtained from Eq. (44a) for u¯T-induced moisture–vortex instability (red) and for βq-induced moisture–vortex instability (blue). For the red line uT = −5 m s−1 (βT = 0.3 × 10−11 m−1 s−1) and βq = 0. For the blue line uT = 0 and βq = 0.3 × 10−11 m−1 s−1. For both lines τc = 6 min and m = 0.2.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

In spite of the similar growth rates, the vertical structure of the waves that grow due to these two types of moisture–vortex instabilities are different. Figure 9b shows that the βq-based moisture–vortex instability has no signature in the midtroposphere, i.e., ψB=0. In comparison, the u¯T-only instability exhibits a structure that is comprised of nearly equal parts barotropic and baroclinic structures (Fig. 9a). The baroclinic and barotropic structures are nearly out of phase, so that the resulting wave structure is bottom-heavy (αψ ≃ 0). As a result, u¯T-only moisture–vortex instability exhibits a wave structure that is distinct from baroclinic instability, in which the wave structure tilts with height against the shear (Fig. 5).

Fig. 9.
Fig. 9.

As in Fig. 5, but showing a zonal wavenumber-15 wave that grows from (a) u¯T-based moisture–vortex instability, where u¯T=7m s1, βq = 0, and (b) βq-based moisture–vortex instability, where u¯T=0 and βq = 0.4 m−1 s−1. In both panels m = 0.2 and τc = 6 h.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

c. Gross PV perspective on moisture–vortex instability

We can further understand the moisture–vortex instability mechanism by rearranging the terms in Eq. (28) in such a way that it becomes an equation that describes the evolution of ψT. The rearrangement yields the following equation:
D¯hDt[mh2ψT+(12m)kd2ψT]=Pr+Gr,
where
Pr=(1m)[υTβqυBβm]m[υTβ+υBβTu¯TζBx],
Gr=(tu¯Tx)f0τcM¯qP,
are the terms that describe the propagation and growth of ψT, respectively. The propagation term Pr contains the same terms that we saw in the case of baroclinic instability. However, in baroclinic instability these terms could also cause instability and growth. In moisture–vortex instability, growth depends on terms that have τc, i.e., water vapor (recall that q3=τcP). The negative sign has to do with the definition of ψT, which is positive for upper-tropospheric cyclones and hence is negative for lower-tropospheric cyclones. Thus, instability occurs when a positive moisture tendency is spatially collocated with a cyclonic vorticity tendency. This growth can be enhanced or damped by the baroclinic component of the lower- tropospheric flow, which has the opposite polarity of u¯T (recall that u¯3=u¯Bu¯T).

Figure 10 shows how u¯T can either enhance or suppress moisture–vortex instability when u¯B=0, i.e., u¯3=u¯T. For a westward-propagating disturbance, westerly shear (u¯T>0) advects the lower-tropospheric moisture away from the vortex. This advection increases αP and thus weakening or even preventing moisture–vortex instability. In the case of easterly shear (u¯T<0), the lower-tropospheric westerlies advect the anomalous rainfall toward the center of low pressure, reducing αP. To first order, Gr is represented by the denominator terms in αP. Thus, moisture–vortex instability occurs as long as αP < 90°.

Fig. 10.
Fig. 10.

Schematic describing how the mean thermal wind u¯T (red arrows) affects the phasing between anomalous lower-tropospheric circulation and anomalous rainfall. A westerly (u¯T>0) thermal wind corresponds to mean lower-tropospheric easterlies when u¯B=0. The lower-tropospheric moisture q3 is shown as a blue–green line, and the vertical velocity associated with convection (ωQ) is shown as the teal vertical arrow.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

In observations it is likely that u¯B is nonzero and so u¯3 may not correspond to u¯T. However, the relationship in Fig. 10 is still applicable if the barotropic component of the zonal wind is removed since it does not play a role in determining αP (not shown).

6. Does baroclinic instability weaken when τc > 0?

So far, we have shown that when τc > 0 moisture–vortex instability arises as a mechanism for wave growth. This instability is summarized in the MVI term in Eq. (44a). However, we have not discussed how baroclinic instability is modified by prognostic moisture, and how moisture–vortex and baroclinic instabilities could interact with one another.

Figure 11 shows the contribution of MVI, BCI and BMI to the growth of a wave. BMI and BCI are added since both terms require δi to be nonzero. It is clear that baroclinic instability (δi) and moisture–vortex instability MVI occur separately and barely overlap when considering the approximate relation in Eq. (44a).

Fig. 11.
Fig. 11.

Growth rate contribution from (a),(b) MVI and (c),(d) the modulation of baroclinic instability when τc > 0, which is written as BCI + BMI − δi for the growing modes obtained from Eq. (44a). (a),(b) The section in Fig. 6a where u¯T>0. (c),(d) The section in Fig. 6a where u¯T<0. In all panels the contribution of baroclinic instability δi to growth is shown as contours. The contour interval is 0.2 day−1.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

A comparison of Figs. 11a and 11b reveals that moisture–vortex instability is stronger when u¯T<0 than when u¯T>0. This difference in amplitude of MVI is related to β, which impacts the propagation of the moist wave in an analogous way to what was shown when considering βq-only instability. When β = 0, the MVI term is symmetric with respect to the sign of u¯T (see supplementary material).

When considering the modulation of baroclinic instability when τc > 0 (Figs. 11c,d), we see that baroclinic instability is weakened when u¯T<0. In contrast, there is little modulation of baroclinic instability when u¯T>0. This difference between u¯T<0 and u¯T>0 is due to the BMI term. Because cp is westward at the wavenumbers that correspond to baroclinic instability (see Fig. 4), it follows that BMI is larger in magnitude for u¯T<0 because the two contributions add up. For u¯T>0, cp and u¯T cancel one another and BMI is small.

It is worth noting that the MVI term [Eq. (44b)] and the terms that contribute to the weakening of baroclinic instability [Eqs. (44c) and (44d)] scale with τc. Thus, moisture–vortex instability strengthens and baroclinic instability weakens as τc increases. We can see this change even when considering the full solutions from Eq. (31a) for different values of τc (Fig. 12). As τc increases, the growth rate in the regions that are outside the region of baroclinic instability become larger in amplitude. In turn, growth rates in the region of baroclinic instability weaken. This change is seen for both easterly and westerly u¯T, but it is stronger for easterly u¯T.

Fig. 12.
Fig. 12.

Growth rates for the unstable mode obtained from Eq. (31a) for βq = 0, m = 0, and (top) u¯T>0 and (bottom) u¯T<0 for τc values of (a) 0.1, (b) 0.5, (c) 1, (d) 2, and (e) 4 h. Growth from baroclinic instability δi is shown as contours for each panel. The panels are arranged as in Fig. 11. The zeroth contour is shown as a dot–dashed line.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

The mechanism in which moisture–vortex instability amplifies at the expense of baroclinic instability can be understood by examining how αψ and αP change with increasing τc (Fig. 13). When τc is very small, αψ is near zero everywhere except for the region where baroclinic instability is seen, where the angle quickly increases to ~90°. The angle between P′ and ψT is close to ~90° indicating that precipitation is in quadrature with ψT for nearly all zonal wavenumbers and values of u¯T. As τc increases, αP decreases, as expected from Eq. (37). It decreases the fastest in regions of u¯T<0 where baroclinic instability is observed. The phase angle αψ also decreases rapidly in this region. In contrast, both αP and αψ decrease more slowly in the westerly shear regions where baroclinic instability is observed.

Fig. 13.
Fig. 13.

As in Fig. 12, but showing αψ as shading and αP as contours defined as in Fig. 3. The angle αψ is defined as a tilt against the shear, with increasing αψ meaning increased tilt against the shear. The angle αP is defined as a westward shift of P′ with respect to ψT.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

We can use Figs. 10 and 13 to explain why baroclinic instability weakens more rapidly when u¯T<0. The rapidly decreasing αP favors moisture–vortex instability, since more vortex stretching from convection occurs near the center of a vortex. Furthermore, the reduction of αψ indicates weaker tilt with height in the moist wave, which is less favorable for the generation of available potential energy associated with baroclinic instability (Eady 1949). The reduction in both αP and αψ indicate that the structure of the moist wave shifts toward a structure that is more consistent with moisture vortex instability, i.e., precipitation shifted toward the surface low and small αψ, which favors a vertically stacked, bottom-heavy vertical structure.

We can confirm the results from Fig. 13 by examining the horizontal structure of the growing waves under westerly and easterly u¯T (Fig. 14). When τc = 6 h, the growing wave under westerly u¯T still exhibits a structure that is reminiscent to baroclinic instability. We observe the westward tilt in height of ψ′, and the large-scale adiabatic lifting (ωa) is approximately in phase with the convectively driven ascent (ωQ). In contrast, the structure of the growing wave under easterly u¯T exhibits a structure more reminiscent to moisture–vortex instability, and consistent with the structure of observed and simulated MLPSs (Chen et al. 2005; Adames and Ming 2018b; Clark et al. 2020; Murthy and Boos 2020). Precipitation is shifted toward the vortex and very little wave signature is seen in the upper troposphere. Additionally, ωa is shifted to the west of ωQ. Perhaps the only signature of baroclinic instability seen in this case is the weak eastward tilt with height seen in ψ′. This bottom-heavy structure is maintained for growing modes even when βq is nonzero (see supplementary material).

Fig. 14.
Fig. 14.

(top three rows) As in Fig. 5, but showing the growing zonal wavenumber-8 solutions of Eq. (31a) for (a) u¯T=20m s1 and (b) u¯T=20m s1. For both solutions τc = 6 h, m = 0.2, and βq = 0. (bottom) Adiabatic ωa (pink), diabatic (teal), and total (purple dashed) vertical velocity anomaly averaged over 10°–25°N.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0205.1

7. Synthesis and discussion

In this study, we analyze a linear two-layer QG model with prognostic moisture. The moisture is coupled to precipitation through a simplified Betts–Miller-like scheme. We investigate how moisture impacts the instability and scale selection of waves that propagate in the presence of a thermal wind u¯T. The wave solutions obtained in this study can be described as moist Rossby waves that can grow from two instabilities, whose characteristics are summarized in Table 3:

  • Baroclinic instability: Two counterpropagating waves in the presence of u¯T become phase locked. The waves amplify one another via generation of available potential energy that occurs due to upward motion induced by temperature and vorticity advection, or through vortex stretching from convection induced by the aforementioned processes. The salient feature of this instability is the tilt with height of PV and associated wind fields (Fig. 1a). Large-scale adiabatic lifting (ωa) and convectively driven (ωQ) ascent are in phase.
  • Moisture–vortex instability: Moistening of the lower troposphere by horizontal moisture and temperature advection induce a buildup of moisture, which enhances precipitation. The moisture anomalies exhibit an in-phase component with the vortex, which grows through vortex stretching (Fig. 1b). Unlike baroclinic instability, waves that grow through moisture–vortex instability do not exhibit significant tilts with height. Unlike baroclinic instability, ωa is shifted to the west of ωQ.
Table 3.

Comparison of the two main instabilities discussed in this study.

Table 3.

Both baroclinic and moisture–vortex instability share some similarities. Both instabilities are enhanced by a lower NGMS (m) and when the temperature and moisture gradients (βT and βq) are in the same direction (see supplementary material). However, both instabilities also exhibit distinct differences. Baroclinic instability is strongest when precipitation responds instantaneously to convection (τc = 0) while moisture–vortex instability becomes stronger with increasing τc.3 Furthermore, when τc > 0 the β effect causes moisture–vortex instability to be stronger over easterly shear than over westerly shear. The opposite is true for baroclinic instability. Our study indicates that, in regions of high humidity, moisture–vortex instability may be the preferred instability in regions of easterly shear while baroclinic instability is dominant in regions of westerly shear.

Our results may shed some insights onto the mechanism of growth of several tropical motion systems.

a. Monsoon low pressure systems

The results of this study may be most applicable to MLPSs. As discussed by Cohen and Boos (2016), the horizontal structure of these systems does not exhibit the vertical tilts characteristic of baroclinic instability. Studies by Chen et al. (2005), Adames and Ming (2018b) and Clark et al. (2020) show that these systems exhibit a bottom-heavy, upright structures, consistent with Fig. 9a. Furthermore, large-scale adiabatic lifting in these systems is shifted to the west of the convectively driven ascent (Adames and Ming 2018b), consistent with the shifts shown in Fig. 14b. Thus, our results combined with observed structure of MLPSs suggest that these systems may grow from moisture–vortex instability.

Barotropic instability is also thought to play a role in the growth of MLPSs (Krishnamurti et al. 1976; Diaz and Boos 2019a,b). This study did not examine how moisture–vortex instability is affected in an environment with horizontal barotropic shear. However, we hypothesize that the moist barotropic instability mechanism proposed by Diaz and Boos (2019b) may be a combination of barotropic and moisture–vortex instabilities. Future work should examine possible interactions between these two instabilities.

It is worth pointing out that the phase speed in systems that grow from moisture–vortex instability under easterly shear exhibit weak propagation when the thermal wind is strong (Fig. 6). This propagation is weak even when considering βq. However, both simulated and observed MLPSs exhibit some amount of west or northwest propagation (Boos et al. 2015; Clark et al. 2020). It is possible that MLPSs occur in weaker shear, and can thus propagate against the mean westerlies. Alternatively, a nonlinear mechanism such as beta drift may account for nearly all of the propagation of MLPSs, as posited by Boos et al. (2015).

b. AEWs

At first glance, our results may not seem directly applicable to AEWs. The African easterly jet exhibits a maximum near 600 hPa (Burpee 1972; Cook 1999). Such a jet cannot be described well with the two-layer model described here. However, it is possible that the preference of easterly shear to favor moisture–vortex instability may extend to shallower jets such as the African easterly jet. This hypothesis is supported by recent studies that analyzed the growth of AEWs. For example, Russell et al. (2020) argue that rotational stratiform instability may be the main mechanism that leads to growth in AEWs, playing a larger role than baroclinic and barotropic instability. Their proposed instability and the vertical structure of PV discussed by Russell and Aiyyer (2020) are reminiscent of moisture–vortex instability.

Another study by Núñez Ocasio et al. (2020) found that developing AEWs contain embedded mesoscale convective systems (MCSs) that propagate at the same speed as the wave. Nondeveloping AEWs, on the other hand, do not exhibit such phase locking. They posit that the phase locking between the developing AEWs and the MCSs could be regarded as observational evidence of moisture–vortex instability. The studies by Russell et al. (2020) and Núñez Ocasio et al. (2020) indicate that interactions between convection and PV in AEWs are essential to the growth of AEWs. However, more research is needed to understand the role of moisture–vortex instability in AEWs, and to determine if moisture–vortex instability is related to the rotational stratiform instability described in Russell et al. (2020).

c. Tropical cyclogenesis

The vertically stacked structure of the waves that grow due to moisture–vortex instability in the presence of u¯T is reminiscent of the vertical structure of a tropical cyclone (Marks and Houze 1987; Moon et al. 2020). While the model described in this study may not necessarily apply to tropical cyclogenesis because of the underlying assumptions and linearity, it may nonetheless provide some indicators of the mechanisms in which cyclogenesis can occur under vertical wind shear. For example, it may be possible that cyclogenesis is enhanced by moisture–vortex instability, in which case it may be possible for tropical cyclones to preferentially occur in conditions of easterly shear.

d. A baroclinic and moisture–vortex instability spectrum?

So far, we have discussed baroclinic and moisture–vortex instability as nearly mutually exclusive phenomena. However, our results indicate that a spectrum between baroclinic and moisture–vortex instability may exist (see Fig. 14). Baroclinic and moisture–vortex instability comprise the ends of the spectrum. For the former both |αψ| and αP are approximately 90°. For the latter αψ ≃ 0 and αP < 90° as indicated by Table 3. In reality, atmospheric phenomena could grow from a combination of both, as indicated by the spectrum values of αψ and αP shown in Fig. 13. This spectrum would be analogous to the spectrum of barotropic and baroclinic instability (Krishnamurti et al. 1976; Thorncroft and Hoskins 1994), where the former occurs over strong horizontal shear and the latter over strong vertical shear. The spectrum for baroclinic and moisture–vortex instability would be related to how the thermal wind and the moist enthalpy gradient change the vertical tilt in a disturbance and its phase relationship with the lower-tropospheric moisture. More work is needed to show if this spectrum exists or not.

e. Similarity to the “barotropic governor” effect

The results of this study also show that moisture–vortex instability is enhanced at the expense of baroclinic instability, reminiscent to the so-called “barotropic governor” effect (James 1987). In this effect, the meridional shear in the zonal wind reduces the meridional scale of the most unstable modes, reducing the growth rate from baroclinic instability. Thus, in baroclinic regions, barotropic instability is enhanced at the expense of baroclinic instability. Similarly, the enhanced role of water vapor as τc increases causes the most unstable modes to shift away from the vertically tilted structure that favors baroclinic instability. This shift suggests the possible existence of a “moist governor” effect analogous to the barotropic governor, but acting on the vertical structure of the most unstable modes rather than the horizontal structure.

8. Concluding remarks

In the tropics, baroclinic instability was thought to be important over the South Asian and North African monsoons, both regions that are characterized by easterly vertical wind shear. Recent work has questioned the role of baroclinic instability in disturbances that grow in this region (Cohen and Boos 2016; Russell et al. 2020; Russell and Aiyyer 2020). Results from the simple model analyzed here supports these studies and suggests that moisture–vortex instability may be preferred in these regions instead.

It is important to recognize that the model analyzed here is highly idealized and simplified. The QG approximation may not always be applicable in AEWs and MLPSs (Boos et al. 2015). Future studies may examine idealized models that do not apply this approximation, and perhaps are more tailored to the mean state in which these systems occur. A more refined model can perhaps further elucidate what mechanism leads to the growth of MLPSs and AEWs. These refined models can also elucidate how these systems will respond to climate change (Dong et al. 2020).

Last, the gross PV equation is shown to be a useful quantity in determining the relative role of dry PV and moist enthalpy in a moist, balanced wave. Gross PV is nearly identical to the “moist” PV discussed by Lapeyre and Held (2004). Additionally, as noted by AM18, the gross PV equation bears resemblance to the precipitating QG equations discussed by Smith and Stechmann (2017), and some of the results discussed here resemble studies that use these equations (Wetzel et al. 2017, 2020). The gross PV equation also bears resemblance to the “equivalent” Ertel PV employed by some studies (Rotunno and Klemp 1985; Martin et al. 1992; Cao and Cho 1995; Marquet 2014). It is likely that gross PV is related to these other PV quantities, but a rigorous derivation showing this has not been performed. Future work will seek to elucidate the relationship between these quantities. If they are indeed related, it may be possible to develop diagnostic tools that help us improve the simulation and forecasting of systems such as MLPSs and AEWs.

Acknowledgments

ÁFA was supported by the National Science Foundation’s Grant AGS-1841559. ÁFA would like to thank Yi Ming and Spencer Clark for conversations that motivated the development of the two-layer model presented here. Conversations with Anantha Aiyyer also motivated this study and prompted the discussion on the barotropic governor effect. The author also thanks three anonymous reviewers for constructive comments that improved the contents of the manuscript.

APPENDIX

Applicability and Approximate Solutions to the Linear Moist QG Model

a. Applicability of the QG approximation to MLPSs and AEWs

The applicability of the QG approximation is contingent on the Rossby number (Ro) being much smaller than unity (Ro ≪ 1). The Rossby number is typically written as
Ro=u¯f0L.
In both the South Asian and African monsoons, u¯~10m s1. The zonal scale of monsoon low pressure systems has been estimated to be 2000 km (Godbole 1977; Sikka 1977; Lau and Lau 1990). Combining results yields Ro = 0.12 suggesting that the QG approximation may be applicable in this case. It is worth noting that for stronger monsoon depressions, which exhibit stronger winds and a smaller horizontal scale, the QG approximation is not necessarily applicable, as discussed by Boos et al. (2015).

Previous research has estimated the Rossby number for African easterly waves to be near 0.25 (Grist et al. 2002). African easterly waves exhibit wavelengths that range from 2000 to 5000 km and can span a wide range of latitudes, from 5° to 20°N, with centers of action both north and south of the easterly jet (Burpee 1972; Reed et al. 1977; Diedhiou et al. 1999). If we assume the center of these waves lies near 10°N, we find that the Rossby number for individual vortices ranges from 0.12 to 39. While the low end of this range would validate the use of the QG approximation, the higher range does not. While the results of this study may provide some qualitative insights about the growth of AEWs, the results may not always be valid for these waves.

b. Limiting cases of baroclinic instability

There are two limiting cases to Eq. (38) that are worth discussing.

1) Dry baroclinic instability
In a dry atmosphere, m = 1, which causes Eq. (38) to reduce to
Bd=0,
which yields the dispersion
ω0d=β(2k2+kd2)2k(k2+kd2)±δd,
where
δd=[β2kd44k2(k2+kd2)2u¯T2k2(kd2k2)k2+kd2]1/2.
Equation (A3) describes the propagation of waves that may grow through dry baroclinic instability if δd is imaginary. It is well documented that when u¯T=0 Eq. (A3) yields
ω0d(1)=βk,ω0d(2)=βkk2+kd2,
which describe a barotropic Rossby wave and a baroclinic Rossby wave, respectively.
2) Moist baroclinic instability
Another limiting case worth examining is when the troposphere is saturated (m = 0), which yields the following dispersion relation:
Bm=0.
In such a case the wave frequency is equal to
ω0m=(β2k+βqk2kd2)±δm,
where
δm=[(βkd2βqk2)24kd4k2+u¯Tk2(βT+βq)kd2]1/2.
We can consider two cases that elucidate the limit m → 0. In the absence of a thermal wind, Eq. (A7) reduces to
ω0m(1)=βk,ω0m(2)=βqkkd2,
which describe a barotropic Rossby wave and a baroclinic moisture wave. The moisture wave solution is that described by Sobel et al. (2001) in the limit of τc = 0 and m = 0. It propagates due to the vortex stretching that occurs from convection that results from anomalous meridional moisture advection (Pυ3yq¯3).

As in the dry baroclinic instability case, instability is possible only in the presence of a thermal wind (u¯T0). Recall that, even though the second rhs term in Eq. (A8) has a plus sign that βT=u¯Tkd2 and u¯T are of opposite signs. As a result, in the absence of βq, the second rhs term in Eq. (A8) is always negative. Additionally, β adds a low wavenumber limit to baroclinic instability since it acts to keep the square root positive.

The contribution of βq to instability is more complicated. The βq can enhance instability if it is of the same sign as βT, that is, if the temperature and moisture gradients are in the same direction. This is generally the case in the midlatitudes, where both βq and βT are negative, and in the South Asian summer monsoon, where they are both positive. However, if βq and βT are of opposite polarities then baroclinic instability is weakened. It is possible to completely eliminate baroclinic instability if βq is of the opposite polarity as βT and has a greater magnitude. In such a case βm and u¯T will be of the same sign and δm will be real for all zonal wavenumbers.

In addition to either enhancing or weakening the effectiveness of u¯T in inducing baroclinic instability, βq can also either enhance or weaken the stabilizing effect of β. If βq is positive, the first rhs term in Eq. (A8) becomes smaller. The converse is true for a negative βq.

3) General case with no thermal wind
In the case of no thermal wind, the solutions take the form of
ω0(1)=βk,ω0(2)=mβk+(1m)βqkmk2+kd2,
which is the barotropic Rossby wave solution and the moist Rossby wave solution described in AM18.

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1

The planetary vorticity in the northern outer tropics (10°–30°N) ranges from 2.5 to 7.2 × 10−5 s−1. While these values smaller than those of the midlatitudes, they are not small enough to invalidate the QG approximation, though it is indeed less robust than in the midlatitudes.

2

The symmetry between easterly and westerly u¯T is specific to the two-layer model used in this study, in which the depths of the two layers are equal and u¯T does not vary in space and time. If the two layers have different pressure depths or if a horizontal gradient in u¯T is included, a symmetry between westerly and easterly u¯T will not be seen even when τc = 0.

3

AM18 showed that moisture–vortex instability is strongest when τc ≃ 6 h. While not shown in this study, we found that this result remains true for a two-layer QG model.

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