Reproducibility of Equatorial Kelvin Waves in a Super‐Parameterized MIROC: 2. Linear Stability Analysis of In‐Model Kelvin Waves

While low‐resolution climate models at present struggle to appropriately simulate convectively coupled large‐scale atmospheric disturbances such as equatorial Kelvin waves (EKWs), superparameterization helps better reproduce such phenomena. To evaluate such model differences based on physical mechanisms, a linearized theoretical framework of convectively coupled EKWs was developed in a form readily applicable to model evaluation by allowing background stability and diabatic heating to have arbitrary vertical profiles rather than assuming simplified ones. A system of linearized equations of convection‐coupled gravity waves was derived as a two‐dimensional model of the convectively coupled EKWs. In this work, the basic states are taken from observations, CTL‐MIROC and SP‐MIROC experiments introduced in Part 1. The tendency of convectively coupled gravity waves to grow faster under top‐heavy heating is confirmed for realistic stratification profiles, as found in previous studies under constant stratifications. A comparison of linear unstable solutions with basic states taken from SP‐MIROC and CTL‐MIROC shows that the top‐heavy heating profile in SP‐MIROC largely contributes to the enhancement of the EKW‐like unstable modes, while subtle differences of stratification profiles considerably affect EKW behaviors. The bottom‐heavy heating bias in the CTL‐MIROC likely originates from insufficient modeling of subgrid stratiform precipitation in tropical organized systems. It is desirable to incorporate such stratiform components in cumulus parameterizations to achieve better EKW reproducibility.

Earlier theoretical studies of convection-wave coupling around the equator were focused on the MJO rather than the EKW.Emanuel (1987) assumed that the energy pumped into the atmosphere through the wind-induced surface heat exchange (WISHE) is immediately consumed by convection, thereby always satisfying an assumption of quasi-equilibrium in each column.Growth rates in zonal-meridional two-dimensional linearized equations were larger for higher wavenumbers; this result was inconsistent with the large-scale nature of the MJO.Emanuel (1987) argued that this defect originated from the immediate vertical redistribution of surface heat fluxes before induced anomalies could be conveyed anywhere horizontally.In reality, gravity waves smoothen out buoyancy anomalies to meridional extents on the order of the equatorial deformation radius.Thus, he argued that only the lowest-wavenumber modes, which had spatial structures and phase speeds similar to MJOs, were physically valid in his framework under the strict quasi-equilibrium assumption.
Subsequent studies aimed at resolving the problem related to high wavenumbers by either coupling the system to the stratosphere and allowing some wave energy to escape upward (Yano & Emanuel, 1991), or relaxing the convective quasi-equilibrium assumption (Emanuel, 1993;Emanuel et al., 1994;Neelin & Yu, 1994).By allowing the convection to lag behind large-scale forcing, higher-frequency modes can be damped, leaving behind MJO-like small zonal wavenumber (1-4) disturbances.These studies modeled the MJO using simple mechanisms, but higher-wavenumber EKWs were generally beyond their scope.
In contrast, Mapes (2000) modeled the convection-coupled EKW (CCEKW) in a linearized set of two-layer equations in the zonal direction.He assumed that the amount of convective heating depends both on the convective available potential energy (CAPE) and convective inhibition (CIN).He considered two types of latent heating sources: deep convective heating and stratiform heating lagging behind the deep convection by 3 hr.He found that all the variability was damped out when CAPE dominated the heating.In contrast, in the CINcontrolled regime, stratiform precipitation allowed Kelvin-like waves to grow.Khouider and Majda (2008) investigated similar convection-wave coupling in linearized three-dimensional equations with the beta-plane approximation of the Coriolis parameter.They simplified the vertical structures of disturbances using a Galerkin projection by retaining the first two baroclinic modes but discarding the others.After considering many types of diabatic forcing, such as radiative cooling, deep convective heating and stratiform heating, they concluded that Kelvin, Yanai, and M = 1 westward inertia-gravity waves grow.Han and Khouider (2010) introduced background shear to the framework of Khouider and Majda (2008).They found that vertical wind shear with upper-level easterly tends to destabilize westward-propagating waves while suppressing the growth of eastward-propagating ones, including EKWs.Meridional shear, if strong enough, can destabilize mixed Rossby-gravity waves (MRGs) and equatorial Rossby waves without any diabatic heating.In contrast, vertical shear cannot trigger such shear instability without diabatic heating.Stechmann and Hottovy (2017) proposed another framework for three-dimensional convection-coupled equatorial waves (CCEWs), where some components of the diabatic forcing were stochastic rather than deterministic.Under their stochastic equations, MJOs, EKWs, westward inertia-gravity waves, and equatorial Rossby waves appeared as peaks in the spatio-temporal spectrum of precipitation.They also obtained the eigenmodes of the MJO and the CCEKW from their equations by excluding the stochastic components in the heating terms.
Although the aforementioned three-dimensional theories (Han & Khouider, 2010;Khouider & Majda, 2008;Stechmann & Hottovy, 2017) could reproduce various equatorial waves in a unified manner, their vertical truncation to a few modes made them unsuitable for evaluating numerical simulations with detailed vertical structures.
In contrast, Ž. Fuchs et al. (2012) linearized vertically resolved Boussinesq equations in the x z plane and investigated the behaviors of convection-coupled gravity waves (CCGWs).Since the equations for pure EKWs can be split to geostrophic balance in the meridional direction and the nonrotating gravity wave in the zonalvertical plane, CCGWs in the x z plane are considered reasonable proxies for convective-coupled EKWs in three-dimension (Mapes, 2000;ž. Fuchs & Raymond, 2007;Raymond & Fuchs, 2007).Ž. Fuchs et al. (2012) assumed uniform background static stability and parameterized the vertical profile of diabatic heating as sinusoidal-exponential functions, which were fixed to zero at both the domain top and bottom and peaked at a configurable altitude between them.They assumed that the vertical integral of diabatic heating is governed by two factors: precipitable water and CIN.Specifically, they assumed that the positive anomaly of precipitable water enhances precipitation, while larger CIN suppresses it and results in negative heating anomalies.Under these conditions, two growing modes emerged: a stationary "moisture mode" and the CCGW.This moisture mode was fed by positive feedback among precipitable water anomaly, latent heating anomaly, and moisture imports accompanying updrafts.The CCGW mode, which corresponds to CCEKWs in three dimensions, grew faster under top-heavy heating profiles.They also performed numerical simulations of the linearized equations in three dimensions under beta-plane rotation and found that the behavior of the CCGW mode was similar to that expected in the two-dimensional theory.Although Ž. Fuchs et al. (2012) demonstrated that a top-heavy diabatic heating profile supports stronger CCEKWs, the "heating top heaviness" in their theory could not be directly compared to that in reality because they imposed a constant N 2 and idealized sinuous heating profiles, which are not applicable for the tropical troposphere.
These works revealed various features of convection-coupled equatorial waves, but their applications were limited by the simplified vertical structures.All variables were truncated to a few vertical modes in some studies (Han & Khouider, 2010;Khouider & Majda, 2008;Mapes, 2000;Stechmann & Hottovy, 2017), while a constant background static stability was assumed in others (ž.Fuchs & Raymond, 2007;Raymond & Fuchs, 2007;Ž. Fuchs et al., 2012).In reality, the squared Brunt-Väisälä frequency (N 2 ) is less than 0.5 × 10 4 s 2 around 200 hPa, while it is greater than 1.5 × 10 4 s 2 in the lower free troposphere (shown later).Thus, for the quantitative assessment of CCGWs in reality, theories assuming constant N 2 throughout the troposphere are not ideal.Furthermore, this constraint prevents a detailed comparison of CCGWs between simulated and observed environments; such a comparison is thought to be essential for understanding the mechanisms of model biases and mitigating them.Kiladis et al. (2009) investigated the structures of CCEWs by comparing composites constructed from satellite observations, model reanalyzes, and radiosonde observations.They found a tilted vertical structure of temperature anomalies in EKWs.The anomalies are tilted westward below 300 hPa and eastward above 300 hPa, resulting in a boomerang-like shape.Around the bending altitude or the "elbow" of the boomerang-shaped temperature anomaly (∼300 hPa), the updraft and warm anomalies are positively correlated, implying the generation of the available potential energy (APE).The westward and eastward tilting of gravity wave phases are associated with downward and upward energy transport, respectively, which also suggests APE generation around the elbows of the waves.Nakamura and Takayabu (2022) quantified the APE budget of EKWs, demonstrating the APE production by diabatic heating in phase with warm anomalies around 300 hPa.Because this APE-producing altitude is located in the upper troposphere, top-heavy heating profiles are favorable for the CCEKW growth as the APE production is maximized.Superparameterized (SP)-GCMs, proposed around 2000 (Grabowski, 2001;Grabowski & Smolarkiewicz, 1999) and implemented recently in multiple GCMs (Hannah et al., 2020;Khairoutdinov et al., 2005), tend to maintain the amplitude of EKWs stronger than the original GCMs and closer to observations (Benedict & Randall, 2011;Hannah et al., 2020).Existing theories suggest that the top-heavier heating profile in SP-GCMs contributes to reinforced EKWs.However, background values of N 2 typically differ among conventional GCMs, their SP counterparts, and the observed values, thereby complicating the discussion.It is expected that the top heaviness of latent heating, in terms of CCGWs, depends on the profiles of both latent heating and N 2 because the amplitudes of heating-induced motion is regulated by the static stability.Thus, it is desirable if a framework to evaluate the favorableness of both heating and N 2 profiles for CCGWs is available.
In this study, a theoretical framework to evaluate the growth of the CCEKW was constructed in a form directly applicable to model outputs.Specifically, the CCGW theory presented by Ž. Fuchs et al. (2012) was expanded to arbitrary vertical profiles of N 2 and diabatic heating.This expansion made it possible to compare simulated EKWs in terms of not only their statistical features but also their physical mechanisms.We used the fields simulated in the standard MIROC6 (Tatebe et al., 2019) and its SP version (SP-MIROC) described in our companion paper (Yamazaki & Miura, 2024a, Part 1 hereafter), as well as observational data sets, as the basic states for the linear analyses.Thus, this study serves not only to develop a theoretical framework of the CCEKW, but also to investigate the EKW enhancement in SP-MIROC introduced in Part 1.The rest of the paper is organized as follows.The equations and data for the stability analyses are introduced in Section 2. The results are provided in Section 3 and discussed in Section 4. A summary of this paper is given in Section 5.

Basic Equations
A set of linearized primitive equations in the x-log(p) coordinate is considered.Here, x denotes the zonal coordinate, and p is the pressure.For simplicity, the meridional coordinate is not considered.This simplification to the two-dimensional (2D) coordinate possibly has only a minor impact because the 3D linear model and 2D linear theory by Ž. Fuchs et al. (2012) produced similar results in terms of spatial structures and tendency to favor topheavy heating.The vertical coordinate was defined as The prognostic variables were u, w*, T, and Φ, representing horizontal wind, vertical wind, temperature, and geopotential, respectively.The basic state is motionless and has a horizontally uniform temperature profile T.
where H(x, z*, t) is a parameterized diabatic heating function discussed later; B′ and N* 2 are defined using the gravity acceleration (g), potential temperature (θ) and its basic state θ) as shown below: Ž. Fuchs et al. (2012) parameterized the column-integrated diabatic heating using two terms: one term was proportional to the anomaly of precipitable water, thereby allowing for stationary "moisture modes," and the other mimicked CIN, thereby destabilizing the CCGWs (Mapes, 2000).Yasunaga and Mapes (2012) observationally supported larger contribution of the precipitable water anomaly to rotational waves linked to the "moisture mode" than to divergent ones, such as the CCGW.To focus on CCGWs, only the CIN term is used in this study.Raymond and Fuchs (2007) formulated the CIN term using the buoyancy anomaly in the lower free troposphere and the moist entropy anomaly in the boundary layer, which is closely related to surface fluxes and gives birth to the WISHE mechanism.However, Ž. Fuchs et al. (2012) noted that the WISHE mechanism did not significantly influence their model results.Thus, the diabatic heating anomaly is parameterized in a very simple formula using a fixed height of 2.5 km indicating the CIN threshold layer: where h(z) is an arbitrary vertical heating profile normalized by temperature anomalies at z = 2.5 km to model the tendency of precipitation-related heating to be stronger when CIN represented by B′(z = 2.5 km) is smaller.The validity of this assumption is discussed in Section 4.1.
By assuming a sinusoidal dependence of all prognostic variables in the x direction with a wavenumber k and eliminating u′ and Φ, we transform Equations 1-4 to the following form.
Journal of Advances in Modeling Earth Systems 10.1029/2023MS003837 YAMAZAKI AND MIURA Eigenmodes of Equations 8 and 9 were obtained by vertically discretizing the system to 200 levels from 1,000 to 20 hPa and reducing the system to an eigenvalue problem of a matrix.The eigenvectors represent the spatial structures of eigenmodes, and eigenvalues represent phase speeds and growth rates.See Appendix A for details of the discretization procedure.

Basic States
Equations 8 and 9 require the basic-state stratification T,N * 2 ) and a normalized latent heating profile h(z).Four data sources were used to set up these profiles: the ERA5 reanalysis (Hersbach et al., 2020) for stratification, the TRMM 3G25 V7 product (Shige et al., 2004(Shige et al., , 2007(Shige et al., , 2009;;Tao et al., 2016) for latent heating, and CTL-MIROC and SP-MIROC models (as described in Part 1) for both stratification and latent heating.All profiles were averaged in 10°S-10°N, 0°E 360°E.The ERA5 stratification was averaged for the period 1981-2010, while the stratification in CTL-MIROC and SP-MIROC was averaged for 5-year model runs.Heating profiles were constructed from TRMM estimation or MIROC outputs by linearly regressing latent heating to temperature anomalies defined as 4-6-day band-passed temperatures at z = 2.5 km.Regressed heating profiles are then multiplied by 4 for better agreement to composited amplitudes by Nakamura and Takayabu (2022); the ratio of peak heating rate to the low-level temperature anomaly was approximately 5 day 1 (ex.peak heating of 5 K day 1 when the temperature anomaly is 1 K).Here, only the 4-6-day temporal filter is applied to temperature and heating, while the amplitude by Nakamura and Takayabu ( 2022) is from variables filtered by an EKW-specific spatio-temporal filter.This difference may be responsible for the difference in heating amplitudes.In preliminary experiments without the 4× amplitude adjustment, the coupling between heating and waves was insufficient and prevented EKW-like modes from outgrowing unnatural perturbations.Note that diabatic heating other than latent heat of water is not included in this study.Although the amplitudes of other heating sources are smaller than that of the latent heating, the contribution of other sources to EKWs is an interesting topic for future studies.The combination of the ERA5 stratification and TRMM heating is called the "Standard" experiment modeling the real world.The stratification and heating profiles derived from those sources are shown in Figure 1 (a, b).The stratification of ERA5, CTL-MIROC, and SP-MIROC are roughly similar, whereas the peak heights of the latent heating are notably different: the TRMM-estimated heating peaks at 400 hPa, while it peaks at 600 hPa in CTL-MIROC and at 500 hPa in SP-MIROC.In this study, the terms "top-heavy" and "bottom-heavy" are used in a relative sense.Thus, the SP-MIROC that peaks at 500 hPa is top-heavy in comparison to the CTL-MIROC profile with a peak at 600 hPa.
To understand the effect of heating amplitude, the TRMM-derived heating profile was multiplied by two in the "Double" experiment and by 0.5 in the "Half" experiment, respectively.Furthermore, to investigate the sensitivity of CCGWs to top heaviness in realistic stratification, the TRMMestimated heating profiles were vertically scaled by 1.1 in the "Top-heavy" experiment and by 0.9 in the "Bottom-heavy" experiment.The profiles employed in these sensitivity experiments and the original TRMM heating profile are compared in Figure 1c.The heating profile used for the Bottomheavy experiment would be described as top heavy in convention because it peaks around 450 hPa, which is lower than the TRMM-derived peak but higher than the 0°C level.However, we refer to the sensitivity experiment as the Bottom-heavy experiment in this paper for convenience in comparison to the Standard experiment.To compensate for the changes in columnintegrated heating in these modified heating profiles, the values of the rescaled profiles were divided by 1.1 for the Top-heavy experiment and by 0.9 for the Bottom-heavy experiment, resulting in different peak values (Figure 1c).
To test for necessity of realistic stratification profiles, a constant stratification of N 2 = 10 4 s 2 was deployed in a sensitivity experiment called "Simple," similar to the basic state assumed by Ž. Fuchs et al. (2012).In this experiment, the domain top was lowered to 100 hPa because the stratospheric high stability could not be represented under a constant N 2 .The sinusoidalexponential expression of the heating profile employed by Ž. Fuchs et al. ( 2012) is adopted in the Simple experiment: where the parameter a ≡ 0.4 determines the peak altitude of the heating.a = 0.4 corresponds to a peak heating altitude of around 230 hPa (Figure 1c).Here, the heating peak is configured to be much higher than that in the real world because no growing modes emerged at lower heating peaks, such as 260 hPa.The threshold of the heating peak for disturbance growth is different from that obtained by Ž. Fuchs et al. (2012) because the Boussinesq approximation adopted by them is not employed in this study.Removal of the Boussinesq approximation requires stronger circulation in the upper troposphere to satisfy the continuity (Equation 3), which strengthens the stabilizing effect of N* 2 w*′ in higher altitudes.This requires the heating to be top-heavy to compensate for the stronger stabilizing effect in the upper troposphere under non-Boussinesq equations.
To investigate more deeply the differences in EKWs between CTL-MIROC and SP-MIROC, mix-up sensitivity experiments were conducted.Heating from CTL-MIROC and stratification from SP-MIROC were combined in the H CTL N 2 SP experiment, and vice versa in the H SP N 2 CTL experiment.In total, 10 basic states were constructed (Table 1) and spatial structures and growth rates of disturbances were compared among those basic states.

Wave Composites
To compare the spatial structures of the linear growing modes with the actual EKWs, we performed composite analyses for reanalyses and model simulations.The analysis procedures were identical to those described in Part 1.We used the NOAA Interpolated Outgoing Longwave Radiation (OLR) (Liebmann & Smith, 1996) and ERA5 temperatures and winds; the OLR and ERA5 data were collected daily from 2001 to 2005, and they served as an observational reference.Daily fields from 5-year AGCM experiments using CTL-MIROC and SP-MIROC were used for the in-model composites.Those fields were first filtered using a EKW-specific spectral window (zonal wavenumber: 4 to 8, period: 4-6 days, equivalent depth: 10-100 m).Then, filtered variables were composited with respect to zonal phases of EKW-filtered OLR.See Nakamura and Takayabu (2022) and Part 1 for detailed procedures.The compositing results (Figure 2) clearly show that the two model runs and the realworld reanalysis reproduce a boomerang-shaped structure of the temperature and wind anomalies in EKWs.This feature is consistent with those reported in previous studies (Kiladis et al., 2009;Nakamura & Takayabu, 2022) and implies that wave energies are radiated from the bending altitude.

Results
The dispersion relations of the fastest-growing modes are shown in Figure 3  Figure 3d reveals that the growth rate depends strongly on the amplitude of the heating as the results of the "Double" and "Half" heating experiments show considerably faster and slower growth, respectively, compared to that in the "Standard" experiment.
Structures of the fastest-growing modes at K = 6 are shown in Figure 4.The zonal wavenumber of K = 6 was selected because it is located in the middle of the EKW filter range used to create the observational composites.
Changing the values of K hardly affects the spatial structures of the linear growth modes (figure not shown).
Figure 4 shows that the growing modes in the basic states in all experiments, except for the "Simple" experiment, reproduce boomerang-shaped structures of temperature and wind perturbations; these results are in agreement with the composites of the actual in-model EKWs (Figure 2).In contrast, under the "Simple" basic state, buoyancy anomalies are tilted westward throughout the troposphere, and the boomerang-shaped structure is not observed at all.Furthermore, the heating peak in the "Simple" basic state is excessively high (around 230 hPa), and when the peak altitude is lowered, no growing disturbances emerged.This result demonstrates that a constant N 2 profile cannot support the realistic growth of EKWs with observed heating.This means that a realistic representation of the basic stratification is essential for appropriately evaluating the environmental favorability for EKWs.
A comparison of the Double, Standard, and Half basic states (Figures 4a-4c) indicates that the amplitudes of diabatic heating hardly affects perturbation structures.Thus, the heating amplitude serves to manipulate the growth rates, while phase speeds and spatial structures are less sensitive.
A comparison of the Standard, Top-heavy, and the Bottom-heavy basic states highlight the sensitivity of the CCGW to heating altitudes in a realistic stratification.The boomerang-shaped structure is common to these three states (Figures 4d,4e,4f); however, the growth rate varies systematically: the higher the heating peak, the faster is the growth of the perturbation (Figure 3f).This result allows us to extend the conclusion reported by Ž. Fuchs et al. (2012) to realistic stratification profiles.
The CTL-MIROC and SP-MIROC basic states result in growing modes (Figures 4h and 4i) featuring boomerang-like structures, although detailed structures are distorted from the Standard basic state and in-model composites (Figure 2).Waves consisting the in-model composites are likely in quasi-equilibrium affected by nonlinear terms, while our growing modes are derived from a purely linear system.The discrepancy of the structures between our linear modes and wave composites suggests that nonlinear effects play a non-negligible role in determining the dominant structure of EKWs.CTL-MIROC and SP-MIROC yield similar phase speeds (Figure 3a), while the growth rates in SP-MIROC lie between those in CTL-MIROC and Standard, regardless of the zonal wavenumbers (Figure 3b).This result indicates that the SP-MIROC environment is more favorable for CCGW growth than that of CTL-MIROC but is less favorable than the reality; this inference is consistent with the in-model CCEKW amplitudes (Part 1).The mixed-up experiments showed that changing the heating profile from CTL-MIROC to SP-MIROC ("CTL-MIROC" to H SP N 2 CTL ; H CTL N 2 SP to "SP-MIROC") considerably increases the growth rate (Figure 3b).This finding indicates that the heating profile is responsible for the vigorous perturbation growth in SP-MIROC.The amplitude of heating is smaller in SP-MIROC than in CTL-MIROC (Figure 1b), which reduces the growth rate.However, the higher heating peak in SP-MIROC makes it more favorable for CCGW growth.
The results of the mixed-up experiments further indicate that the stratification profiles, as well as heating, can affect the growth rate.A combination of the heating profile from SP-MIROC and stratification from CTL-MIROC results in growth rates highly similar to those obtained under the all-SP-MIROC basic state (Figure 3b, red lines).In contrast, growth rates under the H CTL N 2 SP basic state are systematically lower than those in the all-CTL-MIROC basic state (Figure 3b, blue lines).Thus, the favorability of the basic state for the CCGW can depend on subtle differences in stratification, as well as heating profiles.This highlights the importance of accurate stratification for evaluating the CCEKW.
It would be convenient for theoretical analyses to represent the growth modes by a sum of a few dry gravity waves without heating.To test for the feasibility of such decomposition, structures and phase speeds of dry gravity waves were derived from the ERA5 stratification using the linear system described in Section 2, but setting to h (z) = 0 throughout the atmosphere.However, no dry gravity wave modes (Figure 5) manifest significant similarity to the TRMM-heated Standard basic state (Figure 4b).This finding indicates that the contribution of diabatic heating to the CCEKW is quite large.

Heating Parameterization in the Linearized CCGW Theory
The formulation of diabatic heating (Equation 7) in the linearized system in this study is highly simplified.We examined its validity by comparing it with satellite-based composites reported by Masunaga (2012) and with inmodel statistics of this study.Masunaga (2012) used satellite observation data sets to create lag composites between precipitation events and environmental anomalies.According to his results, precipitation is negatively correlated with temperature at 1,000-700 hPa and positively correlated with temperature at 600-300 hPa.This relationship holds true in the ERA5 reanalysis and MIROC experiments, as the precipitation anomalies are negatively correlated with temperature anomalies in the lower free troposphere (Figure 6).The heating parameterization in this study (Equation 7) assumes that the diabatic heating, which is mostly contributed by precipitation, is negatively correlated with temperature anomalies at 2.5 km.Since the 2.5-km altitude corresponds to pressure between 700 hPa and 800 hPa, this assumption is consistent with the observational study (Masunaga, 2012), the ERA5 reanalysis, and the MIROC-based model simulations (Figure 6).
While the heating profile h(z) is prescribed in this study to maintain the linearity of the equations, heating does not actually occur independent of wave-scale updrafts.Instead, vertical velocities associated with gravity waves should interact with the heating profiles and constitute joint eigenmodes of the actual CCGWs.As shown in Figure 7, when the heating peak was lifted in the linear system of this study, the peak updraft altitude shifted

Journal of Advances in Modeling Earth Systems
10.1029/2023MS003837 upward more than the heating peak did.While vertical velocities above 200 hPa appear to be less affected, they would contribute little to the perturbation-heating coupling because almost no heating occurs at such high altitudes.Thus, if the vertical velocities can provide feedback to the heating profiles, a strong response of the updraft profile can inflate the heating top heaviness, thereby increasing the sensitivity of CCGWs to the mean heating profiles.A nonlinear model will be needed to explore such coupling in detail.

Effect of Stratification on EKW Growth
Equation 9 indicates that the tendency of the buoyancy anomaly is influenced by both vertical advection and diabatic heating.Smaller values of N 2 at an altitude are expected to allow for stronger vertical winds, which will have an effect similar to that of stronger heating at that altitude.Thus, the uppertropospheric weak stratification in CTL-MIROC (Figure 1a) may act similarly to stronger heating in high altitudes, raising the effective heating peak and contributing to the slightly stronger EKW growth in the all-CTL-MIROC basic state than in the H CTL N 2 SP experiment.
To further verify the hypothesis of effective top-heavy modification by the stratification difference in CTL-MIROC, we performed another set of sensitivity experiments designed from the Simple basic state.In these experiments, the heating profile was the same as that in the Simple experiment (Figure 1c), but the stratification was modified so that N 2 increases or decreases, below or above the heating peak at 240 hPa (Figure 8).
The results summarized in Table 2 indicate that an increase of N 2 below the heating peak (400 hPa-Positive) and a decrease in N 2 above the peak (170 hPa-Negative) boost the growth rate from 0.02 day 1 to around 0.03 day 1 .In both of the cases, N 2 above the heating peak is relatively smaller than that below the peak.Hence, the competing heating profile becomes effectively top-heavy.In contrast, a decrease in N 2 below the heating peak (400 hPa-Negative) and an increase above the peak (170 hPa-Positive) suppress the growth rate to below 0.02 day 1 .This finding further supports the idea of effective modification of top heaviness by stratification: N 2 above the heating peak being relatively larger than below results in perturbation weakening, similarly to the behavior observed in the case of the lowered heating peak.

Process Contributing to the Bottom-Heavy Heating Bias in CTL-MIROC
The condensation heating profiles are known to differ between stratiform and convective precipitation (Houze Jr, 1982, 1989, 1997;Johnson, 1984;Mapes & Houze, 1995).Stratiform precipitation heats the upper troposphere and cools the lower troposphere, resulting in a top-heavy heating profile.In contrast, convective precipitation heats the entire troposphere, but the heating is mainly centered in the mid-level around 500 hPa, resulting in more bottom-heavy heating than in the case of stratiform precipitation.Thus, it is natural to hypothesize that convective precipitation is too dominant in CTL-MIROC, causing a bottom-heavy heating bias, as pointed out by Nigam et al. (2000) and Lin et al. (2004) for multiple GCMs.
According to satellite estimations, convective precipitation contributes to only around half of the total amount of precipitation in the tropics (Schumacher & Houze, 2003), indicating a significant contribution of stratiform precipitation to top-heavy heating.In contrast, almost all the precipitation in the tropical wet areas originates from cumulus parameterization in CTL-MIROC (figure not shown), both in the mean state and in the EKW timescale (4-6 days).The dominance of the cumulus parameterization is very reasonable because the model is incapable of resolving cumulus clouds or mesoscale convective systems (MCSs) because of its low resolution.However, the cumulus scheme (Chikira & Sugiyama, 2010) adopted in MIROC6 is designed to represent only the plume dynamics rising from the boundary layer, without accounting for the stratiform precipitation component.Thus,  To simulate top-heavy heating naturally in the conventional GCMs, it is desirable to handle organized systems involving stratiform precipitation within the cumulus parameterization, which dominates tropical rainfall in low-resolution models.Moncrieff et al. (2017) introduced multiscale coherent structure parameterization (MCSP), in which momentum transport and stratiform top-heavy heating associated with slantwise convection typical in MCSs are modeled.As expected, implementing MCSP in CAM5.5 (Moncrieff et al., 2017) and E3SMv1 (Chen et al., 2021) resulted in stronger EKWs.Although the MCSP currently contains an uncertain parameter controlling the convective-stratiform partitioning, such MCS-aware parameterizations are promising for mechanism-oriented improvements of tropical phenomena such as EKWs in the conventional low-cost GCMs.

Summary
A linearized theoretical framework of the convection-coupled EKWs was constructed in a form directly applicable to model evaluation, thereby enabling model intercomparison of not only the black-box results but also the physical mechanisms behind the phenomenon.
A set of linearized equations of the CCGW was devised to obtain a two-dimensional model for the CCEKW by extending the idealized framework of Ž. Fuchs et al. (2012) to arbitrary stratification and diabatic heating profiles.
The tendency of top-heavy heating to grow CCGWs, which was demonstrated by Ž. Fuchs et al. (2012) under constant N 2 throughout the domain, was confirmed to be valid under realistic stratification profiles as well.A comparison of linear unstable solutions with basic states taken from CTL-MIROC and SP-MIROC showed that the top-heavy heating profile in the latter largely contributed to the enhanced EKW-like modes.Further sensitivity experiments revealed that a bias of upper-troposphere stratification in CTL-MIROC partly compensated for the bottom-heavy heating bias.Furthermore, under a simplified configuration with constant N 2 and solid tropopause, unrealistically top-heavy heating is required to grow any disturbance.These results indicate that a realistic representation of stratification in linear analyses, which is made possible by this study, is necessary to evaluate the environmental favorability for CCEKWs.
Results of linear stability analyses presented herein, combined with MIROC simulation results (Part 1), lead to the conclusion that the enhancement of EKW activities in SP-MIROC is enabled by top-heavy latent heating calculated in the CRM.Over the rainy areas in the tropics, CTL-MIROC depends strongly on a cumulus parameterization scheme designed for convective precipitation, which has a lower heating peak.In contrast, SP-MIROC depends more on stratiform precipitation, which has a top-heavy heating profile, over the rainy tropics.This suggests that in CTL-MIROC, inadequate representation of subgrid stratiform precipitation accompanying organized cloud systems, such as MCSs, is the primary cause of weak EKWs.Hence, it is desirable to incorporate stratiform components in MCSs to cumulus parameterizations (e.g., Moncrieff et al., 2017) to reproduce top-heavy heating profiles and realistic amplitudes of EKWs in the conventional GCMs.
The linear model presented in this study can be extended in many ways, for example, by providing feedbacks of perturbation fields to heating profiles, nonzero basic-state winds, meridional coordinates, and finite-amplitude nonlinear interactions.Deterministic linear systems, such as the one used in this study, are inherently incapable of directly predicting the equilibrium amplitudes of perturbations, even though the growth rate are in qualitatively good correlation with the observed amplitudes in this study.Furthermore, our linear model failed to reproduce the exact structure of EKWs in MIROC-based experiments, as noted in Section 3. In contrast, fully nonlinear systems have to handle all multiscale interactions in order to precisely represent only one kind of perturbation and will likely need to be solved in a GCM-like 3D numerical model, which is not ideal for a GCM evaluation framework.Meanwhile, the stochastic linear equations adopted by Stechmann and Hottovy (2017) successfully reproduced the waves with finite amplitudes by allowing unstable convective forcing to be activated only intermittently under the stochasticity.Though the validity of this stochasticity-driven amplitude throttling as a proxy for nonlinear interactions is not yet clear, it will be interesting to evaluate CCEW amplitudes in realistic basic states using stochastic forcing.Such a finite-amplitude framework, if verified, can enable more comprehensive quantitative comparisons of vertical wave structures.

Appendix A: Discretization of the Linearized EKW Model
Here, a vertical domain extending from 1,000 hPa to 20 hPa 0 ≤ z * ≤ z * max = H ln 1000 hPa 20 hPa ) is discretized to n = 200 levels with uniform intervals: z * i = (i 1)Δz * (i = 1,2,…,n), where Δz * ≡ z * max / (n 1).The vertical derivatives are discretized using second-order accurate central-differences.Then, Equations 8 and 9 become: where O, A 1 , A 2 , B, and C are n × n matrices and the row (h 1 h 2 ⋯ h n ) in C corresponds to the vertical index corresponding to z = 2.5 km.Vertical velocities on the upper and lower boundaries are set to zero, as implied by the formulation of A 1 and A 2 .Since the 2n × 2n matrix L serves as the time derivative operator, its eigenvalues and eigenvectors are those of the time evolution as defined in Equations 8 and 9.This system has 400 eigenmodes because two variables are discretized to 200 layers.Westward-propagating modes are discarded because they have corresponding eastward-propagating modes with the same vertical structure, and EKWs always propagate eastward.

Figure 1 .
Figure 1.Vertical profiles of (a) N 2 and (b) normalized heating rate h(z) for the Standard (black solid lines), CTL-MIROC (blue broken lines), and SP-MIROC (red broken lines) experiments, respectively.In panel (b), 95% confidence intervals of the h(z) profiles are shown as shadings for the MIROC experiments.In panel (c), normalized heating profiles are shown for the Simple (brown broken line), Top-heavy (red solid line), Standard (black solid line), and Bottom-heavy (blue solid line).
with respect to zonal wavenumbers K ≡ 2πR e k, where R e is the radius of the Earth.Regardless of values of K, the phase speeds are similar (15-23 m s 1 ) among all basic states (Figures 3a, 3c, 3e) and are consistent with the observed EKWs.The growth rates depend mildly on K (Figures 3b, 3d, 3f), with the results of the Standard experiment showing a peak around K = 6.

Figure 2 .
Figure 2. EKW composites for (a) ERA5, (b) CTL-MIROC, and (c) SP-MIROC.Shading represents the temperature anomalies normalized by their standard deviations, which are displayed at the top of each panels Vectors represent zonal and vertical wind anomalies.

Figure 3 .
Figure 3. Dependence of (a, c, e) phase speeds and (b, d, f) growth rates on zonal wavenumbers.

Figure 4 .
Figure 4. Fastest-growing, eastward-propagating eigenmodes at K = 6 in (a) Half, (b) Standard, (c) Double, (d) Bottom-heavy, (e) Standard, (f) Top-heavy, (g) Simple, (h) CTL-MIROC, and (i) SP-MIROC experiments, respectively.Note that the results of the Standard experiment are shown in two panels (b, e) for easier comparison with the heating experiments for amplitude (a, c) and peak altitude (d, f).The normalized temperature perturbations B′ are shown by shading and wind perturbations (u′, w′) are indicated using arrows.

Figure 5 .
Figure 5. Normalized temperature perturbations B′ of eastward-propagating eigenmodes under ERA5 stratification and no heating.Only the top-16 modes in terms of phase speeds are shown.

Figure 8 .
Figure 8. Vertical profiles of N 2 in additional sensitivity experiments based on the "Simple" basic state.

Table 1
Configurations of the Sensitivity ExperimentsNote.Details of the profile sources are given in the body text.
The heating profile is taken from the Simple basic state in all experiments.MIROC depends more heavily on convective precipitation than ideal, resulting in heating profiles that are biased to bottom-heavy ones, especially over MCS-dominated areas.

Modeling Earth Systems
YAMAZAKI AND MIURA