## 1. Introduction

Atmospheric dynamics in the tropics are characterized by the predominance of organized convection on a wide range of scales, spanning from mesoscale systems to synoptic- and planetary-scale convectively coupled waves, such as Kelvin waves and the Madden–Julian oscillation (MJO) (Nakazawa 1974; Hendon and Liebmann 1994; Wheeler and Kiladis 1999). Despite continued efforts by the climate community, present coarse-resolution general circulation models (GCMs) poorly represent variability associated with tropical convection (Slingo et al. 1996; Moncrieff and Klinker 1997; Scinocca and McFarlane 2004; Lau and Waliser 2005; Zhang 2005). One of the main sources of error in these models arises from deficiencies in the treatment of cumulus convection (Moncrieff and Klinker 1997; Lin et al. 2006), which has to be parameterized in coarse-resolution GCMs. However, marked improvement have been made in a few GCM simulations recently (Khouider et al. 2011; Del Genio et al. 2012; Crueger et al. 2013; Deng et al. 2015; Ajayamohan et al. 2013, 2014). Given the importance of the tropics for climate prediction and numerical weather prediction (NWP), the search for new strategies for parameterizing the unresolved effects of tropical convection has been a key focus of researchers during the last few decades.

Several methods have been developed to address the multiscale nature of the problem. Cloud-resolving models (CRM) on fine computational grids, as well as high-resolution numerical weather prediction (NWP) models with improved convection parameterizations, have succeeded in representing some aspects of organized convection (ECMWF 2004; Moncrieff et al. 2007; Slawinska et al. 2014). In addition, superparameterization (SP) methods (Grabowski and Smolarkiewicz 1999; Grabowski 2001, 2004; Randall et al. 2003; Majda 2007) and sparse space–time SP (Xing et al. 2009; Slawinska et al. 2015) use a cloud-resolving model in each column of a large-scale GCM to explicitly represent small-scale processes, mesoscale processes, and interactions between them. However, these methods are not currently computationally viable for application to large ensemble weather prediction or climate simulations. SP methods also have difficulty resolving propagating convective systems because the small-scale CRM is typically performed in a two-dimensional periodic domain. This can distort important convective features like mesoscale downdrafts and convective momentum transport (Pritchard et al. 2011; Moncrieff and Waliser 2015); however, there are some recent strategies to address the latter effect (Tulich 2015). Thus, the search for computationally inexpensive and realistic convection parameterizations that are seamlessly scalable between medium- and coarse-resolution GCMs remains a central unsolved problem in the atmospheric community (Arakawa 2004).

A closely related issue to the parameterization problem is the development of theories relating cumulus convection and the large-scale variables. An early theory for this cross-scale interaction is the convective instability of the second kind (CISK) idea (Charney and Eliassen 1964). CISK describes a two-way feedback between cumulus convection and wind convergence in the planetary boundary layer. In the original formulation, this convergence is caused by Ekman pumping due to the large-scale geostrophically balanced circulation, but wave–CISK is a theory that is more relevant to nonbalanced equatorial flows (Lindzen 1974).

Both forms of CISK are heavily criticized in the literature in favor of an alternative known as the quasi-equilibirum (QE) hypothesis (Arakawa and Schubert 1974; Emanuel et al. 1994; Arakawa 2004). For a more neutral contemporary perspective on this controversy, the reader is referred to Smith (1997). In the broadest sense, QE supposes that, over large spatial scales, cumulus convection acts to remove static instability in an atmospheric column. The static stability is typically a function of the humidity and temperature fields; therefore, for the present purposes, we define a QE parameterization as any scheme that relates total precipitation to the humidity and temperature alone. In the QE context, the hypothesis of wind-induced surface heat exchange (WISHE) provides a mechanism for the interaction between the large-scale circulation and cumulus convection (Emanuel et al. 1994). While this mechanism is well established for tropical cyclones, it is unclear to what extent WISHE is relevant to dynamics in equatorial regions (Grabowski and Moncrieff 2001).

For each of the physical hypotheses above, there is a corresponding set of operational cumulus parameterizations. Broadly speaking, convection schemes can be divided into those based on moisture-convergence closures (Kuo 1974), the moist adjustment idea (Manabe et al. 1965), and the QE hypothesis (Arakawa and Schubert 1974; Betts and Miller 1986). CISK thinking informed the moisture-convergence schemes. The QE hypothesis at its core is a statement about statistical equilibrium, but the atmosphere, in reality, is far from equilibrium. Moreover, the QE hypothesis breaks down as the current GCM grid sizes approach the cumulus scale. One generic design principle for treating nonequilibrium systems in atmosphere–ocean science is the addition of stochastic perturbation (Buizza et al. 1999; Palmer 2001; Lin and Neelin 2003; Khouider et al. 2003; Majda et al. 2008; Majda and Stechmann 2008). In particular, one of the more promising approaches has been the use of Markov chain lattice models to represent unresolved subgrid variability (Khouider et al. 2003). This type of lattice model is an extension of an Ising spin-flip model used for phase transitions in material science (Majda and Khouider 2002; Katsoulakis et al. 2003b) and has been successfully used to improve simple convection parameterizations (Khouider et al. 2003; Majda et al. 2008; Khouider et al. 2010; Frenkel et al. 2012, hereafter FMK12, 2013, hereafter FMK13). Another stochastic cumulus convection parameterization is that of Plant and Craig (2008).

In addition to stochastic parameterizations, there have been large improvements in deterministic parameterizations. Some drivers of these improvements include large field campaigns, such as the TOGA COARE (Moncrieff and Klinker 1997), and an enhanced understanding of organized convection. In particular, a clearer understanding of equatorial convectively coupled waves (Wheeler and Kiladis 1999; Kiladis et al. 2009; Straub and Kiladis 2002) has informed the development of the multicloud parameterizations (Khouider and Majda 2006b,a, 2007, 2008a,b; Khouider et al. 2010; FMK12). The multicloud parameterizations take advantage of the observed self-similarity and vertical structure of equatorial waves (Wheeler and Kiladis 1999) and have been successfully blended with the Ising model stochastic parameterization approach (FMK12; FMK13). Especially in FMK13, it is shown that blending the multicloud ideas with stochastic modeling allows for rich multiscale behavior in nonlinear simulations with only a single spatial dimension. It is also shown that stochasticity enhances the intermittency of single-column model simulations (see FMK13, their Fig. 4). Finally, both the deterministic and the stochastic multicloud model (SMCM) can realistically replicate aspects of convectively coupled waves and intraseasonal oscillation in a prototype GCM setting (Khouider et al. 2011; Deng et al. 2015; Ajayamohan et al. 2013, 2014).

In this study, we revisit the controversy between CISK and QE, motivated by recent work with observations from Darwin, Australia, that has established a strong link between the large-scale convergence field and local convection (Davies et al. 2013). Moreover, this study failed to find a strong link between CAPE and local precipitation. Motivated by these observations, several studies have attempted to infer causality by fitting multicloud-based stochastic models to the estimated cloud fraction fields (Peters et al. 2013; Dorrestijn et al. 2015; De La Chevrotière et al. 2014). They found that large-scale pressure velocity at 500 hPa from a reanalysis product is a better predictor of convection over Darwin than the corresponding moist thermodynamic state. These data have also been used to evaluate several operational convective mass-flux trigger functions (Suhas and Zhang 2014). Moreover, there is evidence that the transition from shallow to deep convection is linked to the large-scale vertical moisture advection (Hagos et al. 2014, and references therein). Finally, the hypothesis that the diabatic heating is separable with respect to the vertical velocity is a key assumption in the derivation of steady-state slantwise layer overturning models which successfully describe many aspects of mesoscale convective systems (Moncrieff 2010, and references therein).

The observational studies above are based on diagnostics from a single location, but the idea of convergence coupling has been primarily criticized on a dynamical basis (Emanuel et al. 1994). To address these concerns, the aim of this paper is to develop a prototype nonlocal stochastic convection parameterization that takes into account the effects of large-scale convergence and avoids the pitfalls of conventional moisture-convergence closures (Kuo 1974). The primary aim here is to explore the dynamical consequences of this additional physical assumption. Because the SMCM shows realistic variability in computationally inexpensive one-dimensional simulations (Frenkel et al. 2013), it is an ideal test bed for these ideas. It also provides a more flexible framework than the deterministic multicloud model does for coupling cloud transitions to arbitrary scalar fields, such as the vertical velocity. There has also been some recent work on implementing the SMCM with convergence coupling in the ECHAM GCM (Peters et al. 2015).

Specifically, a flexible framework for including the effects of convergence coupling in the SMCM is developed. Using this framework, it is shown that coupling the interaction of congestus and deep clouds to the large-scale convergence leads to realistic variability. This approach blends the convergence-coupling idea with the CAPE-coupling approach used in past work (Khouider and Majda 2006a,b, 2008a,b; FMK12; FMK13). In some respects, the nonlocal convergence coupling introduced here accounts for nonlocal interactions between microlattice convective sites, and it complements recent work along these lines (Khouider 2014).

In the present paper, we find that, in spatially extended idealized Walker cell simulations, this deep-convergence-coupled SMCM shows an overall increase in variability about the mean and an enhanced low-frequency variability. In particular, the coupling enhances the persistence of moist gravity waves in the dry regions flanking the central warm pool. These waves have an approximate phase speed of 10 m s^{−1} and significantly warm and dry the atmosphere in their wake. There is observational evidence that Kelvin waves do indeed propagate with remarkable persistence across planetary zonal distances (Kiladis et al. 2009; Straub and Kiladis 2002; Wheeler and Kiladis 1999).

Moreover, an intermittent regime-switching behavior arises on intraseasonal time scales that switches the system between epochs of regular and irregular walker cell variability as seen in CRM simulations (Slawinska et al. 2015, 2014). However, these benefits are only conferred when the transition of congestus to deep convection is convergence coupled. Naively replacing the model’s implicit low-level moisture-convergence coupling with an explicit dry-convergence coupling leads to degeneracies, such as extreme sensitivity to numerical resolution. Gains from convergence coupling only occur when modeling the formation of deep rather than shallow clouds. This is consistent with observations (Davies et al. 2013).

The paper is outlined as follows. In section 2, two prototype parameterizations are developed that, respectively, couple congestus and deep clouds to the large-scale convergence. Then section 3 describes the setup for the idealized Walker circulation experiments. Results for the schemes are shown in section 4. Particular attention is given to the deep-convergence-coupled results in section 4a, and the degenerate congestus-convergence-coupled scheme in section 4b. Concluding remarks are given in section 5.

## 2. Stochastic multicloud model

The multicloud model (Khouider and Majda 2006b,a, 2008b) and its stochastic variant (FMK12; FMK13; Khouider et al. 2010; Deng et al. 2015) have been successful in replicating the observed dynamics of organized tropical convection by coupling heating rates to the large-scale thermodynamic state. As hinted in the introduction, the multicloud models accomplish this by capturing transitions between congestus, deep, and stratiform heating, as seen in Fig. 1. While the deterministic multicloud models accomplish this using a diagnostic nonlinear switch function to transition between congestus and deep heating (Khouider and Majda 2006b), the SMCM explicitly treats the cloud fractions of these three archetypal heating patterns as prognostic variables (FMK12; FMK13; Khouider et al. 2010). It accomplishes this via a computationally efficient coarse-grained continuous-time Markov chain for the fraction of congestus *σ*_{c}, deep *σ*_{d}, and stratiform *σ*_{s} sites in a given numerical grid cell (Katsoulakis et al. 2003b,a; Khouider and Majda 2008b). Like the convective inhibition models of Majda and Khouider (2002) and Khouider et al. (2003), this setup distinguishes between the processes that lead to the formation of convection sites and those that alter the magnitude of the heating. Therefore, stochastic convergence coupling does not necessarily entail a wave-CISK-type instability in the SMCM.

The SMCM also allows for realistic physically motivated interactions between cloud types, which are easily coupled in an explicit fashion to any deterministic quantity of choice. Using such a model, we can hope to address the validity of the convergence-coupling hypothesis. As such, extending the work of (Peters et al. 2013) to the prognostic spatially extended SMCM provides an ideal test bed for gauging the validity of the convergence-coupling hypothesis in state-of-the-art convection schemes.

Along these lines, section 2a contains abbreviated description of the two-baroclinic-mode dynamical core. The stochastic coupling to the large-scale thermodynamics and convergence is introduced in section 2b.

### a. Dynamical core and convection closure

Khouider and Majda (2006b,a, 2008b), Khouider et al. (2010), FMK12, and FMK13 assume three heating profiles associated with the main cloud types that characterize organized tropical convective systems (Johnson et al. 1999): cumulus congestus clouds that heat the lower troposphere and cool the upper troposphere through radiation and detrainment; deep convective towers that heat the whole tropospheric depth; and the associated lagging-stratiform anvils, which heat the upper troposphere and cool the lower troposphere through evaporation of stratiform rain. In its simplest form, the multicloud model captures these three modes of heating using the first two vertical modes of a constant stratification Boussinesq system. Therefore, the simplest version of the dynamical core of the multicloud parameterizations consists of two coupled and forced shallow water systems. To simplify the current study, the meridional dependence of the equations is ignored, and the simulations are performed in a single ring of latitude. In CRM simulations (Slawinska et al. 2014) and the past work on the multicloud model, this one-dimensional setup has proved sufficient to generate a wide array of interesting variability. This is especially true in simulations with a nonuniform SST pattern.

The velocity *u*_{j} and potential temperature *θ*_{j} equations are derived in standard fashion by Galerkin projection of the rigid-lid dry Boussinesq equations onto the first two baroclinic modes *Z*(*z*) and *Z*(2*z*), where *z* ≤ *H*_{T}. The velocity equation is damped by a simple drag law with a time scale of *τ*_{R} = 75 days .

The boundary layer equivalent potential temperature *θ*_{eb} is composed of forced convective downdrafts *D* and the evaporation *E* and has no advective contributions linear or otherwise. The multicloud formulation enters through the three heating rates *H*_{c}, *H*_{d}, and *H*_{s}, which represent congestus, deep, and stratiform convective heating, respectively. These and other important diagnostic quantities are given in Table 1.

The reader will note that the heating rates *H*_{d} and *H*_{c} are each the product of cloud fraction fields *σ*_{d} and *σ*_{c} and some measure of the energy available for convection. The energy available for congestus and for deep heating are distinct but closely related quantities that depend only on the thermodynamic degrees of freedom *θ*_{eb}, *q*, *θ*_{1}, and *θ*_{2}. While diagnostic closures adequately model congestus and deep heating, FMK13 showed an improvement when using a lag-differential equation to model the stratiform heating, and this is the form used in (7). In that equation, *τ*_{s} is the time scale over which *H*_{s} relaxes to a fraction of *H*_{d} determined by the stratiform cloud fraction *σ*_{s}. On the other hand, the cloud fractions fields are treated as stochastic processes that evolve according to a set of intuitive rules described in the next section (Khouider et al. 2010; FMK12; FMK13).

### b. Stochastic coupling

The evolution of the cloud fractions is given by a continuous-time Markov chain, which has transition rates that depend on the large-scale variables of the system. This approach to stochastic parameterization is introduced in Majda and Khouider (2002) and Khouider et al. (2003) and can be roughly thought of as introducing a state-dependent multiplicative “noise.” However, because the Markov chain is defined on a discrete state space, the simulated pathways cannot be described using a Langevin equation with white noise (Gardiner 2009).

The stochastic parameterization attempts to model subgrid-scale dynamics explicitly by defining a lattice within each coarse grid cell. The underlying PDE [(1)–(7)] is discretized onto a regular numerical mesh, and each grid cell is further divided into a rectangular *ℓ* × *ℓ* lattice. Each element of this lattice is occupied by either a congestus, deep, or stratiform cloud or by a clear-sky site, which are represented, respectively, by the integers 1, 2, 3, and 0 (clear sky). A continuous-time Markov chain that allows for transitions between these four states at a certain rate is defined on this discrete state space. These sorts of models have been used in material science and chemistry to model the reaction of different chemical species (Gillespie 1977), but the approach here is to couple the transition rates between clouds to a PDE [(1)–(7)] via the large-scale resolved variables.

The SMCM allows for a few different transitions between the cloud types. With the associated transition rate in parentheses, these transitions are as follows:

- Formation of a congestus cloud from clear sky (
*R*_{01}) - Formation of a deep cloud from clear sky (
*R*_{02}) - Conversion from a congestus to a deep cloud (
*R*_{12}) - Conversion from a deep to a stratiform cloud (
*R*_{23}) - Decay of congestus (
*R*_{10}), deep (*R*_{20}), or stratiform clouds (*R*_{30})

*θ*

_{eb}will tend to promote active convection, and high tropospheric temperatures will tend to discourage it.

*R*

_{12}) is a critical transition for convergence coupling. However, it is possible that interesting results can be obtained by making a similar modification to other transitions. The transition from deep to stratiform clouds (

*R*

_{23}) is a particularly interesting candidate for such an alteration, especially given the importance of wind shear in organizing convection. However, for the sake of simplicity, we constrain our current analysis to transitions in the lower troposphere that pertain directly to the WISHE–CISK debate. Along these lines, a modification can be introduced so thatwhere

*W*

_{ij}is a proxy that depends on the large-scale convergence.

### c. Transition rates with convergence coupling

*W*

_{ij}in (8), it is useful to note the precise form of the vertical velocity

*w*in any two-baroclinic-mode Boussinesq model. Mass continuity requires that

*w*at a height

*z*be given bywhere

**u**= (

*u*,

*υ*) is the horizontal velocity.

*R*. Specifically, the large-scale convergence for given location and height is given bywith the backward-centered difference operator

*x*) := 1 − exp(−

*x*). The function Γ(

*x*) is designed to normalize the tendency of each factor and satisfies 0 < Γ(

*x*) ≤ 1. The only difference between these rates and those of FMK13 is the additional factor Γ(

*W*

_{ij}). The other transitions are left unaltered, and a comprehensive list of the transition rates is available in Table 2.

Stochastic transition rates with multiplicative convergence coupling. Compare to Table 2 from FMK13.

*W*

_{01},

*W*

_{12}, and

*W*

_{02}represent the large-scale convergence evaluated at the vertical level relevant to the transition. Specifically,where

*z*

_{ij}is the vertical level governing the transition,

*τ*

_{w}is the strength of the convergence coupling, and

Default constants and parameters common to all multicloud simulations discussed in this report. The shills divide deterministic and stochastic parameters from FMK13, convergence-coupling parameters, and numerical parameters, respectively. Many of the parameters are chosen to satisfy constraints at radiative–convective equilibrium (RCE). Largely reproduced from FMK13.

In the formulation above, *R*_{01}, *R*_{02}, and *R*_{12} each involve the product of three different factors (cf. Table 2). One factor Γ(*W*_{ij}) is related to the large-scale convergence, while the other two are related to the grid-scale thermodynamics. This setup is general, but it is not clear that all three transitions considered in (11)–(13) should be coupled to the large-scale convergence simultaneously. To address this ambiguity, this paper will study three different kinds of stochastic setups.

#### 1) Thermodynamics-only coupling (THERMO)

A thermodynamics-only setup is a base case that yields results that are nearly identical to FMK13. This setup is obtained by setting *τ*_{w} = 0, which implies that

#### 2) Convergence coupling (CCON)

Another possible setup is one that couples the formation of congestus clouds to the large-scale convergence alone. In FMK13 and other works (Khouider and Majda 2006a,b; Khouider et al. 2010), it is shown that the thermodynamics-only SMCM already features an implicit low-level moisture-convergence mechanism resulting from the second baroclinic contribution to (5) and the moisture dependence of *C*_{l}. Wholesale replacing this implicit moisture-convergence mechanism with an explicit dry-convergence coupling fundamentally alters the underlying cloud formation mechanism of the SMCM and provides an interesting, albeit unrealistic, test bed for the convergence-coupling idea. The CCON setup yields intriguing improvements for some parameter regimes but, as expected, suffers from extreme sensitivity to these same parameters.

In particular, this setup fixes *W*_{02} = *W*_{12} =

#### 3) Deep convergence coupling (DCON)

These degeneracies are not present when the transition from congestus to deep clouds is coupled to the large-scale convergence. As will be seen in subsequent sections, this DCON stochastic setup allows for the benefits of the THERMO simulations while altering the dynamics of large-scale convectively coupled waves in a realistic and intriguing fashion. The DCON setup consists of fixing *W*_{01} = *W*_{02} = *C*, *C*_{l}, and *W*_{12} to vary.

## 3. Idealized Walker circulation simulations

The past work on convergence coupling is typically focused on its role mediating interactions between convection, tropical cyclones (Charney and Eliassen 1964), and synoptic-scale equatorial waves (Lindzen 1974). Therefore, we expect the convergence coupling designed here to show interesting characteristics in simulations with an imposed large-scale circulation. In the SMCM, a planetary-scale SST pattern that mimics the so-called Indonesian warm pool will force an idealized version of the Walker circulation. This is a standard test bed for simplified convection parameterizations (Khouider et al. 2003; FMK13).

*A*

_{SST}elsewhere, as in Khouider and Majda (2006b, 2008b), FMK12, and FMK13. Unless otherwise stated,

*A*

_{SST}= 5 K. This setup mimics the Indian Ocean–western Pacific warm pool and has yielded interesting Walker-like circulations in FMK13.

Time series of 1000 days are generated for each formulation of the transition rates using a time step of 30 s and a total of 1000 grid cells spread over a 40 000-km equatorial domain. The number of stochastic elements per coarse grid cell is *ℓ*^{2} = 30^{2} = 900. Unless otherwise stated, the convergence-coupling strength is fixed at *τ*_{w} = 10 h, and the interaction radius is fixed at *R* = 240 km. These and other parameters are given in Table 3.

The numerical method used here is the same as that used in FMK13. Namely, an operator-splitting strategy is used, which alternates solutions of the hyperbolic terms, source terms, and stochastic process. The conservative terms are discretized and solved by a nonoscillatory central differencing scheme, while the remaining deterministic forcing terms are handled by a second-order Runge–Kutta method (Khouider and Majda 2005a,b). The stochastic component of the scheme is resolved using Gillespie’s (1975) exact algorithm. For more details on the algorithm, see (Khouider et al. 2010; FMK12; FMK13).

Here, we note that combining convergence coupling for the congestus and deep transitions invariably results in a strong numerical instability for reasonable values of Δ*t* and *τ*_{w}. This is why we only consider the CCON and DCON stochastic setups rather than some combination of the two.

## 4. Results

First, we will provide a qualitative overview of the dynamics of the three stochastic setups. The anomalies from the temporal mean of the first baroclinic velocity *u*_{1} for THERMO, DCON, and CCON are available in Fig. 2, and the corresponding climatological mean and variance are shown in Fig. 3.

All stochastic setups show interesting variability about the mean, but the nature of the variability is subtly altered between the simulations. All three simulations show small-scale wave activity in the center of the domain (e.g., the high-SST region), corresponding to a background of convective activity. However, the simulations differ in the behavior of large convectively coupled waves (CCWs) at the edges of the elevated-SST region. The THERMO simulation has the same behavior as previously seen in (FMK12; FMK13) with strong second baroclinic heating around 20 000 km, which transitions to deep heating in the form of CCWs around 25 000 km (not shown). In the *u*_{1} field, the most salient feature is the strong and regular convectively coupled waves that depart the warm pool region every 12 days in strictly alternating order.

Adding convergence coupling to either the congestus (CCON) or deep (DCON) transitions results in breakdown of this order. The CCON regime represents a more drastic alteration and features strong CCWs on many different scales interacting with each other without the emergence of a clear periodicity. The DCON regime provides a more subtle alteration that causes the regular waves to leave the warm pool at double the period (24 days) and to propagate farther into the dry region. Occasionally, one of these waves will circle the domain entirely to reinteract with the warm pool, as can be seen around day 930. This interaction initiates a transition to a more chaotic regime for long periods of time. This enhanced persistence of the CCWs is the primary result of this study.

The simulations have a roughly comparable mean *u*_{1} climatology (6 m s^{−1}), except for the CCON simulation, which has a peak mean *u*_{1} of ~4 m s^{−1}. On the other hand, the second baroclinic *u*_{2} structure changes substantially between the simulations. While the DCON setup is quite similar to the base-case THERMO setup, the CCON simulation does not feature the characteristic double peak in the second baroclinic velocity component.

The total variability about the mean also differs subtly between the setups. The THERMO and DCON schemes show a triple-peaked variability structure that is due to a triple peak in convective heating seen in past results (FMK12; FMK13). On the other hand, the CCON setup shows larger variability throughout the domain but with a much flatter peak. However, as will be shown later, the CCON setup is degenerate and extremely sensitive to parameters, and we emphasize that it is important to favor realistic over larger variability.

### a. Deep-convergence-coupled DCON

In the formulation above, two key parameters were introduced: *R* and *τ*_{w}. Of these two, the parameter *τ*_{w} explicitly tunes the strength of the convergence coupling through (14), while *R* has a more subtle effect. In this section, we study the effect of varying *τ*_{w}, which we will often refer to as the “convection strength” or “strength parameter” in the context of the deep-convergence-coupled DCON simulations. In particular, we perform simulations fixed at *R* = 240 km and for *τ*_{w} = 0, 1, 10, 100, and 1000 h. Of course, *τ*_{w} = 0 implies that the convergence coupling is disabled, which is the same as the THERMO setup following (14).

This deep-convergence-coupled setup shows improved low-frequency variability and intermittent dynamics, as is readily visible in the anomalous *u*_{1} Hovmöller diagrams shown in Fig. 4. From left to right with increasing *τ*_{w}, the Hovmöller diagrams represent a continuum from order to disorder. As discussed in the previous section, the majority of the variability in the THERMO simulation (*τ*_{w} = 0) is composed of large CCWs that emanate from the warm pool region at regular ~12-day intervals. Moreover, these CCWs alternatively propagate eastward and then westward in perfect sequence. For *τ*_{w} = 1, this structure is still somewhat visible, but the coherence and regularity of these waves is weakened. With *τ*_{w} = 10, large CCWs similar to those in THERMO but with a 2-times-longer time scale alternatively propagate east/west until one circles the domain and interacts once more with the warm pool region (see day 930). This interaction then initiates a series of many small and large CCWs, which are released from the center of the domain. This behavior is markedly more chaotic and features variability on longer time scales then the regular east-then-west waves in the THERMO. The effect is increasingly pronounced for *τ*_{w} = 100 and 1000.

The *u*_{1} climatology shown in the left panel of Fig. 5 reflects this increased variability outside the warm pool region. While the mean fields of the convergence-coupled simulations do show a slightly stronger circulation between 15 000 and 25 000 km, the result is quite subtle. On the other hand, there is a large increase in variability with *τ*_{w}, especially outside of the warm pool region. This is evidently due to the propagating CCWs visible in Fig. 4.

We note here that, for a range of parameter values, simulations that disabled the thermodynamic coupling for deep convection—by fixing the CAPE proxy *C* at a constant—displayed aphysical behaviors. These solutions consist of a single large-amplitude quasi-steady wave that continuously circles the domain and weakly interacts with the SST gradient (not shown). This behavior is very similar to the behavior of the fixed cloud fraction model, which indicates that, for the current scheme and parameter values, thermodynamic and convergence coupling should be combined. In more comprehensive models, which can create convergence through many mechanisms, such as meridional advection and boundary layer processes, a successful convergence-only stochastic coupling might be possible.

#### 1) Enhanced persistence of equatorial waves

In this section, we present qualitative and quantitative evidence for the enhanced persistence of the CCWs in the regions between 25 000 and 35 000 km and between 5000 and 15 000 km. We will here refer to these regions as the flanks of the warm pool.

The large CCWs in the flank regions of the convergence-coupled simulations have an interesting phase speed and wave structure. Figure 6 shows a zoomed-in Hovmöller diagram of one such example in the *τ*_{w} = 10 simulation. The wave is generated in the warm pool, and, as it propagates eastward, its phase speed is reduced when it exits the warm pool region around 25 000 km. Moreover, the wave appears to be partially sustained by reciprocal interactions with the warm pool via fast-moving gravity waves. This is a consequence of the convergence coupling, which enables the interaction of dry waves with moist waves. The dynamical fields are averaged along the traveling wave in the two marked segments, and the resulting wave structure is plotted in Fig. 7. As the wave leaves the warm pool and slows, it transitions from congestus-dominated to deep-dominated heating. This occurs because the available energy in the dry region for congestus convection is much lower than that available for deep convection.

*H*

_{c}to deep heating

*H*

_{d}. This effect can be quantified by seeing how well

*H*

_{c}for a particular spatial location

*x*

_{0}predicts

*H*

_{d}in other spatial locations. In particular, the diagnostic we use is the lagged correlation function given by

To identify waves propagating in the flank regions, a seed location of *x*_{0} = 30 000 km is used. The results for the THERMO and DCON simulations are available in Fig. 8. These plots are quite similar to the Hovmöller diagrams shown in Figs. 3 and 4, but they filter for CCWs in the flank region and represent an average over many individual wave events. For both THERMO and DCON, the large CCW near the *x*_{0} is clearly visible as a line of high-correlation coefficients that extends toward the center of the domain. In the THERMO simulation, the waves appear to dry and decohere around the seed of *x*_{0} = 30 000 km. For the DCON simulation, these waves propagate with the same phase speed for an additional 7 days until the correlations vanish around 35 000 km because this result is an average over all flank-region CCWs and provides a quantitative confirmation of the qualitative results in Figs. 4 and 6.

Another interesting consequence of the convergence coupling is that it appears to reduce long-distance lagged correlations. In the THERMO results, the seed location strongly correlates with a wave around 15 000 km on the other side of the warm pool. This is likely because the flank-region CCWs in the THERMO scheme are much more strongly linked to convective activity in the warm pool region between 15 000 and 25 000 km. There are no similar long-distance correlations in the DCON scheme, so the DCON scheme appears to discourage this link. Put another way, convergence coupling encourages interaction with local atmosphere in the flank regions, rather than slaving it to the convective activity in the warm pool.

#### 2) Sensitivity to SST gradient

A simple way to enhance persistence in an idealized Walker cell simulation is to make the dry regions moister by reducing the amplitude of the imposed SST pattern. Here, this is accomplished by reducing *A*_{SST} from 5 to 4.5 K, which amounts to a 1-K reduction in the difference between moist- and dry-region

The lagged correlation results for the THERMO and DCON setups are available in Fig. 9. The wave persistence is indeed enhanced in the THERMO simulation with a weaker warm pool, but there are still important differences between the setups. The wave in the THERMO simulation shows a broader correlation structure in time near *x*_{0} = 30 000 km, and it shows correlations with a westward-propagating wave at a lag of 5 days. Qualitatively, the correlation structure is similar to that seen in Fig. 8.

The DCON simulation, on the other hand, shows a very localized wave that does not correlate with any westward waves. This mirrors the results in the previous section. Moreover, the wave shows strong lag correlations with an eastward-traveling wave at *x* = 0 km at a lag of 7.5 days. This eastward-traveling wave is likely generated in the dry region by convergence because of dry gravity waves emanating from the CCW around *x* = 30 000 km. This reemergence of moist waves is not present in the THERMO simulations, and it is clear that the enhanced persistence via convergence coupling is a distinct effect.

#### 3) Moisture budget

*P*is revealing. Unlike, the wave propagation diagrams in Fig. 8, this quantity is calculated for each spatial location separately and is given byFigure 10 contains this quantity for the standard THERMO and DCON simulations.

Both simulations have a similar structure, with three general types of relationship between *P*. First, moisture convergence is negatively correlated with precipitation in the warm pool region, which is a result of the heavy congestus and stratiform heating in this region. In the dry regions near 0 km, the moisture convergence and precipitation are positively correlated for several lags. Finally, the flank regions (e.g., 27 000 km) show an interesting regime where *ρ*(*x*, ±*τ*) < 0 for *τ* > 0.25 days, but *P* are positively correlated for shorter lags. This last moisture budget regime corresponds to the passage of a wave like that seen in Figs. 6, 7, and 8. For the current purposes, CISK is defined by a two-way feedback between moisture convergence and precipitation. However, in the flank-region regime, moisture convergence predicts total heating, but total heating is anticorrelated with precipitation for larger lags.

As the THERMO simulations show, even a QE-based scheme can have regions where the moisture budget shows some characteristics of CISK (e.g., 0 km) and other regions where surface fluxes are of primary importance (e.g., warm pool). Moreover, because the same three regimes are present in both the THERMO and DCON simulations, it is clear that the method of convergence coupling considered here does not fundamentally change the thermodynamics of the scheme. In other words, DCON does not act like the Kuo-like moisture-convergence closure that was criticized in (Emanuel et al. 1994). The DCON scheme simply alters the location of these three moisture budget regimes, and the flank-region moisture budget regime covers a much larger swath of the domain (e.g., 25 000–35 000 km). This is precisely the same region where the CCW propagation was enhanced (cf. Fig. 8).

### b. Congestus-convergence-coupled CCON

At this point, we digress to explore an interesting negative result. Namely, we claim that coupling congestus clouds to the convergence is highly unrealistic and should be avoided in the development of prototype cumulus parameterizations. Given the results of Figs. 2 and 3, one might naively expect that the CCON regime, which replaces the moist convergence mechanism with a dry-convergence mechanism, will perform comparably to a deep-coupled regime. This is, however, not the case, because unlike the deep-coupled setup, the CCON setup shows strong sensitivities to the key parameters *τ*_{w} and *R*.

The CCON setup is overly sensitive to the convergence-coupling strength parameter *τ*_{w}, a fact that the climatology of *u*_{1} available in Fig. 11 clearly demonstrates. As *τ*_{w} is increased, the strengths of the mean circulation and the variability are noticeably decreased. The CCON parameterization appears to shut down the circulation for these large values of *τ*_{w}. One potential explanation for this malignant behavior can be seen be varying the parameter *R*.

The interaction radius *R* explicitly controls the scales over which the large-scale convergence field interacts with the grid-scale convection. Ideally, one would prefer the dynamics to internally set this scale. In other simulations (not shown), the DCON scheme was insensitive to this parameter. Moreover, convection in the warm pool region is essentially unaltered in the DCON simulation compared to the control (THERMO). On the other hand, convection in the warm pool region in the CCON simulation is strongly dependent on *R*.

In this section, we demonstrate the interaction radius sensitivity by performing numerical experiments for interaction radii *R* = 80, 160, 240, and 480 km with a fixed value *τ*_{w} = 10 h. The zoomed-in 15-day snapshots of the congestus cloud fractions *σ*_{c} shown in Fig. 12 provide a possible explanation for this. Compared to the THERMO simulation, the cloud fractions are smaller overall and show a noisier background state. In particular, the cloud fractions are slaved to dry waves that emanate from the larger CCWs and propagate with a speed of 25 m s^{−1}. These dry waves carry elevated cloud fractions that occasionally interact to produce large-enough cloud fractions to initiate a large CCW. Both the dry and convectively coupled waves noticeably increase in horizontal extent (decrease wavenumber) with *R*.

These larger waves appear to interact more strongly with one another than with the mean circulation, which increases the variability but decreases the strength of the climatological circulation. This profound sensitivity to the parameters *R* and *τ*_{w} reflects the explicit role convergence coupling plays when attached to the formation of congestus clouds. In fact, this scheme shows evidence of grid-scale convection as *R* is decreased, which is a hallmark of CISK. This is in contrast to the attractive results seen above for the DCON formulation.

## 5. Conclusions

In this study, we have modified the stochastic multicloud model to include the nonlocal effects of convergence coupling. This is motivated by recent work showing the importance of the convergence coupling in column multicloud models run in a diagnostic setting (Peters et al. 2013; Dorrestijn et al. 2015). However, these diagnostic studies cannot address the dynamical criticisms of convergence coupling provided by (Emanuel et al. 1994) and others. The present study addresses these traditional criticisms by implementing convergence coupling in a fully prognostic spatially extended setting.

We conclude that the addition of convergence coupling does have beneficial effects if implemented in a suitable way. To be specific, coupling the transition from congestus to deep clouds to both the large-scale convergence and local CAPE enhances the persistence of convectively coupled waves in nonlinear idealized warm pool simulations. Because there is no rotation in the model, these waves are analogous to equatorial Kelvin waves in the real atmosphere. Therefore, the soft nonlocal convergence coupling presented here potentially describes the remarkable ability of atmospheric Kelvin waves to sometimes propagate unimpeded across the eastern Pacific and the Andes mountain range (Straub and Kiladis 2002; Kiladis et al. 2009).

This scheme also shows an attractive but subtle sensitivity to the convergence-coupling strength *τ*_{w}, which results in chaotic time series with rich low-frequency content. This behavior likely results because the convergence coupling enables a reciprocal interaction of dry and moist waves. The interaction of moist and dry waves is a mechanism well-known for creating gregarious multiscale organized convection (Mapes 1993; Stechmann and Majda 2009). Moreover, it is desirable that the setup shows low sensitivity to the tuning parameter *R*, the interaction radius. Indeed, extreme sensitivity to this and other parameters is a key symptom of the degenerate congestus-convergence-coupled scheme. The latter serves as an example of how not to implement convergence coupling, as shown in section 4b.

The deep-convergence-coupled setup does not fundamentally alter the thermodynamics of convection compared to the original multicloud formulation. This form of nonlocal convergence coupling is not a moisture-convergence closure like the Kuo schemes, and it does not show unattractive CISK-like behavior, as shown in section 4a. Indeed the current results complement the evidence that the transition from shallow to deep convection is promoted by vertical moisture transport (Hagos et al. 2014). On the other hand, coupling the congestus clouds to the convergence field shows unattractive characteristics reminiscent of CISK. This reflects the intuition that the formation of congestus clouds is driven by boundary layer dynamics rather than the free-tropospheric convergence field.

It is unknown how the addition of rotation and another horizontal spatial dimension will affect these results on nonlocal convergence coupling, so extending the present work to a more realistic atmospheric simulation is an interesting avenue of future research. There is existing work on implementing the stochastic multicoud model in a full atmospheric GCM (Ajayamohan et al. 2013, 2014; Deng et al. 2015; Peters et al. 2015) that can be leveraged for these purposes. Also, while the model studied here includes the effects of stratiform heating, the formation of stratiform clouds is not explicitly coupled to the winds in any way. Coupling the stratiform clouds to vertical velocity and/or shear in idealized Walker circulation simulations is another interesting research direction.

In summary, this paper indicates that nonlocal convergence coupling potentially plays an important role in mediating interactions between convection and a large-scale SST-driven circulation. This mechanism is distinct from the wind-induced surface heating mechanism. We stress here that, in models with nonhomogeneous SSTs, the relationships between terms in the moisture budget often depend on the region. In moist regions with high SST, surface heat fluxes can play a key role, but in drier/colder regions, precipitation is frequently associated with large-scale moisture convergence. Coupling the transition from congestus to deep clouds appears to beneficially alter dynamics in these drier regions, while leaving convection in the moist regions largely untouched.

## Acknowledgments

The authors thank Boualem Khouider for stimulating discussion on this subject and Olivier Pauluis for helping improve the presentation of these results. The authors also thank Mitchell Moncrieff and an anonymous reviewer for their comments, which helped improve the manuscript. The research of A. J. M. is partially supported by the Office of Naval Research MURI award Grant ONR-MURI N-000-1412-10912. Y. F. is a postdoctoral fellow supported through the above MURI award, and N.D.B. is supported as a graduate student on the MURI award.

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