Interaction of the Convective Energy Cycle and Large-Scale Dynamics

Jun-Ichi Yano; Robert S. Plant

doi:10.1175/JAS-D-23-0066.1

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Fig. 1.

Plots of the idealized mean circulation: (a) the zonal wind u; (b) the vertical velocities $\bar{w}$ (solid), w_c (long dashed) and −w_R (short dashed); (c) the barotropic height field h_b.

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Fig. 2.

Examples of the solitary wave solutions (7.8a)–(7.8c) with $\hat{α} = 1$ : (a) eastward propagating with ω = 1 and (b) westward propagating with ω = −1: the horizontal coordinate is $\hat{α} ξ = \hat{α} {\hat{ϵ}}^{- (1 / 2)} x$ with the unit scale of about 50 km. Plotted are the zonal wind u′ (solid), convection anomaly $w_{c}^{'}$ (long dashed), and the height h′ (short dashed).

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Editorial Type:: Article

Article Type:: Research Article

Interaction of the Convective Energy Cycle and Large-Scale Dynamics

Jun-Ichi Yano

Jun-Ichi Yano ^aCNRM, UMR 3589 (CNRS), Météo-France, Toulouse, France

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and

Robert S. Plant

Robert S. Plant ^bDepartment of Meteorology, University of Reading, Reading, United Kingdom

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Online Publication:: 17 Nov 2023

Print Publication:: 01 Nov 2023

DOI:: https://doi.org/10.1175/JAS-D-23-0066.1

Page(s):: 2685–2699

Received:: 07 Apr 2023

Final Form:: 25 Jul 2023

Accepted:: 05 Sep 2023

Published Online:: 17 Nov 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The importance of the convective life cycle in tropical large-scale dynamics has long been emphasized, but without explicit analysis. The present work provides it by coupling the convective energy cycle under the framework of Arakawa and Schubert’s convection parameterization with a shallow-water analog atmosphere. A careful derivation of the system is first presented, because it is rather missing in the literature. The squared frequency of linear convectively coupled waves is given by a squared sum of the dry gravity wave and the convective energy cycle frequencies, shortening the period of the convective cycle through the large-scale coupling. In a weakly nonlinear regime, the system follows an equation analogous to the Korteweg–de Vries equation, which exhibits a solitary wave solution, with behavior reminiscent of observed tropical westerly wind bursts.

Significance Statement

The present work suggests that a nonlinear description of a large-scale tropical system with an explicit convective life cycle may provide a simple model of tropical westerly wind bursts. At the same time, an important lesson to learn is that, if the focus of a study is on the global scale of the atmosphere, it is wise not to try to include a convective life cycle explicitly into the model. Such a configuration will simply be dominated by the short convective-scale variabilities, which one would wish to filter out.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jun-Ichi Yano, jun-ichi.yano@cnrs.fr

Abstract

Significance Statement

Corresponding author: Jun-Ichi Yano, jun-ichi.yano@cnrs.fr

Keywords: Tropics; Convection; Dynamics

1. Introduction

It is commonly accepted that tropical atmospheric dynamics is essentially described by the interactions between large-scale equatorial waves and small-scale convection: cf. critical reviews in introductions of Yano and Tribbia (2017), Yano and Wedi (2021), and further references therein. A standard approach has been to introduce parameterized convection to the large-scale dynamics under a general framework of convective quasi equilibrium (cf. Yano and Plant 2012a), which assumes that small-scale convection is in equilibrium with the large-scale dynamics in a certain manner. This general conceptual framework can cover a wide range of formulations, including the original one by Arakawa and Schubert (1974), but also a more straightforward assumption of convective neutrality of the large scale, originally suggested by Betts (1986), observationally supported by Xu and Emanuel (1989), and applied to theoretical studies by Emanuel (1987) and Neelin et al. (1987). More classical approaches of wave CISK (Hayashi 1970; Lindzen 1974) as well as moisture-based closures (e.g., Kuo 1974; Tiedtke 1989) can also be included in this category in the present context.

All of these approaches have in common that they do not introduce an explicit process characterized by a convective time scale. Here, we interpret the convective quasi equilibrium (cf. Yano and Plant 2012a) in a slightly more general manner, also encompassing the moisture- and convergence-controlled perspectives (e.g., Kuo 1974; Lindzen 1974), sitting back from the existing conceptual controversies on tropical moist large-scale dynamics (cf. Emanuel et al. 1994; Tomassini 2020). We also adopt the view that the moisture-based description falls into a general category of quasi-equilibrium closures, as pointed out by Yano and Plant (2016).

At the same time, there has been a persistent feeling in the tropical community that a finite time scale for the life cycle of small-scale convection plays a critical role in the tropical large-scale dynamics. This feeling may be, for example, reflected upon through brief, albeit rather obscure discussions on the convective life cycle leading to his Eqs. (2.2) and (3.6) in Kuo (1974), the emphasis on mesoscale processes for convection parameterizations in the review by Houze and Betts (1981), and probably most succinctly summarized by an argument of activation control by Mapes (1997).

The most straightforward way to include a convective time scale within a parameterization is to introduce it as a finite-time adjustment process toward an equilibrium. A parameterization by Betts (1986) follows this approach, although his main focus in the formulation is in defining an equilibrium profile. Neelin and Yu (1994) and Yu and Neelin (1994) introduced this finite-time convective adjustment in the context of large-scale dynamic studies. Similar approaches are adopted by, e.g., Frierson et al. (2004), Stechmann and Majda (2006), Bouchut et al. (2009), and Lambaerts et al. (2011). However, these convective adjustment approaches are still short of introducing a life cycle of convection: adjustment only describes a monotonic approach toward an equilibrium, without going through anything like a cycle. A simple model for the convective life cycle was introduced by Yano and Plant (2012b).

Yano and Plant (2012b) showed that a basic behavior of atmospheric deep convection, especially its tendency for following a cycle of discharge and recharge (cf. Bladé and Hartmann 1993), can be described by an energy cycle, as originally introduced by Arakawa and Schubert (1974) as their Eqs. (132) and (140), but by adding simple closures to this system [cf. Eq. (2.5) below]. A key simplification in the formulation of Yano and Plant (2012b) is to consider only a single, deep convection mode so that the integral kernel, defined by Eqs. (B36) and (B37) in Arakawa and Schubert (1974), reduces to a single scalar parameter.

The purpose of the present study is to couple this convective energy cycle system with a simple large-scale dynamics described by a shallow-water analog, and to present its basic behavior. The most fascinating finding from this study is the existence of a solitary wave solution under weak nonlinearity, whose behavior is reminiscent of observed tropical westerly wind bursts (cf. Hartten 1996; Yano et al. 2004).

For this goal, the first half of the paper (sections 2–4) constitutes a careful derivation of the model formulation. In section 2 we emphasize that the representation of the convective life cycle introduced in the present study is built on a solid basis of the mass-flux convection parameterization formulation (cf. Yano 2014; Plant and Yano 2015): the section introduces this formulation and defines it in full detail. Here, we also emphasize that we consider only a single deep convective mode in the present study, although a generalization is conceptually straightforward (cf. Yano and Plant 2012c). Section 3, in turn, discusses how convection and the large-scale dynamics interact thermodynamically. For a clear elucidation of the large-scale dynamics, we adopt a shallow-water analog model. Such a model is more than often introduced in heuristic manner without careful derivation. Here, we demonstrate explicitly how a shallow-water analog atmospheric model can be derived in a deductive manner from the primitive equation system in section 3.

A key ingredient for successfully reducing the large-scale moist dynamics into a shallow-water analog, without explicit moisture equation, is an assumption of the parcel–environment quasi equilibrium [Zhang 2002, 2003; Donner and Phillips 2003: see also section 7.4 of Yano (2023, manuscript submitted to J. Adv. Model. Earth Syst.) for further discussions], as going to be emphasized in section 2. A particularly important contribution of the present study is a consideration on the nonlinear advection terms in terms of their consistencies with the energy budget. Finally, a complete formulation of the system is presented at the end of section 4 in a nondimensional form. The derived system is analyzed over sections 5–7 in three steps: steady solutions (section 5), linear waves (section 6), and a weakly nonlinear analysis (section 7). The paper is concluded by section 9 after further discussion in section 8.

Due to a drastic reduction and idealization of the dynamics into a shallow-water analog, the present study stays away from full phenomenologies of the tropical atmosphere. For the same reason, various recent developments in tropical convection dynamics, reviewed by, e.g., de Rooy et al. (2013), Yano et al. (2014), and Tomassini (2020), are not taken into account either. Nevertheless, the present study suggests a potentially significant importance of the nonlinearities introduced for tropical convective wave dynamics. A more general purpose of the present work is to suggest diverse possibilities of implementing the same formulation into full atmospheric models, through presenting the formulation in a lucid manner in its simplest form. Moreover, the present idealized study with drastic simplifications already suggests what can be expected in full numerical model runs, with theoretical explanations, as summarized in the final section.

2. Convective energy cycle system

Following Yano and Plant (2012b), the convective energy cycle system is given by

\frac{d K}{d t} = A M_{B} - D,

(2.1a)

\frac{d A}{d t} = - γ M_{B} + F,

(2.1b)

with the convective kinetic energy K and the cloud work function A as prognostic variables. Full derivations of Eqs. (2.1a) and (2.1b), respectively, are found in Yano (2015a) and appendix B of Arakawa and Schubert (1974). See also Yano and Plant (2015) for a more careful derivation for the latter.

We define K and A in terms of vertical integrals by

K = \int_{z_{B}}^{z_{T}} σ \frac{ρ}{2} w_{c}^{2} d z,

(2.2a)

A = \int_{z_{B}}^{z_{T}} η b d z .

(2.2b)

Here, notably, σ is the fractional area occupied by convection, η is a normalized vertical profile of convective mass flux, M_B is an amplitude of the convective mass flux, b is the convective buoyancy, ρ is the air density, w_c is the convective vertical velocity, and z is the vertical coordinate.

Note that the first term on the right-hand side of Eq. (2.1a) constitutes a vertical integral of the buoyancy production of the kinetic energy in the terminology of boundary layer meteorology (cf. Sorbjan 1989): essentially it is a vertical integral of the vertical buoyancy flux w_cb. This term is expressed in terms of the two factors A and M_B by means of separating the total convective mass flux, M = ρσ_cw_c, into its amplitude M_B and a normalized vertical profile η:

M = η (z) M_{B} .

(2.2c)

Substitution of Eq. (2.2c) into the vertical integral of the buoyancy-production term leads to the final expression of the first term on the right-hand side of Eq. (2.1a). See Yano et al. (2005) for a more careful derivation.

Under the general definition of Eq. (2.2c), the vertical profile η of convection is not specified, and it can be defined in any manner as required: from a cloud-resolving model diagnosis (cf. Yano et al. 2005) or from any cloud model [e.g., under an entraining-plume hypothesis, as assumed in Arakawa and Schubert (1974)]. For the same reason, we do not require a precise definition of the amplitude M_B for the present purposes. Arakawa and Schubert (1974) define M_B as the convective mass flux at the convection base, whereas Yano et al. (2005) define it through a normalization condition on the profile η.

By factorizing out an intensity of convection M_B, it transpires that A measures the efficiency with which available potential energy is converted into kinetic energy through buoyancy production. For this reason, Yano et al. (2005) propose to call it the potential energy convertibility (PEC). It reduces to the cloud work function introduced by Arakawa and Schubert (1974) when an entraining plume profile is assumed for η. It further reduces to the convective available potential energy (CAPE), when we set η = 1. In the following, A is referred to as the cloud work function with a reminder of the more general term, PEC, in parenthesis. It is occasionally referred to as CAPE, whenever the context makes that necessary.

The buoyancy b is defined as that which actually drives convection, as explicitly simulated in cloud-resolving models. For this reason, Yano et al. (2005) diagnose b from cloud-resolving model outputs. Alternatively, in the context of convection parameterization, as in Arakawa and Schubert (1974), it is defined by the cloud model adopted (e.g., entraining plume). Finally, in defining CAPE, the buoyancy b is defined simply by parcel lifting.

We assume that the convective damping D is expressed by a Rayleigh damping:

D = \frac{K}{τ_{D}}

(2.2d)

with the damping time scale, τ_D ∼ 10³ s, as directed diagnosed from cloud-resolving model simulations by Yano (2015a); γ measures the efficiency with which convection consumes the cloud work function (PEC) A with time, corresponding to the kernel

K

introduced by Arakawa and Schubert (1974), but reducing it to a scalar by only considering a single deep convective mode here.

The large-scale forcing F was taken to be a prescribed constant in Yano and Plant (2012b) in order to consider the convection dynamics in a stand-alone manner. For the present purpose of considering a coupling of this energy cycle system with the large-scale dynamics, the large-scale forcing must evolve following the evolution of the large-scale state. Thus, we define it by

F ≃ \int_{z_{B}}^{z_{T}} \frac{g η}{\bar{T}} (w \frac{\partial \bar{θ}}{\partial z} - Q_{R}) d z,

(2.2e)

as an approximation of Eq. (B33) of Arakawa and Schubert (1974), where g is the acceleration due to gravity,

\bar{T}

the large-scale temperature, w the large-scale velocity,

\bar{θ}

the large-scale mean potential temperature, and Q_R the radiative heating rate. It is important to note that we neglect a contribution of boundary layer processes to the large-scale forcing in the definition (2.2e). This simplification is consistent with that which Arakawa and Schubert (1974) adopted in their quasi-equilibrium diagnosis, as more specifically termed parcel–environment quasi equilibrium by Zhang (2002, 2003) and Donner and Phillips (2003). The latter studies, moreover, show how well this equilibrium state is established observationally: see also section 7.4 of Yano (2023, manuscript submitted to J. Adv. Model. Earth Syst.) for further discussions.

Finally, the vertical integrals in Eqs. (2.2a), (2.2b), and (2.2e) are, in principle, performed from the convection base z_B to its top z_T. However, for the sake of simplifying the coupling with the large-scale dynamics, we reset them to be the surface, z_B = 0, and the top of the atmosphere z_T. By adopting an equivalent vertical coordinate in the large-scale dynamics (cf. section 3) z_T can easily be reinterpreted as the top of the troposphere. This approximation is justified by focusing on deep convection in the present study.

For achieving the simplest possible coupling, we still assume that the radiative heating rate Q_R is prescribed, but modify the first term in the definition (2.2e) above, by following the evolution of the large-scale vertical velocity w. Anticipating the shallow-water reduction of the large-scale dynamics in the next section we assume a normalized vertical profile of the vertical velocity w to be W so that

w = \tilde{w} (x, t) W (z) .

(2.3a)

Here,

\tilde{w} (x, t)

designates the horizontal dependence of the large-scale vertical velocity, and x is the only horizontal coordinate. Throughout the paper, vertical profiles are designated by uppercase letters, and keep in mind that all of the vertical profiles are defined to be nondimensional, and also normalized to O(1). Furthermore, the tilde sign is added to distinguish the horizontal components until the end of section 3.

As a result, the large-scale forcing may be rewritten as

F = μ \tilde{w} + F_{R},

(2.3b)

where

\begin{array}{l} μ = \int_{z_{B}}^{z_{T}} \frac{g η}{\bar{T}} W \frac{d \bar{θ}}{d z} d z \\ \sim \frac{g H}{T_{0}} \frac{d \bar{θ}}{d z} \sim \frac{10 m s^{- 2} \times 30 K}{300 K} \\ \sim 1 m s^{- 2} \end{array}

(2.4a)

measures the efficiency with which large-scale ascent generates the cloud work function (PEC) A. The second term in Eq. (2.3b),

F_{R} = - \int_{z_{B}}^{z_{T}} \frac{g η}{\bar{T}} Q_{R} d z,

(2.4b)

measures the rate at which the cloud work function (PEC) is generated by radiative cooling.

It may be worthwhile to recall that Arakawa and Schubert (1974) discuss the system (2.1a) and (2.1b) in introducing the quasi-equilibrium hypothesis for closing their mass-flux convection parameterization formulation. Here, their original system [their Eqs. (132) and (140)] is simplified by considering only a single deep convective mode, by following Yano and Plant (2012b). A system with two convective modes, deep and shallow, is considered by Yano and Plant (2012c).

Finally, for closing the system, as in Yano and Plant (2012b), we assume a relation

K = β M_{B},

(2.5)

where β is a constant estimated to be β ∼ 10⁴ m² s⁻¹ for deep convection, as assumed here. Yano and Plant (2012b) justify this assumption by quoting earlier cloud-resolving simulation analyses (Emanuel and Bister 1996; Shutts and Gray 1999; Parodi and Emanuel 2009). Keep in mind that the parameter β depends on the convection depth in general.

3. Large-scale system

As a first step in constructing a large-scale system to be coupled with the convective energy cycle system introduced in the last section, we consider the large-scale heat equation in section 3a, because it is the key equation to achieve a coupling of the two scales. The formulation is completed more formally by introducing the normal mode decomposition of the linear primitive equation system in section 3b. The presentation is rather backward, because the first subsection has to quote some of the results to be obtained in the following subsection. Nevertheless, we present in this order for the sake of making the physical motivations clear before presenting a more complete mathematical formulation. The system is assumed linear throughout this section. Nonlinear advection terms will be considered later in section 4e.

a. Large-scale heat equation

A major feedback of convection to the large-scale state is found in the heat equation, which may be written as

\frac{\partial θ}{\partial t} + w \frac{d \bar{θ}}{d z} = Q_{c} + Q_{R},

(3.1a)

where Q_c is the convective heating rate, approximately given by

Q_{c} = σ w_{c} \frac{d \bar{θ}}{d z} .

(3.1b)

Equations (3.1a) and (3.1b) are directly obtained from Eq. (41) of Arakawa and Schubert (1974) by neglecting the effect of detrainment for simplicity. Please also refer to Eq. (3.6a) of Yano (2015b) for a full derivation. Recall that Q_R is the radiative heating.

Because the convective dynamics is described in terms of a single vertical mode, it is appropriate to reduce the large-scale dynamics similarly. For this reason, we have already assumed only a single vertical mode for the large-scale dynamics by writing the vertical velocity in the form of Eq. (2.3a) in section 2, and equivalently, the potential temperature is represented by

θ = \tilde{θ} (x, t) Θ (z) .

(3.2)

Here, Θ is a nondimensional, normalized vertical profile and

\tilde{θ}

describes the horizontal dependence. We also set

σ w_{c} = \frac{η}{ρ_{0}} M_{B} = η {\tilde{w}}_{c},

where ρ₀ is the surface density.

As a standard procedure for projecting an equation onto a given vertical mode, we multiply Eq. (3.1a) by Θ, and integrate it vertically. As a result, we obtain

\frac{\partial \tilde{θ}}{\partial t} + \frac{θ^{*}}{z_{T}} \tilde{w} = \hat{η} {\tilde{w}}_{c} - {\hat{Q}}_{R}^{*} w_{R},

(3.3)

where

\frac{θ^{*}}{z_{T}} = ⟨ W Θ \frac{d \bar{θ}}{d z} ⟩ = \frac{θ_{0} h_{E}}{z_{T}^{2}},

(3.4a)

\hat{η} = ⟨ η Θ \frac{d \bar{θ}}{d z} ⟩,

(3.4b)

{\hat{Q}}_{R}^{*} = - ⟨ Θ Q_{R} ⟩ .

(3.4c)

Here, we define the angle brackets as an integral operator

⟨ * ⟩ = \frac{1}{z_{T}} \int_{0}^{z_{T}} * d z,

setting z_B = 0 in the vertical integrals, as already discussed in section 2. We have also assumed that Θ is normalized by

⟨ Θ^{2} ⟩ = 1.

We further introduce θ* as a characteristic scale for θ. An alternative representation is also given in Eq. (3.4a) in terms of a reference value of potential temperature θ₀ and an equivalent depth h_E: this form will prove convenient later.

It can be shown from Eq. (3.10b) to be derived in next subsection that the vertical-wind profile W is related to the potential-temperature profile Θ by

W = \frac{θ^{*}}{z_{T}} {(\frac{d \bar{θ}}{d z})}^{- 1} Θ .

(3.4d)

Additionally, the nondimensional radiative vertical velocity w_R has been introduced in Eq. (3.3), in order to represent a possible horizontal distribution of radiation. This study assumes the radiation to be horizontally homogeneous, and thus, we will simply set it to unity in the following, but explicitly reintroduce it whenever important to indicate the role of radiation in a given equation.

With the final goal of reducing the system to a shallow-water analog in mind, it is convenient to replace the potential temperature

\tilde{θ}

in the heat Eq. (3.3) by the height field

\tilde{h}

. These two variables are linked together through hydrostatic balance, as will be obtained in Eq. (3.11b) below:

\tilde{h} = - \frac{h_{E}}{θ^{*}} \tilde{θ} = - \frac{z_{T}}{θ_{0}} \tilde{θ} .

(3.5)

As a result, the heat equation reduces to

\frac{\partial \tilde{h}}{\partial t} - \hat{S} (\tilde{w} - α {\tilde{w}}_{c}) = {\hat{Q}}_{R} .

(3.6)

Here, the introduced nondimensional parameters are estimated as

\hat{S} = \frac{h_{E}}{z_{T}} \sim 10^{- 2},

(3.7a)

α = \frac{z_{T}}{θ^{*}} \hat{η} = \frac{z_{T}^{2}}{θ_{0} h_{E}} \hat{η} \sim 1,

(3.7b)

{\hat{Q}}_{R} = \frac{h_{E}}{θ^{*}} {\hat{Q}}_{R}^{*} = \frac{z_{T}}{θ_{0}} {\hat{Q}}_{R}^{*} .

(3.7c)

Recall that

\hat{η}

has already been defined by Eq. (3.4b). The orders of magnitude estimates in (3.7a) and (3.7b) are based on h_E ∼ 10² m, z_T ∼ 10 km, θ₀ ≃ 300 K, and θ* ∼ K.

b. Normal-mode decomposition of the linear primitive equation system

A thermodynamic formulation for a shallow-water analog atmosphere has been introduced in the last subsection, in which the large-scale heat equation reduces to a height equation for shallow water. To complete the construction of a shallow-water analog of the tropical atmosphere large-scale dynamics, we now consider a full, linear primitive equation system to see how the vertical profiles of the variables may be defined consistently. These profiles are usually called normal modes (cf. Kasahara and Puri 1981).

We consider a linear horizontally one-dimensional system with the Boussinesq approximation:

\frac{\partial u}{\partial t} = - \frac{\partial ϕ}{\partial x},

(3.8a)

\frac{\partial ϕ}{\partial z} = g \frac{θ}{θ_{0}},

(3.8b)

\frac{\partial θ}{\partial t} + w \frac{d \bar{θ}}{d z} = Q,

(3.8c)

\frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} = 0.

(3.8d)

Here, θ₀ is a constant reference potential temperature already introduced in Eqs. (3.4a) and (3.5), u is the horizontal velocity, and ϕ is the geopotential. The total diabatic heating has been set to Q = Q_c + Q_R by following the last subsection.

To apply the above system to a realistic atmosphere, the system is best reinterpreted as a consequence of transforming the pressure coordinate, p, into an equivalent geometrical coordinate z by the relation dp = −ρ₀gdz, with ρ₀ a reference density, but with a minor modification to the hydrostatic balance (3.8b) of multiplying by an additional factor $ρ_{0} θ_{0} / ρ \bar{θ}$ on the right-hand side. Keep in mind that all of the vertical integrals considered in the convective energy cycle formulation must also be reinterpreted accordingly.

We introduce a separation of variables by Eqs. (2.3a) and (3.2), as well as

u = Φ \tilde{u}, ϕ = Φ \tilde{ϕ}, Q = Θ \tilde{Q} .

(3.9a,b)

By substituting Eqs. (2.3a), (3.2), (3.9a), and (3.9b) into Eqs. (3.8a)–(3.8d), we find that the vertical profiles must mutually satisfy the relations

z_{T} \frac{d Φ}{d z} = - Θ,

(3.10a)

Θ = \frac{z_{T}}{θ^{*}} \frac{d \bar{θ}}{d z} W,

(3.10b)

Φ = z_{T} \frac{d W}{d z} .

(3.10c)

The two scales z_T and θ* have been introduced so that all the vertical profiles consistently remain nondimensional, and also of the order unity. Recall that θ* has already been introduced by Eqs. (3.4a) and (3.5).

By further substituting (3.10a) and (3.10c) into (3.10b), we find

[\frac{d^{2}}{d z^{2}} + \frac{1}{z_{T}} (\frac{1}{θ^{*}} \frac{d \bar{θ}}{d z})] W = 0.

Here, z_Tθ* constitutes an eigenvalue in this equation. A more commonly accepted form is obtained by rewriting the above to

[\frac{d^{2}}{d z^{2}} + \frac{1}{h_{E}} (\frac{1}{θ_{0}} \frac{d \bar{θ}}{d z})] W = 0

with the equivalent depth,

h_{E} = \frac{θ^{*}}{θ_{0}} z_{T},

constituting the standard eigenvalue of this problem [cf. Eq. (3.4a)]. It can be seen that the equivalent depth is the scaled-down version of the vertical scale by the relative fluctuation of the buoyancy with respect to the reference state.

Consequently, the equations for the horizontal components are given by

\frac{\partial \tilde{u}}{\partial t} = - \frac{\partial \tilde{ϕ}}{\partial x},

(3.11a)

\tilde{ϕ} = - \frac{g z_{T}}{θ_{0}} \tilde{θ} = - \frac{g h_{E}}{θ^{*}} \tilde{θ},

(3.11b)

\frac{\partial \tilde{θ}}{\partial t} + \frac{θ^{*}}{z_{T}} \tilde{w} = \tilde{Q},

(3.11c)

\frac{\partial \tilde{u}}{\partial x} + \frac{\tilde{w}}{z_{T}} = 0.

(3.11d)

By further setting,

\tilde{ϕ} = g \tilde{h}

, rewriting Eq. (3.11c) in terms of

\tilde{h}

, we recover Eq. (3.6) already introduced. By eliminating the vertical velocity with the help of the mass continuity (3.11d), we find that the governing equation set for the horizontal components constitutes an analog of the shallow-water system with the equivalent depth, h_E playing the role of the depth:

\begin{array}{l} \frac{\partial \tilde{u}}{\partial t} = - \frac{\partial \tilde{ϕ}}{\partial x}, \\ \frac{\partial \tilde{ϕ}}{\partial t} + g h_{E} \frac{\partial \tilde{u}}{\partial x} = - \frac{g z_{T}}{θ_{0}} \tilde{Q} . \end{array}

4. Nondimensionalization

For ease of further analyses, we now nondimensionalize the system derived over sections 2 and 3.

a. Convective energy cycle system

To nondimensionalize the convective energy cycle, we first note that the equilibrium state is given at the convective scale by

A = A_{0} \equiv {β / τ_{D} \sim 10 J kg}^{- 1},

(4.1a)

M_{B} = M_{0} \equiv F_{R} / γ \sim 10^{- 2} kg m^{- 2} s^{- 1},

(4.1b)

where F_R is the radiative contribution to convective forcing. Estimates are based on the values of β ∼ 10⁴ m² s⁻¹, τ_D ∼ 10³ s, F_R ∼ 10⁻² m⁻² s⁻³, and γ ∼ 1 m⁻⁴ s⁻² kg⁻¹ by following Yano and Plant (2012b). Setting, for now, the large-scale equilibrium to be simply quiescent,

\tilde{w} = \tilde{h} = 0

, we find that the convection-base mass flux is further constrained to satisfy

M_{B} = \frac{ρ_{0} {\hat{Q}}_{R}}{α \hat{S}}

(4.1c)

from Eq. (3.6). Recall that

{\hat{Q}}_{R}

is a measure of the radiative cooling rate, as defined by Eq. (3.7c). Obviously, this value must also agree with F_R/γ given by Eq. (4.1b).

We nondimensionalize the large-scale vertical velocity by

\tilde{w} = w_{0} {\tilde{w}}_{*},

where the subscript * suggests a nondimensionalized horizontal dependence, and w₀ is the scale of the vertical velocity. Keep in mind that the subscript * will be tentative, and it will be removed as soon as the nondimensionalization is accomplished.

The appropriate time scale τ_c and vertical-velocity scale w₀ for nondimensionalization are given by

τ_{c} = {(β / F_{R})}^{1 / 2} \sim 10^{3} s,

(4.2a)

w_{0} = F_{R} / μ \sim 10^{- 2} m s^{- 1} .

(4.2b)

The convective-scale variables are nondimensionalized into k_c and a by setting

M_{B} = M_{0} k_{c},

(4.3a)

A = \frac{τ_{D}}{τ_{c}} A_{0} a,

(4.3b)

such that the resulting nondimensionalized equations are

\frac{\partial k_{c}}{\partial t^{'}} = a k_{c} - \frac{k_{c}}{τ_{D}^{*}},

(4.4a)

\frac{\partial a}{\partial t^{'}} = - k_{c} + w + w_{R},

(4.4b)

where the dependent variables are defined by the following:

k_c = w_c: convective kinetic energy (or the convective mass flux). As required, we use these two notations in an interchangeable manner, i.e.,
$k_{c} = w_{c}$ (4.5)
depending on the context. Recall that this relation (4.5) directly follows from the original closure assumption (2.5).

a: the cloud work function (which may conceptually be interpreted as a convective potential energy);
$τ_{D}^{*} = τ_{D} / τ_{c}$ : a nondimensional damping time scale;
w_R (=1): a normalized radiative vertical velocity.

In Eqs. (4.4a) and (4.4b), the subscript * indicating nondimensional variables has already been removed. Note further that a prime sign is added to the nondimensional time t′ because a different nondimensionalization of time will be introduced for the large-scale dynamics in the next subsection.

b. Large-scale system

We nondimensionalize the large-scale system by introducing the scales u₀, h₀, τ_L, and L, marking the nondimensional variables with the subscript * for now; thus, e.g.,

\frac{\partial}{\partial x} = \frac{1}{L} \frac{\partial}{\partial x_{*}} .

By substituting into Eqs. (3.11a)–(3.11c), we find that convenient nondimensionalization scales are

h_{0} = h_{E}, u_{0} = c_{g}, τ_{L} = L / c_{g},

(4.6a,b,c)

where

c_{g} = {(g h_{E})}^{1 / 2}

is the gravity wave speed, and the characteristic horizontal scale L is left to be determined. We set L = 3 × 10³ km provisionally, for the purpose of some numerical estimates.

After removing the tilde signs, and removing the subscripts * from nondimensional variables, the resulting nondimensional set of equations are

\frac{\partial u}{\partial t} = - \frac{\partial h}{\partial x},

(4.7a)

\frac{\partial h}{\partial t} + \frac{\partial u}{\partial x} = - Q,

(4.7b)

w = - {\hat{r}}_{L} \frac{\partial u}{\partial x} .

(4.7c)

Here,

Q = \hat{α} (w_{c} - w_{R}) = \hat{α} w_{c} - {\hat{Q}}_{R},

(4.8a)

{\hat{r}}_{L} = \frac{c_{g} z_{T}}{w_{0} L} \sim 10,

(4.8b)

\hat{α} = α / {\hat{r}}_{L},

(4.8c)

and

{\hat{r}}_{L}

may be considered an effective aspect ratio of the system. Alternatively, it can be interpreted as a ratio of two characteristic horizontal scales:

{\hat{r}}_{L} = L_{D} / L,

where

L_{D} = \frac{c_{g} z_{T}}{w_{0}} \sim 3 \times 10^{4} km .

Also keep in mind that the total depth of the shallow water is h_T = 1 + h.

Recall from Eq. (3.6) that α, defined by Eq. (3.7b), controls the relative contributions of large-scale and convective-scale velocities to the stratification, also referring to Eq. (3.7a). The parameter $\hat{α}$ introduced by Eq. (4.8c) thus measures the efficiency of convection in modifying the stratification of the atmosphere, while 1 − α may be considered a nondimensional measure of the effective stratification (or gross moist stability; Neelin and Held 1987). In particular, when α = 1, the convective atmosphere is effectively neutrally stratified. Here, w₀ is a characteristic scale of the large-scale vertical velocity and, by nondimensionalization, the radiatively driven vertical velocity is w_R = 1.

c. Two time scales

To couple together the two systems for convection and the large scale, we need to take care of the two different time scales adopted for the systems in nondimensionalization, τ_c [Eq. (4.2a)] and τ_L = L/c_g [Eq. (4.6c)]. The ratio of the two is

{\hat{r}}_{c} = τ_{c} / τ_{L} \sim 10^{- 2} .

(4.9a)

For consistency in the following, we adopt τ_L as the common time scale for both the convective and large scales. As a result, the nondimensional equation set for the convective scale, which was originally nondimensionalized by τ_c, must be modified into

{\hat{r}}_{c} \frac{\partial k_{c}}{\partial t} = a k_{c} - \frac{k_{c}}{τ_{D}^{*}},

(4.9b)

{\hat{r}}_{c} \frac{\partial a}{\partial t} = - k_{c} + w + w_{R}

(4.9c)

by changing the nondimensionalization time scale from τ_c to τ_L. Note that for a large-scale horizontal scale of L ≃ 30 km,

{\hat{r}}_{c} ≃ 1

, and the two time scales match.

d. Coupling problem

Through the considerations over the last subsections, we have arrived at a complete nondimensional set of equations given by (4.7a)–(4.7c) and (4.9b), (4.9c). However, this equation set is probably better considered to be redundant, because the large-scale height h which is also related to the potential temperature θ by Eq. (3.5), is effectively equivalent to the convective-scale cloud work function (PEC), a. To appreciate this point, consider that a is defined by a buoyancy integral. By neglecting contributions from the boundary layer, and also neglecting the virtual effect, the buoyancy integral is determined exclusively by the environmental potential temperature. Thus, to a good approximation, a is nothing other than an alternative measure of the tropospheric potential temperature, which itself is a measure of the buoyancy in the present system. Clearly, we do not wish to carry two separate buoyancy measures h and a under a single-layer description. Thus, we need to remove this redundancy by establishing an equivalence between the two, such that they are governed by an identical equation.

This is accomplished in the following manner, by introducing two additional constraints. By comparing between the right-hand side of Eq. (4.9c) and the definition (4.8a), we find that

{\hat{r}}_{c} \hat{α} \frac{\partial a}{\partial t} - \hat{α} w = - Q,

(4.10a)

also recalling that k_c = w_c. For comparison, the height Eq. (4.7b) is rewritten with the help of Eq. (4.7c) as

\frac{\partial h}{\partial t} - \frac{w}{{\hat{r}}_{L}} = - Q .

(4.10b)

These two expressions suggest that the two variables become equivalent by setting

h = {\hat{r}}_{c} \hat{α} a .

(4.11a)

Furthermore, for consistency of the large-scale vertical advection term (second on the left-hand side) in both Eqs. (4.10a) and (4.10b), a further constraint is required to establish the equivalence

\hat{α} = 1 / {\hat{r}}_{L} .

(4.11b)

By further referring to the definition of

\hat{α}

in Eq. (4.8c), this condition simply reduces to

α = 1.

(4.11c)

Recall from section 4b that the parameter α measures the efficiency of convection in modifying the stratification of the atmosphere.

The equivalence between CAPE (PEC) and the height in the shallow-water analog atmosphere has been pointed out by Mapes (1998). We just establish this connection in a more formal manner. As a result, there is no longer a need to consider the time evolution of PEC, a, separately.

Consequently Eq. (4.9b) describes the convective-scale process, alongside the equation set (4.7a)–(4.7c) for the large scale. With the help of Eq. (4.11a), the PEC can be eliminated from Eq. (4.9b), which becomes

\hat{ϵ} \frac{\partial k_{c}}{\partial t} = \hat{α} h k_{c} - \frac{k_{c}}{{\tilde{τ}}_{D}},

(4.12a)

where

{\tilde{τ}}_{D} = τ_{D}^{*} / {\hat{r}}_{c} {\hat{α}}^{2} = τ_{D} / {\hat{r}}_{c} {\hat{α}}^{2} τ_{c} \sim 10^{4},

(4.12b)

\hat{ϵ} = {\hat{r}}_{c}^{2} {\hat{α}}^{2} \sim 10^{- 6} .

(4.12c)

Large and small values for these two parameters suggest shorter time scales involved with convection compared to those of the large scale.

e. Full system with nonlinearity

It remains to add nonlinearity to the linear version of the large-scale system derived so far, Eqs. (4.7a)–(4.7c). This final step turns out to be rather involved, and the details are presented in the appendix. Therein, we examine the physical consistency of the included nonlinear terms with the energy cycle of the system. Based on those examinations, we adopt the final large-scale equation set to be

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = - \frac{\partial h}{\partial x},

(4.13a)

\frac{\partial h}{\partial t} + \frac{\partial u}{\partial x} = - Q,

(4.13b)

w = - {\hat{r}}_{L} \frac{\partial u}{\partial x} .

(4.13c)

Thus, the nonlinear advection term has been added only to the momentum equation, Eq. (4.13a), but not to the continuity (heat) equation, Eq. (4.13b).

In summary, the full nonlinear system consists of Eqs. (4.13a)–(4.13c) and (4.12a). Recall that the heating term Q in Eq. (4.13b) is defined by Eq. (4.8a), and also recall the relation (4.5).

5. Steady solutions

We first examine the steady solutions. There are the two reasons to do so. First, this is an important prerequisite to perform the perturbation analyses (both linear and nonlinear) in subsequent sections. Moreover, steady solutions of the system are of interest in their own right. Especially, a solution with a steady circulation may provide an idealized analog of the Hadley–Walker circulation. However, only an outline of the problem is presented in this short subsection with the main goal of proceeding to the perturbation analyses next.

The steady heat budget of the system is obtained by substituting Eqs. (4.13c) and (4.8a) into Eq. (4.13b):

\bar{w} - α {\bar{w}}_{c} + {\hat{r}}_{L} {\hat{Q}}_{R} = 0

\bar{w} - α {\bar{k}}_{c} + α w_{R} = 0 .

(5.1)

Here, the overbars are added to denote a steady state. Also keep in mind that we retain two notations with k_c = w_c.

The equilibrium state of convection is obtained from (4.12a) as

{\bar{k}}_{c} = {\bar{w}}_{c} = 0

(5.2a)

\bar{h} = 1 / \hat{α} {\tilde{τ}}_{D} \sim 10^{- 3} .

(5.2b)

In the following, we take the second choice, (5.2b), which is only a matter of adding a constant height on perturbations. The first choice, (5.2a), is less interesting with no possibility of convection in the basic state.

From the heat balance (5.1), we see that

\bar{w}

and

{\bar{w}}_{c}

can be chosen freely so long they are consistent with the dynamics. To seek a more specific solution, we set

\bar{u} = {\bar{u}}_{0} \sin k x,

(5.3a)

with

{\bar{u}}_{0}

a constant (Fig. 1a). Its substitution into the continuity Eq. (4.13c) leads to

\bar{w} = - {\bar{w}}_{0} \cos k x,

(5.3b)

with

{\bar{w}}_{0} = {\hat{r}}_{L} k u_{0}

. Furthermore, from Eq. (5.1),

{\bar{w}}_{c} = w_{R} - \frac{{\bar{w}}_{0}}{α} \cos k x .

(5.3c)

To maintain the convective vertical velocity to be always positive definite, i.e.,

{\bar{w}}_{c} \geq 0

, we require

w_{R} \geq {\bar{w}}_{0} / α

. If we further assume the minimum convective velocity to be zero, we obtain

{\bar{w}}_{0} = α w_{R}

(Fig. 1b).

Fig. 1.

Plots of the idealized mean circulation: (a) the zonal wind u; (b) the vertical velocities $\bar{w}$ (solid), w_c (long dashed) and −w_R (short dashed); (c) the barotropic height field h_b.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0066.1

Finally, the steady nonlinear momentum equation,

\frac{\partial}{\partial x} \frac{{\bar{u}}^{2}}{2} = - \frac{\partial \bar{h}}{\partial x},

must be satisfied. However, here we face a problem: by the convective equilibrium condition, we have already set

\bar{h}

to be constant by Eq. (5.2b), and thus, the right-hand side vanishes from the above, and there is no term to balance with the nonlinear advection on the left-hand side. We circumvent this difficulty by noting that the nonlinear advection term arising from a baroclinic circulation, actually projects onto a barotropic mode, and thus, the height perturbation required to balance the right-hand side is also of a barotropic mode:

\frac{\partial}{\partial x} \frac{{\bar{u}}^{2}}{2} = - \frac{\partial {\bar{h}}_{b}}{\partial x},

with the subscript b standing for the barotropic mode, but also suggesting that this mode arises directly from the surface boundary effect, e.g., the SST distribution, partially reminiscent of the idea of Lindzen and Nigam (1987; see also section 4.4 of Yano 2023, manuscript submitted to J. Adv. Model. Earth Syst.). The barotropic height field which balances with the nonlinear term is given by (Fig. 1c)

h_{b} = \frac{u_{0}^{2}}{4} \cos 2 k x .

The short analysis of this section outlines very crudely how a consistent theory for steady tropical circulations can be developed in the context of a shallow-water analog formulations: for further details and discussions we refer to, e.g., Gill (1980), Lindzen and Nigam (1987), Neelin and Held (1987), and section 4 of Yano (2023, manuscript submitted to J. Adv. Model. Earth Syst.).

6. Linear analysis

For performing perturbation analyses in the following two sections, we assume a homogeneous basic state with no large-scale circulation, i.e., $\bar{u} = \bar{w} = 0$ . The basic-state height is defined by Eq. (5.2b), and from Eq. (5.1), ${\bar{w}}_{c} = w_{R} = 1$ , also recalling α = 1 [cf. Eq. (4.11c)].

The resulting set of linear perturbation equations is

\frac{\partial u^{'}}{\partial t} = - \frac{\partial h^{'}}{\partial x},

(6.1a)

\frac{\partial h^{'}}{\partial t} + \frac{\partial u^{'}}{\partial x} + \hat{α} w_{c}^{'} = 0,

(6.1b)

\hat{ϵ} \frac{\partial w_{c}^{'}}{\partial t} = \hat{α} h^{'},

(6.1c)

with the prime sign denoting perturbation variables and

w_{c}^{'} = k_{c}^{'}

We further assume a solution of the form,

\sim e^{i (k x + ω t)}

. Then, the linear frequency is given by

ω^{2} = k^{2} + {\hat{α}}^{2} / \hat{ϵ}

ω^{2} = k^{2} + \frac{1}{{\hat{r}}_{c}^{2}} .

(6.2)

Note that only a neutral wave solution is available, and the standard gravity wave solution is recovered by setting

{\hat{r}}_{c} \to \infty

. Since

{\hat{r}}_{c} = τ_{c} / τ_{L}

this limit corresponds to setting the convective time scale much longer than that of the large scale. Rather unintuitively, the presence of finite convective time scale (i.e., τ_c finite) increases the frequency of the mode to be larger than that of the dry gravity wave: by further decreasing τ_c, the waves propagate faster. Note that in absence of a large-scale circulation, the system reduces to a linear version of the convective discharge–recharge system (cf. Yano and Plant 2012b);

\begin{array}{l} \frac{\partial h^{'}}{\partial t} + \hat{α} w_{c}^{'} = 0, \\ \hat{ϵ} \frac{\partial w_{c}^{'}}{\partial t} = \hat{α} h^{'} . \end{array}

This leads to an oscillating solution with

ω = \hat{α} / {\hat{ϵ}}^{1 / 2} = 1 / {\hat{r}}_{c} = τ_{L} / τ_{c}

. Effectively, the dispersion (6.2) is composed of the squared sum of the dry and convective frequencies.

7. Weakly nonlinear analysis

As an extension to the analysis of the last section, we now take into account a weak nonlinearity. For the purpose of developing a weakly nonlinear formulation in a formal manner, we introduce an explicit perturbation parameter, which we choose to be

\hat{ϵ}

, bearing in mind the numerical estimate of (4.12c). We also focus on the situation in which the system satisfies the free-ride balance

\frac{\partial u^{'}}{\partial x} + \hat{α} w_{c}^{'} = 0,

(7.1)

i.e., a balance between the vertical heat advection (first term) and the diabatic heating (second term; cf. Fraedrich and McBride 1989) to the leading order of Eq. (4.13b), as observed in the tropical atmosphere (cf. Fig. 1 of Yano 2001). This state, alternatively called the weak temperature gradient approximation (Sobel et al. 2001), may also be considered to be a quasi-equilibrium closure under the given shallow-water formulation (cf. section 6.1 of Yano and Plant 2012a).

To obtain (7.1) to the leading order, the variables must be rescaled. It is found that appropriate rescalings are^¹

h = \bar{h} + {\hat{ϵ}}^{3} h^{'},

(7.2a)

w_{c} = {\bar{w}}_{c} + \hat{ϵ} w_{c}^{'},

(7.2b)

u = {\hat{ϵ}}^{3 / 2} u^{'},

(7.2c)

and

\partial / \partial t = \hat{ϵ} \partial / \partial τ,

(7.2d)

\partial / \partial x = {\hat{ϵ}}^{- (1 / 2)} \partial / \partial ξ .

(7.2e)

Thus, a longer time and shorter horizontal scales are introduced compared to the original nondimensionalization scales. Recall that

\bar{h}

is defined by Eq. (5.2b).

After substituting these rescalings into the full set of equations, we obtain to the leading order of Eqs. (4.13a) and (4.12a):

\frac{\partial u^{'}}{\partial τ} + u^{'} \frac{\partial u^{'}}{\partial ξ} = - \frac{\partial h^{'}}{\partial ξ},

(7.3a)

\frac{\partial w_{c}^{'}}{\partial τ} = \hat{α} h^{'} .

(7.3b)

From Eqs. (7.1) and (7.3b), we find

w_{c}^{'} = - \frac{1}{\hat{α}} \frac{\partial u^{'}}{\partial ξ},

(7.4a)

h^{'} = \frac{1}{\hat{α}} \frac{\partial w_{c}^{'}}{\partial τ} .

(7.4b)

Substituting those expressions into Eq. (7.3a), we obtain a single equation for u′:

\frac{\partial u^{'}}{\partial τ} + u^{'} \frac{\partial u^{'}}{\partial ξ} - {\hat{α}}^{- 2} \frac{\partial^{3} u^{'}}{\partial ξ^{2} \partial τ} = 0.

(7.5)

Let us examine the linearized equation briefly:

\frac{\partial}{\partial τ} (1 - {\hat{α}}^{- 2} \frac{\partial}{\partial ξ^{2}}) u^{'} = 0,

which has the dispersion relation

ω (k^{2} + {\hat{α}}^{2}) = 0.

(7.6)

Thus, possible solutions are ω = 0 and

k^{2} = - {\hat{α}}^{2}

. Keep in mind that the horizontal wavenumber, k, is defined in terms of the rescaled horizontal scale. Thus, only evanescent waves are available in the linear limit with the frequency left undetermined. As argued in, e.g., Yano and Flierl (1994) and Yano and Tribbia (2017), linear evanescent waves can be consistent solutions only if nonlinearity becomes important at a certain part of the system.

To solve the nonlinear Eq. (7.5), it is worthwhile to note that it has a similar form to the Korteweg–de Vries equation [cf. sections 13.11 and 13.12 of Whitham (1974), Part 2, epilogue of Lighthill (1978)]:

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{\partial^{3} u}{\partial x^{3}} = 0.

The latter is known to have a soliton solution:

u = 12 k^{2} {sech}^{2} [k (x - x_{0} - 4 k^{2} t)] .

Here, recall that

sech x = \cosh^{- 1} x

, and k and x₀ are arbitrary constants, which adjust the solution form. Thus, we anticipate that a solution with a similar form may also be available with Eq. (7.5). To seek this possibility, we set

u^{'} = u_{0} {sech}^{2} [k (ξ - ξ_{0}) - ω τ],

with u₀, k, and ω the parameters to be determined. Its substitution into Eq. (7.5) yields

u_{0} = 6 ω / \hat{α},

(7.7a)

k = \hat{α} / 2,

(7.7b)

while ω remains an arbitrary constant. The final solutions are

u^{'} = \frac{6 ω}{\hat{α}} {sech}^{2} φ,

(7.8a)

w_{c}^{'} = \frac{6 ω}{\hat{α}} {sech}^{3} φ \sinh φ,

(7.8b)

h^{'} = \frac{6 ω^{2}}{{\hat{α}}^{2}} (- 3 {sech}^{4} φ + 2 {sech}^{2} φ),

(7.8c)

with

φ = \frac{\hat{α}}{2} (ξ - ξ_{0}) - ω τ .

(7.9)

Note that the wavenumber k of the solitary wave solution is controlled by

\hat{α}

, which is proportional to the ratio of the two horizontal scales, i.e.,

\hat{α} = α / {\hat{r}}_{L} = α L / L_{D}

. Also recall the stretching factor

{\hat{ϵ}}^{- (1 / 2)}

applied to the horizontal coordinate. Thus, a characteristic horizontal scale of this solitary wave is inferred by writing

\hat{α} ξ = \hat{α} {\hat{ϵ}}^{- (1 / 2)} x .

From Eq. (4.12b),

\hat{ϵ} = {\hat{r}}_{c}^{2} {\hat{α}}^{2}

, so that

\hat{α} ξ = \frac{x}{{\hat{r}}_{c}} = x \frac{τ_{L}}{τ_{c}} = \frac{L x}{c_{g} τ_{c}},

also recalling the definitions (4.9a) and (4.6c). Bearing in mind that Lx is the dimensional length of the system, a characteristic wavelength of the solitary wave solution is identified as c_gτ_c ∼ 50 km: though slightly of a small side, nevertheless this roughly corresponds to the mesoscale. It also follows that the obtained solution may be interpreted as a prototype for mesoscale convective organization. Note further that the velocity and the height, respectively, are scaled by the factors

ω / \hat{α}

and

ω^{2} / {\hat{α}}^{2}

. Thus, the wave amplitude increases with its frequency ω and in a more acerbated manner for the height than the velocities.

Examples of the solutions with ω = 1 (Fig. 2a) and ω = −1 (Fig. 2b) are shown in Fig. 2 with the horizontal coordinate given by $\hat{α} ξ = \hat{α} {\hat{ϵ}}^{- (1 / 2)} x$ . Here, the eastward- (ω = 1) and westward-propagating (ω = −1) solutions are characterized by westerly and easterly solitary bursts u′ (solid), respectively. Especially, the zonal-wind structure of the eastward-propagating wind burst is reminiscent of observed westerly wind bursts (e.g., Hartten 1996; Yano et al. 2004). By mass continuity, the large-scale ascent is followed by descent in the direction of propagation of the burst. By the free-ride balance (7.1), the convection anomaly, $w_{c}^{'}$ (long dash), also follows the same structure. The positive convective anomaly ahead of the propagation direction is induced by a positive height (CAPE) anomaly h′ (short dash) affront, which is followed by a negative anomaly due to the stabilization by convection. Stabilization leads to a negative convective anomaly, which is followed by a recovery of the positive height (CAPE) anomaly.

Fig. 2.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0066.1

8. Further discussion

Atmospheric precipitating convection goes through a distinguished life cycle from a genesis to decay, and thus, it is natural to expect that the convective life cycle may play an important role in its coupling to large-scale dynamics, especially over the tropics. From this perspective, the basic assumption of convective quasi equilibrium adopted in convection parameterizations is unsatisfactory, because this approximation totally neglects life cycles associated with parameterized convection.

The present work shows what happens when a life cycle of convection is explicitly taken into account as a part of the large-scale dynamics. A qualitative consequence, even without performing any calculations, can even be intuitively expected: the short periodicity of convective life cycles dominate aspects of the coupled dynamics. This expected tendency is demonstrated explicitly by a linear analysis, which shows that the squared frequency of a linear wave is obtained by a squared sum of the characteristic frequency of the convective life cycle and a dry gravity wave frequency, under an analysis assuming no Coriolis force.

The convective life cycle used in the present study is based on the convective energy cycle originally introduced by Arakawa and Schubert (1974), in seeking a basis for a closure of their mass-flux parameterization. The energy cycle is closed by following Yano and Plant (2012b). The large-scale dynamics adopted is a shallow-water analog.

The high-frequency characteristic of convectively coupled waves obtained with explicit convective life cycles is in marked contrast to the typical characteristic under standard formulations with a convective quasi-equilibrium assumption. In the latter case, convection is found to slow down the dry large-scale waves by decreasing the effective stratification of the atmosphere. This behavior arises because any explicit periodicities associated with convection are effectively eliminated by averaging them out through the convective quasi-equilibrium assumption. The approach of the present paper explicitly retains such a high convective-scale frequency, and thus, this frequency is added to a full spectrum of the whole system.

The implications of the obtained result may further be translated in the following manner in the context of convectively coupled equatorial waves: any wave modes explicitly controlled by a convective life cycle must propagate faster than the dry counterparts. The fact that the observed convective equatorial waves propagate rather slower than the dry counterparts suggests that an explicit convective life cycle does not play a significant role in these waves. Such slowdown of the wave propagations is most conveniently explained in terms of the convective quasi-equilibrium framework, as just suggested.

An explicit emergence of the convective-scale high frequencies into the large-scale dynamics is obviously an unfavorable feature, if the focus of modeling is on the long-time-scale phenomena. Recall that the presence of explicit convection leading to a numerical instability as found by Kasahara (1961) is one of the original motivations to introduce convection parameterization into global models, as reviewed by Kasahara (2000).

A more attractive feature emerges when the system is scaled down to a mesoscale regime, also introducing a weak nonlinearity. This rescaling is performed in such a manner that the free-ride balance (Fraedrich and McBride 1989; see also Sobel et al. 2001) is obtained to the leading order. The analysis leads to a nonlinear equation analogous to the Kortweg–de Vries equation, and like the latter, it contains a solitary wave solution. The obtained mesoscale solution is reminiscent of tropical westerly wind bursts. In this manner, the present study suggests that a parameterized convective life cycle as considered herein may provide a prototype model for mesoscale convective organization.

Although an analysis with the rotation effect is still to be performed, it is evident that the eastward-propagating solitary gravity wave solution obtained can be reinterpreted as a Kelvin wave in the presence of rotation so long as we can assume that the equatorial deformation radius is much larger than the longitudinal wavelength. Nevertheless, a full analysis of this system with the rotation effect will be worthwhile to explore rich possibilities of nonlinear interactions between convective life cycles and the equatorial waves.

This investigation may be considered a natural extension of dry solitary equatorial waves as investigated by Boyd (1980, 1983, 1984, 1985). Furthermore, a possibility of interpreting the Madden–Julian oscillation as a solitary nonlinear Rossby wave has already been suggested by Wedi and Smolarkiewicz (2010) and Yano and Tribbia (2017). The present study further suggests that the tropical westerly wind bursts may also be interpreted as a type of nonlinear solitary waves. Such potentially crucial importance of nonlinearities for convectively coupled equatorial waves needs to be further investigated.

9. Conclusions

The most important lesson to learn from the present study is that, as extensively discussed in the last section, if the focus is solely on the global scale of the atmosphere, then one should not try to include a convective life cycle explicitly into a model, how attractive this approach might appear to be at first sight.

On the other hand, for those who wish to investigate tropical atmospheric dynamics in its full spectrum, the convective energy cycle system coupled with large-scale dynamics does provide an attractive option to pursue. Although only a preliminary investigation has been performed, an identified solitary wave solution, reminiscent of tropical westerly wind bursts, already suggests a rich behavior of this system under full nonlinearity. Nevertheless, a full investigation is still awaited, as already suggested from several perspectives in the last section. We should also keep in mind that convection is still parameterized, using a mass-flux-based formulation under an assumption of parcel–environment quasi equilibrium.

This asymptotic expansion form can be derived by setting them more generally as $h = \bar{h} + {\hat{ϵ}}^{α_{1}} h^{'}$ , $w_{c} = {\bar{w}}_{c} + {\hat{ϵ}}^{α_{2}} w_{c}^{'}$ , $u = {\hat{ϵ}}^{α_{3}} u^{'}$ , $\partial / \partial t = {\hat{ϵ}}^{α_{4}} \partial / \partial τ$ , and $\partial / \partial x = {\hat{ϵ}}^{α_{5}} \partial / \partial ξ,$ , with unspecified power exponents α₁, α₂, α₃, α₄, and α₅. Substituting into Eqs. (4.12a), (4.13a), and (4.13b), insisting that Eq. (4.13b) reduces to Eq. (7.1), all the terms in Eq. (4.13a) are retained to leading order, and the time derivative of k_c in Eq. (4.12a) does not vanish at leading order, we determine those exponents to be α₁ = 3, α₂ = 1, α₃ = 3/2, α₄ = 1, and α₅ = −1/2.

Data availability statement.

No data are used in the present study.

APPENDIX

Energy Cycle Analysis

The purpose of this appendix is to identify the physically most consistent form of nonlinearity for the shallow-water analog system from the point of view of the energy cycle of the system. The most straightforward way to add nonlinearity to the linear large-scale system (4.7a)–(4.7c) would be in the identical form as that which appears in the actual shallow-water system:

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = - \frac{\partial h}{\partial x},

(A.1a)

\frac{\partial h}{\partial t} + \frac{\partial}{\partial x} u (1 + h) = - Q,

(A.1b)

w = - {\hat{r}}_{L} \frac{\partial u}{\partial x} .

(A.1c)

Here, we are going to show that this form leads to a physically unacceptable interpretation from the point of view of the energy cycle. We show further that the problem arises with the postulated nonlinear contribution to Eq. (A.1b) but that the nonlinear advection term in Eq. (A.1a) may be retained.

a. Kinetic energy

To derive the kinetic-energy budget, we first rewrite the momentum Eq. (A.1a) in a flux form by multiplying it by h_T = 1 + h, and adding by Eq. (A.1b) multiplied by u:

\frac{\partial u h_{T}}{\partial t} + \frac{\partial}{\partial x} u^{2} h_{T} = - \frac{\partial}{\partial x} y h_{T}^{2} 2 - u Q .

(A.2)

Multiplying Eq. (A.1a) by uh_T and Eq. (A.2) by u, we obtain the budget:

\frac{\partial}{\partial t} \frac{h_{T}}{2} u^{2} + \frac{\partial}{\partial x} \frac{h_{T} u^{3}}{2} = - u \frac{\partial}{\partial x} \frac{h_{T}^{2}}{2} - \frac{u^{2}}{2} Q .

(A.3)

Here is the first key point to note: from a physical consideration, we expect that the large-scale kinetic energy would not directly be modified by a convective process or by diabatic heating. Thus, Eq. (A.3) is not physically consistent by containing a source term due to diabatic heating.

We can trace this physical inconsistency to the fact that the kinetic energy is defined by $h_{T} u^{2} / 2$ above. Although this is a physically consistent definition of kinetic energy in the original shallow-water system, that is no longer the case for this shallow-water analog atmosphere. This conclusion stems from the fact that in the shallow-water analog atmosphere, the height is better interpreted as a representation of the potential-temperature anomaly rather than a representation of a fluid depth, as in the original definition of the shallow-water system.

Based on this consideration, we conclude that the kinetic energy is better defined as u²/2. With this definition, the kinetic-energy budget is obtained by multiplying Eq. (A.1a) by u:

\frac{\partial}{\partial t} \frac{u^{2}}{2} + \frac{\partial}{\partial x} \frac{u^{3}}{3} = - u \frac{\partial h}{\partial x} .

(A.4)

Here, the form of the divergence term is rather unfortunate, and a minor negative consequence from the redefinition.

b. Potential energy

A similar consideration also applies when defining the potential energy of this shallow-water analog system. As already suggested above, the total depth h_T of the system does not have much physical significance: it is better to take the height perturbation, h, as a measure of the potential temperature perturbation θ under Eq. (3.11b). Thus, it also follows that the potential energy is better defined by h²/2 rather than

h_{T}^{2} / 2

. Its budget is obtained by multiplying Eq. (A.1b) by h, so that

\frac{\partial}{\partial t} \frac{h^{2}}{2} + h \frac{\partial}{\partial x} u h_{T} = - h Q .

(A.5)

We may note above that the advection term does not turn into a flux form as expected.

c. Total energy budget

Finally, by taking the sum of Eqs. (A.4) and (A.5), we obtain the conservation law of the total energy as

\frac{\partial}{\partial t} (\frac{u^{2} + h^{2}}{2}) + \frac{\partial}{\partial x} \frac{u^{3}}{3} + h \frac{\partial}{\partial x} u h_{T} + u \frac{\partial h}{\partial x} = - h Q .

To express the last two terms on the left-hand side closer to a flux form, recall that h_T = 1 + h; thus,

h \frac{\partial}{\partial x} u h_{T} + u \frac{\partial h}{\partial x} = \frac{\partial}{\partial x} u h + h \frac{\partial}{\partial x} u h .

We can recognize that the remaining nonflux term on the left-hand side arises from the nonlinear term in the height Eq. (A.1b). This result suggests that it is unphysical to add a nonlinear advection term to the height (heat) equation under the present shallow-water analog formulation. Thus, the choice of the form (4.13b) follows. After this modification, the total energy conservation law reduces to

\frac{\partial}{\partial t} (\frac{u^{2} + h^{2}}{2}) + \frac{\partial}{\partial x} (\frac{u^{3}}{3} + u h) = - h Q .

(A.6)

d. Coupling with convection

The final step is to add the convective kinetic energy to the energy budget (A.6) just obtained. Toward this goal, note first that the term hQ on the right-hand side of the potential energy budget (A.5) can be rewritten with the help of Eq. (4.8a) as

h Q = \hat{α} h k_{c} - h {\hat{Q}}_{R} .

(A.7)

Hence, convective kinetic energy is generated [i.e.,

\hat{α} h k_{c} > 0

on the right-hand side of Eq. (4.12a)] by consuming the potential energy [i.e., hQ > 0 through the same process: the right-hand side of Eq. (A.5)]. By substituting Eq. (A.7) into the right-hand side of Eq. (A.6), we obtain

\frac{\partial}{\partial t} (\frac{u^{2} + h^{2}}{2}) + \frac{\partial}{\partial x} (\frac{u^{3}}{3} + u h) = - \hat{α} h k_{c} + h {\hat{Q}}_{R} .

(A.8)

Taking the sum of Eqs. (A.8) and (4.12a), the total energy budget including the contribution of the convective scale is

\frac{\partial}{\partial t} (\frac{u^{2} + h^{2}}{2} + \hat{ϵ} k_{c}) + \frac{\partial}{\partial x} (\frac{u^{3}}{3} + u h) = h {\hat{Q}}_{R} - \frac{k_{c}}{{\tilde{τ}}_{D}} .

(A.9)

Thus, as a whole the radiation

{\hat{Q}}_{R}

is the only ultimate source of the energy to the system, and the only sink is the dissipative loss

k_{c} / {\tilde{τ}}_{D}

of convective kinetic energy. Note that the large-scale dynamics has been assumed to be dissipationless for simplicity.

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Fig. 1.

Plots of the idealized mean circulation: (a) the zonal wind u; (b) the vertical velocities $\bar{w}$ (solid), w_c (long dashed) and −w_R (short dashed); (c) the barotropic height field h_b.

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Fig. 2.

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Abstract

Significance Statement

Abstract

Significance Statement

1. Introduction

2. Convective energy cycle system

3. Large-scale system

a. Large-scale heat equation

b. Normal-mode decomposition of the linear primitive equation system

4. Nondimensionalization

a. Convective energy cycle system

b. Large-scale system

c. Two time scales

d. Coupling problem

e. Full system with nonlinearity

5. Steady solutions

6. Linear analysis

7. Weakly nonlinear analysis

8. Further discussion

9. Conclusions

Data availability statement.

APPENDIX

Energy Cycle Analysis

a. Kinetic energy

b. Potential energy

c. Total energy budget

d. Coupling with convection

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Interaction of the Convective Energy Cycle and Large-Scale Dynamics

Abstract

Significance Statement

Abstract

Significance Statement

1. Introduction

2. Convective energy cycle system

3. Large-scale system

a. Large-scale heat equation

b. Normal-mode decomposition of the linear primitive equation system

4. Nondimensionalization

a. Convective energy cycle system

b. Large-scale system

c. Two time scales

d. Coupling problem

e. Full system with nonlinearity

5. Steady solutions

6. Linear analysis

7. Weakly nonlinear analysis

8. Further discussion

9. Conclusions

Data availability statement.

APPENDIX

Energy Cycle Analysis

a. Kinetic energy

b. Potential energy

c. Total energy budget

d. Coupling with convection

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