On the Regionality of Moist Kelvin Waves and the MJO: The Critical Role of the Background Zonal Flow

A global model with superparameterized physics is used to shed light on the observed regionality of convectively coupled Kelvin waves and the Madden‐Julian Oscillation (MJO). A series of aquaplanet simulations over zonally uniform sea‐surface temperatures is performed, in which the axisymmetric structure of the background zonal flow [u¯] is altered through nudging, while maintaining a quasi‐fixed rainfall climatology. Results show that nudging [u¯] at the equator to match profiles typical of the Indo‐Pacific or eastern Pacific sectors yields eastward‐moving tropical rain spectra typical of those sectors. Two different mechanistic pathways are identified as being responsible for this mean‐flow dependence, in addition to Doppler shifting effects. The first is through shifts of the Rossby wave critical line in the subtropical upper troposphere that affect the lateral forcing of Kelvin‐mode circulations at the equator by eastward and equatorward‐propagating eddies impinging on the tropics from higher latitudes. The second is through changes in the strength of the mean cyclonic shear in the lower tropical troposphere that affect the degree to which intraseasonal fluctuations in Kelvin‐mode zonal winds modulate the activity of higher‐frequency equatorial Rossby‐type eddies. In cases where the mean low‐level cyclonic shear is enhanced, the strength of this modulation, referred to as “shear‐induced eddy modulation” or SIEM, is also seen to be enhanced, such that MJO‐like modes of variability are rendered either unstable or near neutral, depending on the strength of the shear.

3 of 34 zonal winds. To simplify the problem, they assumed a completely water-covered Earth (i.e., an "aqua-planet") with a zonally uniform SST distribution. Their results showed a strong sensitivity of MJO-like variability in the model to the degree of meridional curvature in the SST profile (see also Jiang et al., 2020;Wang et al., 2018), in addition to whether the profile was maximized or off the equator. Using a very similar model, however, Sooraj et al. (2009) found that the simulated spectrum of tropical variability was strongly sensitive to the vertical structure of the basic state zonal wind near the equator, even in the absence of changes to the underlying SST distribution. In particular, those authors reported an enhancement in eastward-moving intraseasonal zonal wind variability at planetary zonal wavenumbers (k  1-4) when background low-level westerlies beneath upper-level easterlies (like what is observed over the Indo-Pacific) were imposed over a limited portion of the model domain. While the mechanisms responsible for this enhancement were not addressed, evidence was given to suggest an important role of the vertical and/or meridional shear of the background zonal wind.
A physical basis for suspecting that mean-state shear might be critical to the MJO can be found in previous studies that have pointed to interactions between the disturbance's circulation and higher-frequency Rossby-type eddies as being of central importance to its propagation (Andersen & Kuang, 2012;Chikira, 2014;Kiranmayi & Maloney, 2011;Maloney, 2009;Wolding et al., 2016). The reason stems from the nature of these interactions, which leads to the eddies being relatively more active (and hence, more effective at causing lateral mixing of dry air from the subtropics into the tropics) to the west of the MJO's convective envelope, as compared to further east. In theoretical studies that have sought to account for this effect, the approach has been to essentially assume a linear relationship of the form: where primes denote perturbations on intraseasonal time scales, E S is a bulk measure of the anomalous eddy activity, u e  is a bulk measure of the anomalous low-level zonal wind in the vicinity of the equator, and E  is a positive scaling coefficient (Adames & Kim, 2015;Sobel & Maloney, 2013). The rationale stems from observational and modeling work showing that periods of anomalous MJO westerlies tend to be characterized by enhanced mean cyclonic shear and barotropic energy conversion, while the opposite holds true during periods of anomalous MJO easterlies (Andersen & Kuang, 2012;Maloney & Dickinson, 2003). However, in a more recent observational study of the MJO during boreal winter, Wang et al. (2019) obtained evidence that the strength of this modulation, referred to as "shear-induced eddy modulation" (SIEM), is governed in part by the strength of the background cyclonic shear in which the MJO is embedded, owing to the effects of a non-linear eddy-eddy interaction term.
The past several decades have seen considerable progress in our ability to simulate moist tropical waves, without having to rely on problematic convection schemes (Chao & Lin, 1994;Maloney & Hartmann, 2001). Through recent advances in computing power, it is now possible to run global models at horizontal grid spacings fine enough to at least partially resolve the circulations of deep convective cloud systems (Stevens et al., 2019;Wedi et al., 2020). Such high-resolution models, however, remain computationally quite expensive and thus, are not yet practical for highly repetitive/iterative hypothesis testing. As an alternative (Grabowski & Smolarkiewicz, 1999), devised an intermediate approach that has come to be known as "superparameterization" (SP). The idea is to embed a cyclic, two-dimensional cloud-resolving model (2-D CRM) inside each grid box of a relatively coarse-resolution global model. Improved simulation of moist tropical variability, including the MJO, has been a consistent finding in studies comparing SP models to their conventional counterparts (e.g., Hannah et al., 2020;Randall et al., 2003;Tao et al., 2009). This improved simulation has prompted a growing number of authors to adopt such models as tools for studying the origin and dynamics of large-scale tropical wave phenomena (e.g., Andersen & Kuang, 2012;Arnold et al., 2013;Benedict & Randall, 2011;DeMott et al., 2013;Grabowski, 2003;Ma & Kuang, 2016;Pritchard et al., 2014).
In this study, a global model with SP physics is used to explore the hypothesis that much of the observed regionality of Kelvin waves and the MJO can be attributed to regional variations in the background zonal flow, owing to mediation of both tropical-extratropical interactions and convection-wave interactions internal to the tropics. The approach is to perform a series of idealized aquaplanet simulations over zonally uniform SSTs, in which the axisymmetric structure of background zonal flow is altered through nudging. Because only the zonal-mean part of the flow is affected, complications that arise due to introducing zonally

Journal of Advances in Modeling Earth Systems
TULICH AND KILADIS 10.1029/2021MS002528 4 of 34 asymmetric perturbations are avoided. In particular, the approach taken here is designed to ensure the model's simulated rain climatology remains close to that obtained in a free-running "control" integration, enabling isolation of the effects of the background zonal flow.
The next section describes the experimental approach, including the SP model and nudging methodology. Section 3 then describes an analysis of tropical-extratropical interactions in the context of the control integration. This analysis sets the stage for Section 4, which documents a strong sensitivity of the model's eastward-moving tropical spectrum to both the vertical and meridional structures of the background zonal wind. Further tests are described in Section 5, where the mean-flow dependence of the model is studied under eddy damping of the midlatitudes. Section 6 provides a summary and some concluding remarks.

Model Description and Control Simulation
The model is the global SP version of the Weather Research and Forecast model (SP-WRF). A detailed description of the SP-WRF can be found in Tulich (2015), hereafter T15. Briefly, a 2-D CRM version of the standard WRF is embedded inside a 3-D global version of the same model. This seamless coupling is unlike that of most other SP formulations, where models with different vertical grids and dynamical approximations are stitched together. The model includes the effects of convective momentum transport, using a novel scalar-based approach. As shown in T15, the SP-WRF is capable of producing realistic simulations of weather and climate with fidelity comparable to that of other current state-of-the-art global models (see also Figures A1-A3) To provide a baseline for comparison, the SP-WRF is first used to perform a 6-year aquaplanet simulation under zonally uniform SSTs and perpetual equinox conditions. The SST profile is chosen to crudely match observations, using an analytic form designed to ensure a symmetric but otherwise realistic decay with latitude about an equatorial maximum of 28 E C (see Figure 2a). The global model grid spacing is 2.8 E  × 2.8 E  in the horizontal, with 51 vertical levels stretching from the surface to a height of roughly 27 km; the embedded CRMs each have 32 columns with 4-km horizontal grid spacing. The effects of unresolved physics on the CRM grid are handled using the same set of schemes as in T15, except for radiation effects, which are now handled using the Rapid Radiative Transfer Model for GCMs (RRTMG; Iacono et al., 2008). Also, rather than depending on the large-scale flow, the CRM orientation is now treated stochastically, so that no horizontal direction is statistically preferred over any other (see Appendix A for further details).
The model behavior in the above setup, defined as the "control," is found to be broadly realistic. As shown in Figure 2b, for example, the simulated time-and zonal-mean surface rain is similar to that seen on Earth, with a relatively narrow band of intense rain centered at the equator and broader belts of more moderate rain at higher latitudes, reflecting the model's midlatitude storm tracks. The background zonal winds, as shown in Figure 2c, are easterly throughout the depth of the tropical troposphere (with the largest values near the surface), while westerlies prevail at higher latitudes, in association with a pair of well defined subtropical and eddy-driven jets, centered at around 25 E  and 50 E  latitude, respectively. The average spacetime spectrum of tropical rain (Figure 2d) shows evidence of both westward-moving equatorial Rossby-type waves and eastward-moving Kelvin-type waves. The latter are by far the most dominant, however, and exhibit the same sort of non-classical dispersion as seen in the observed Indo-Pacific spectrum of Figure 1a. In Section 3, evidence is given to suggest that this non-classical dispersion is almost certainly a result of external forcing of the tropics by eddies at higher latitudes.

Method for Altering the Background Zonal Flow
To examine how the above spectrum is affected by changes in the background zonal flow, a very strong nudging term is added to the right hand side of the model's prognostic zonal momentum equation, that is, 10.1029/2021MS002528 7 of 34 Besides these near-equatorial changes in the background zonal flow, this study also seeks to quantify the effects of changes in the strength of the climatological subtropical westerly jet. The motivation stems from previous studies showing synoptic-scale Rossby wave trains propagating eastward and equatorward in the subtropics as being potentially important driving agents of Kelvin waves (Huaman et al., 2020;Roundy, 2014;Straub & Kiladis, 2003a) and the MJO (Hall et al., 2017;Hsu et al., 1990;Lin et al., 2009;Matthews & Kiladis, 1999;Ray & Zhang, 2010). Because such eastward-moving wave trains depend crucially on the presence of background westerlies (Yang & Hoskins, 1996), any change in the strength of the subtropical westerly jet will almost certainly affect their ability to potentially modulate eastward-moving convection variability in the tropics.
To explore this idea, a second nudging term is added to the right side of the model's zonal momentum equation, that is,  taken as their simulated climatological averages at 300 hPa for the center of the latitude band of interest. The overall correspondence points to eddies in the subtropics has having significantly larger values of l, as compared to eddies at higher latitudes, with l being larger in both cases for smaller values of E k. As discussed in Appendix B, reasonable agreement is found when comparing these implied regional variations in l versus E k against actual regional horizontal (k-l) wavenumber spectra of 200 E  . The reason for the dominance of signals at 4 E k  and 5 in both cases is presumably tied to the preferred scales of baroclinic instability in the midlatitudes (Pratt, 1977;Randel & Held, 1991).

Composite View of Extratropical Forcing
The similarities between the subtropical vorticity spectrum in Figure 4b and the tropical rain spectrum in Figure 2d suggests that lateral forcing of the tropics by midlatitude eddies may be essential for eliciting the non-classical dispersion, as well as preferred zonal scales, of the simulated Kelvin wave disturbances. Evidence to support this idea can be found in Figure 5a, which depicts the composite 200-hPa horizontal flow and vorticity anomaly patterns associated with individual wave disturbances in filtered tropical rain, where details about the compositing technique are described in the figure caption. The patterns appear very similar to those documented by Straub and Kiladis (2003a, hereafter SK03) in the context of extratropical-forced Kelvin waves over the central tropical Pacific (see their Figure 2 and also Roundy, 2014). A noteworthy feature is the positively tilted Rossby wave train in the extratropics, implying equatorward propagation of wave energy to the west (i.e., "upstream") of the simulated Kelvin wave's convective envelope. The anticyclone on the immediate poleward flank of this envelope, together with straddling cyclones to the east and west, is also reminiscent of SK03's observations. While such off-equatorial gyres are not present in E -plane Kelvin waves (which have no meridional wind), Figure 5b shows that the composite structure in the lower troposphere (850 hPa) is indeed Kelvin-like, with winds that are predominantly oriented in the zonal direction and roughly in-phase with the geopotential height field, very similar to that observed by SK03 (see their Figure 5). The extratropical wave train is centered in this case at around 40 E N and shows little evidence of equatorward propagation, suggesting that the pathway of extratropical forcing lies in the upper troposphere near the level of the subtropical jet core.

On the Mechanism of the Extratropical Forcing
What is the precise mechanism responsible for this apparent extratropical forcing? The working hypothesis here is that the answer involves the effects of transient Rossby wave dissipation due to critical layer absorption in the subtropical upper troposphere. Generally speaking, such absorption occurs when an equatorward-propagating Rossby wave encounters a critical latitude (or line) where the zonal velocity of the wave's crests and troughs matches that of the local background zonal flow, that is, 0 E U c   (Bennet & Young, 1971). A well-known effect of this process is the deposition of the eddy momentum flux, which acts as transient source of zonal momentum, both in a zonal-mean and local perturbation sense (Randel & Held, 1991). The interest here is in the latter sense, specifically in terms of the forcing of individual wavenumbers and frequencies, that is, , where E  denotes the space-time Fourier transform operator and H E   m is the divergence of the horizontal eddy flux of zonal momentum in pressure coordinates. The latter is given by:  denotes the complex conjugate. The fact that E F occurring outside of the tropics can act to excite (dry) equatorial Kelvin modes is well supported by theoretical work of Hoskins and Yang (2000). As those authors point out, the only requirement is that E F projects onto the Kelvin mode's zonal wind eigenstructure.

Journal of Advances in Modeling Earth Systems
TULICH AND KILADIS 10.1029/2021MS002528 10 of 34 To evaluate this forcing pathway in the model, the Kelvin-mode projection of E F is calculated as: where * E  is the meridional trapping-scale parameter of Gill (1980) and r E w is a weighting function intended to isolate the contribution to K E F by "remote" eddies, as opposed to those internal to the tropics, that is, Conversely, the Kelvin-mode projection of E u is calculated as: are identified using a filtered object-based approach, similar to that described in Tulich and Kiladis (2012). Briefly, objects are defined as contiguous regions in the longitude-time domain where filtered rain anomalies averaged between 5 E S and 5 E N exceed a threshold; the object filter retains eastward-moving wavenumbers in the range 1-14, with periods in the range 2-120 days. Composite averaging is performed relative to the set of base points in the longitude-time domain where object-filtered rain anomalies exceed one standard deviation of their respective object's distribution.

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The form of the latter weighting function, together with Equation 10, is very similar to that adopted in previous equatorial wave studies by Yang et al. (2003) and Gehne and Kleeman (2012). Somewhat different from these studies, however, the value of * E  is taken here as 9 E , as opposed to 6 E , corresponding to a dry Kelvin/gravity wave speed * 45 E c  m 1 s E  , as opposed to 20 m 1 s E  . The rationale stems from an analysis in Appendix C, which shows that moist Kelvin wave variability in the model, as well as in observations, is characterized by leading meridional structures in upper-level zonal wind that are well captured by that theoretically expected for dry Kelvin waves with * 9 E    or even larger. The fact these zonal wind structures have non-negligible amplitude in the subtropics, as shown in Figure C1, is a necessary condition for enabling their potential forcing by midlatitude eddies, as embodied mathematically in Equations 8 and 9.
Such remote forcing by itself, however, does not guarantee that Kelvin mode circulations in the tropics will be energetically maintained, since the latter requires a positive correlation between fluctuations in K E u and K E F . The following quantity is therefore of primary interest in this study: where K E P is termed the "remote eddy source" and represents the production of Kelvin-mode kinetic energy (red shading), suggesting that the simulated waves are indeed mechanically forced by Rossby-type eddies in the subtropical upper troposphere, with preferred meridional wavenumbers in the range 7 E l  -9. However, because tropical convection can generally act as a source of Rossby wave energy in the subtropics, especially in the presence of a strong subtropical jet (cf. Sardeshmukh & Hoskins, 1988), the possibility of the reverse forcing pathway cannot be ruled out. More discriminating tests aimed at addressing this issue are described in Section 5.

Remote Eddy Forcing Versus Local Tropical Heating
How does this remote eddy forcing compare to that due to "local" heating internal to the tropics? To address this question, the production of Kelvin-mode available potential energy K E PE is first calculated as: 13 of 34

Vertical Mode Energetics
Another useful way of viewing the energetics of moist tropical waves is in terms of a discrete set of vertical normal modes, where the latter conveys information about the spectrum of vertical wavelengths that are energetically maintained by the forcing. Such information is key for addressing fundamental questions about how the forcing acts to modulate the convection field and vice versa. Here, the vertical orthonormal modes (and their associated "phase speeds" n E c ) are calculated as in Tulich et al. (2007), but with a rigid lid assumed at 150 hPa, as opposed to the model top. The modal forms of K E H and K E P are then calculated respectively as: where ( ) n E  denotes the E nth mode's contribution to the corresponding dynamical field, and the summation is over the same set of wavenumber-frequency bins as described previously. Note that these modal forms are related to their physical-space forms via the Parseval rule, which states that the sum of the former over E n is mathematically equivalent to the mass-weighted vertical average of the latter (Fulton & Schubert, 1985).
Conceptually, the vertical modes are very much like Fourier modes, but with oscillatory structures that deviate somewhat from pure sinusoids, owing to vertical variations in the background static stability. Nevertheless, Tulich et al. (2007) showed that a "bulk" vertical wavelength zn E L can meaningfully be assigned to each mode, based on the analytic expression for Kelvin/gravity waves in a Boussinesq atmosphere with constant static stability, that is, where N 0 2 E u is applied only in the tropics.

Standard IPAC and EPAC Cases
Figures 9a and 9b display the mean eastward-moving tropical rain spectra obtained in the standard IPAC and EPAC cases, respectively. Broad agreement is seen when comparing these spectra against their observed regional counterparts in Figure 1. The spectrum in the IPAC case shows a pronounced MJO-like spectral peak at 2 E k  , in addition to a lobe of relatively slow-moving and dispersive Kelvin waves, much like in the observed regional spectrum of Figure 1a. Conversely, the spectrum in the EPAC case is dominated by faster-moving and more classical Kelvin wave signals, much like in the observed regional spectrum of Figure 1b. This favorable agreement is remarkable and indicates that much of the observed regionality of Kelvin waves and the MJO can be attributed mainly to regional variations in the background zonal flow, as opposed to regional variations in mean-state moisture and/or temperature. Favorable agreement is also To help interpret these results, Figures 9c and 9d compare the space-time spectra of the 1 E n  remote eddy source K1 E P for the two cases. The strong mirroring of the signals in precipitation and K1 E P (e.g., at Figures 9b and 9d) is telling and points to changes in the spectrum of eddy phase speeds (and meridional wavenumbers) capable of forcing Kelvin wave motions at the equator. These changes can be understood, at least in a qualitative sense, by considering the inset in Figure 9c, which shows how the background zonal flow (denoted [ ] u ) is altered in response to the nudging, not only in the deep tropics, but also in the subtropics at upper levels, due to angular momentum being approximately conserved in the poleward flowing branch of the model's Hadley circulation. The net effect, as indicated, is that Rossby waves with a representative phase speed of 18 m 1 s E  have their critical latitude shifted from roughly 16 E  to 13 E  in the IPAC versus EPAC basic states, a shift that apparently enables these relatively fast-moving waves (with larger preferred values of l) to more strongly drive similarly fast-moving Kelvin-mode circulations at the equator in EPAC, via their enhanced projection. Unlike in the CNTL case, however, an important role of the 1 E n  tropical heating source K1 E H is also found for these planetary-scale Kelvin wave signals, in addition to the slower-moving MJO-like signals in the IPAC case (results not shown), suggesting that mechanisms involving convection-wave interactions internal to the tropics may be a further causal factor. Preliminary evidence to support this idea is outlined in the paragraphs below, with additional evidence presented later in Section 5.

The Critical Role of Mean-State Cyclonic Shear at Low Levels
As just mentioned, the net effect of the nudging on the background zonal flow [ ] u extends well into the subtropics in the upper part of the troposphere. The story is quite different in the lower troposphere, however, where the changes in [ ] u tend to be confined to the deep tropics, owing to the mean meridional winds being directed equatorward. This confinement is readily apparent in Figure 10a, which compares the meridional structure of [ ] u at 850 hPa among the three different cases (i.e., CNTL, EPAC, and IPAC). Because strong easterlies prevail outside of the nudging zone, the net effect of the nudging, as shown in Figure 10b, is to enhance the ambient level of cyclonic shear (i.e., [ ]   0 for the northern hemisphere) between roughly 3 E  and 10 E  latitude. The enhancement is especially noteworthy in the IPAC case, where peak values of [ ]  are roughly double those in the EPAC case, while being more than triple those in the CNTL case. To address whether it is this enhancement in meridional shear, as opposed to the introduction of easterly shear in the vertical, that is, responsible for the development of MJO-like variability in the IPAC case, an additional simulation was performed (referred to as "no-shear IPAC"), in which the easterly shear was removed by setting 0 T E U  at all levels below 150 hPa. The resulting spectrum was found to still contain signals resembling the MJO, but with total variance reduced by roughy 20%, apparently due to a corresponding reduction in the   Figure S2). Given this result, it is concluded that the primary reason for the emergence of MJO-like variability in the IPAC case is indeed tied to the introduction of anomalous low-level mean westerlies at the equator and the resultant increase in [ ]  .

Journal of Advances in Modeling Earth Systems
As alluded to in the introduction, one possible explanation for this result is that the SIEM mechanism of MJO propagation (involving the disturbance's modulation of high-frequency eddy activity and the associated lateral mixing of moisture) is amplified in the IPAC case, owing to the effects of non-linearities being magnified by the larger mean cyclonic shear (Wang et al., 2019). To explore this idea, the following linear regression model was constructed using the model output at 850 hPa: where the superscript " E H" denotes application of a 25-day highpass filter, the superscript " E B" denotes application of a 35-120-day bandpass filter, A E  is the antisymmetric component of the vorticity, and E r  is the regression coefficient (which is calculated at each latitude and longitude). The intent is to draw a direct analogy with the linear parameterization of the SIEM mechanism in Equation 1, as first proposed by Sobel and Maloney (2013). Figure 10c compares the meridional distribution of ( E r  ) among the three different cases. The fact that the distributions are everywhere positive in all cases, with peak values located at or near the center of the peak shear zone, is supportive of the choice of 0 E   and the physical concepts underlying this choice. However, rather than being constant, it appears based on Figure 10c that the strength of the SIEM mechanism is indeed strongly dependent on the degree of mean-state cyclonic shear, such that MJO-like modes of variability are evidently rendered unstable in the IPAC basic state, compared to being either near neutral or damped in the remaining two basic states. Additional work is needed to clarify the mechanisms underlying this mean-flow dependence of the SIEM mechanism, which is beyond the scope of this study.
Interactions between convection and radiation are also found to be essential for the simulated MJO-like variability. Repeating the standard IPAC integration, but with the effects of these interactions suppressed (corresponding to the HOMRAD IPAC case in Table 1) yields a tropical rain spectrum that is largely devoid of any sort of coherent intraseasonal modes of variability, being instead dominated by the signals of traditional higher-frequency Kelvin waves with phase speeds in the range 11-15 m 1 s E  (see Figure S3).
This sort of tropical wave response is similar to that seen in previous MJO modeling work by Andersen and Kuang (2012) and Ma and Kuang (2016). Meanwhile, in looking at the column-integrated moist static energy (MSE) budget of the model's MJO analog for the tropical belt 10 E S-10 E N (calculated as in Andersen & Kuang, 2012;see Figures 11a and 11b), the picture is found to broadly resemble that documented for the MJO on Earth by Ren et al. (2021), where longwave radiation feedbacks (denoted "LW") act as the primary amplifier of the disturbance (but tend to oppose its eastward propagation), while the horizontal advection of MSE (denoted "hAdv") acts as the primary driver of propagation (but tends to oppose amplification). This dichotomy can be understood on the basis of Figure 11c, which shows that zonal fluctuations in hAdv tend to be shifted roughly 90 E  to the east of those in both LW and column-integrated MSE, apparently due to the effects of the SIEM mechanism. The overall impression is that of a symbiotic interaction between LW and hAdv, in which both are equally essential for MJO existence and propagation (in accordance with the theory of Sobel & Maloney, 2013), but where the relative importance of hAdv is also crucially dependent on the strength of the mean cyclonic shear at low levels (as implied by Figure 10c).

Weak-Jet Cases
The above findings indicate that the meridional structures of the background zonal flow in both the upper and lower troposphere are important for shaping the model's eastward-moving tropical wave spectrum. To further explore the role of the upper-level mean flow structure, Figure 12 documents the changes that arise in response to an imposed 25% weakening of the subtropical jet. The overall reduction in eastward-moving tropical wave variability and remote eddy production ( in the character of the simulated intraseasonal variability. The changes in the IPAC case are perhaps the easiest to interpret, where the preferential reduction of MJO-like variability at 2 E k  and higher appears to stem from corresponding reductions in the remote eddy source K1 E P . The story in the EPAC case, however, is more complicated: Figure 12d shows that intraseasonal rain signals are actually enhanced at 1 E k  , despite being reduced virtually everywhere else. Interestingly, broadly similar changes are seen in the signals of K1 E P , except for a weak enhancement at 2 E k  , where the latter apparently stems from a reduction in the Doppler-shifted phase speed of the faster-moving eddies involved in forcing these planetary-scale wavelengths (compare the 10 E l  Rossby wave dispersion curves in Figure 12f vs. Figure 9d). The resulting tropical rain spectrum in Figure 12b, which can be regarded as a mixture of relatively fast-moving Kelvin waves and slower-moving MJO-like signals at 1 E k  and 2, is seen to provide even better agreement with the relevant observed spectrum in Figure 1b, in addition to that documented for the Atlantic sector in Figure 1c. The interpretation is that some amount of shielding of the tropics from the effects of faster-moving eddies in the extratropics (i.e., eddies with 15 E c  m 1 s E  ) is also necessary for the simulated MJO-like variability. Presumably, the reason why these intraseasonal signals are more muted in this case, as compared to the weak-jet IPAC case, is tied to the weaker mean cyclonic shear in the lower troposphere, through its effect on the SIEM mechanism, as evidenced in Figures 10b and 10c.
Another potential mechanism for the simulated MJO-like variability is that of "wind-induced surface heat exchange"

Journal of Advances in Modeling Earth Systems
TULICH AND KILADIS 10.1029/2021MS002528 18 of 34 assess the importance of this mechanism, the two WJET cases were repeated, but with the surface sensible and latent heat fluxes being zonally homogenized at each time step within 10 E  of the equator. Results of this experiment (termed HOMFLX, depicted in Figure 13) show that the simulated MJO-like signals are actually enhanced in both cases relative to their original weak-jet counterparts, demonstrating that interactive surface fluxes are not essential to the phenomenon and moreover, tend to have a net damping effect. However, because the signals of low-frequency, westward-moving disturbances in rainfall are also enhanced, the net effect overall is to reduce the percentage of the total intraseasonal (35-120-day) rain variance that is eastward propagating from around 54% to 41% in the IPAC case, and from around 43% to 35% in the EPAC case.  Figures 9a and 9b, but for the "weak-jet" (WJET) variants of the IPAC and EPAC cases, respectively. Middle panels: change in tropical rain variance (WJET minus standard), where red/blue shading denotes positive/negative values with logarithmic intervals spanning the same range as in the top panels; heavy black/ gray contours denote where the signal-to-noise ratio in the weak-jet/standard spectrum exceeds 1.5 (corresponding to the 90% confidence level). Bottom panels: similar to the middle panels, but for the change in K1 E P , where shading levels are the same as in Figures 9c and 9d. These results differ from those of previous aquaplanet studies, in which the mechanism of WISHE has been found to be essential for the emergence of MJO-like modes variability in free-running climate integrations, distinguished by either globally uniform SSTs (Arnold & Randall, 2015;Khairoutdinov & Emanuel, 2018;Shi & Bretherton, 2014) or SSTs that are spatially uniform within the tropical belt 15 E S-15 E N (Shi et al., 2018). A drawback to such idealized calculations, however, is that the simulated mean zonal winds are inevitably quite weak, effectively preventing the types of wave-mean-flow interactions demonstrated as being important here.

Model Behavior Under Suppression of Midlatitude Eddies
A key issue surrounding the above results concerns the extent to which the simulated differences in eastward-moving tropical wave variability can be attributed to mean-state modulation of convection-wave interactions internal to the tropics versus those involving equatorward-propagating eddies in the subtropics. To address this issue, we present a set of simulations similar to those just reported, but where damping of eddy perturbations about the zonal mean is applied poleward of 30 E , to effectively eliminate the forcing of the tropics by midlatitude eddies. This sort of mechanism-denial approach has been become increasingly popular in recent years. Using a global SP model with real-world (as opposed to idealized) lower boundary conditions, for example, Ma and Kuang (2016), hereafter referred to as MK16, showed how damping of midlatitude eddies had little effect on the model's ability to simulate the MJO, provided the underlying basic state was constrained (through a clever combination of nudging and time-invariant forcing) to match that obtained in a free-running reference integration. Here, a similar approach is taken, but where the basic state is maintained strictly through nudging of the zonal-mean temperature, water vapor, and horizontal wind fields. The form of the nudging is the same as in Equation 3, except that nudging timescale E  is increased to 12 h outside of the damping region (| |    30 ) for all variables, except the meridional wind. The goal once again is to ensure that the simulated time-mean and zonal-mean rain climatology remains close to that obtained in the undamped control case for the tropical domain of interest. Following the approach of MK16, the strength of the eddy damping of all prognostic variables, as measured by the inverse damping time scale, is specified to increase linearly with distance from 0 to 1 2.7 E  day between 30 E  and 42 E  and then remain constant thereafter. In the discussion that follows, cases with damping are distinguished from their undamped counterparts using the naming convention, "REF-D," where "REF" identifies the corresponding (undamped) reference integration.  Figures 12a and 12b, but for the sensitivity experiment, HOMFLX, in which the weak-jet IPAC and EPAC cases were repeated, but with the surface sensible and latent heat fluxes being zonally homogenized at each time step within 10 E  of the equator. Inset in each panel denotes the change in rain variance relative to the original weak-jet integration for zonal wavenumbers E k in the range −2 to +6 and frequencies 0.15 E  cpd, following the same plotting convention as in Figures 12c and 12d.

Journal of Advances in Modeling Earth Systems
TULICH AND KILADIS 10.1029/2021MS002528 20 of 34

Effects of Midlatitude Eddy Damping Under the CNTL Basic State
Considering first the eddy-damped variant of the control case (CNTL-D), Figures 14a and 14b show that, while the model's tropical rain climatology remains close to that obtained in the free-running setup (as intended), its spectrum of eastward-moving variability is now markedly different. The waves in this case are almost perfectly non-dispersive, with no indication of the lower-frequency dispersive signals seen previously in Figure 2d. The conclusion is that the latter indeed owe their existence to external forcing of the tropics by midlatitude eddies, while the former do not.
Even in this case, however, Figure 14c shows that the composite upper-level structures of the simulated Kelvin waves are still quite unlike those of linear E -plane solutions, with the same sort of flanking subtropical gyre pattern as seen in the reference composite of Figure 5a. Perhaps the simplest explanation for these flanking gyres is that they arise merely as a passive response to convective heating anomalies of the disturbance. Indeed, a Rossby wave source analysis points to the gyres as being mainly forced by the meridional advection of the climatological absolute vorticity by the disturbance's upper-level divergent wind (results not shown). Rather than acting as a passive response, however, it appears on the basis of Figures 14d and 14e that the gyres, once excited, act to return energy back to their parent Kelvin wave, via the same remote forcing pathway as diagnosed in the undamped reference case. Thus, only a portion of the remote eddy source can generally be attributed to the effects of external forcing from the midlatitudes; the remainder is evidently generated by the disturbance itself.

Effects of Midlatitude Eddy Damping Under the EPAC Basic State
The story is broadly similar when considering the model's response to eddy damping under the EPAC basic state (EPAC-D), defined by mean westerlies aloft. Results in this case, however, show not only the suppression of the slower-moving, dispersive Kelvin wave signals in the synoptic range 4 E k  -6, but also the faster-moving, non-dispersive signals in the planetary-scale range 1 E k  -3 (compare Figure 15a and Figure 9b). Because the phase speed of the missing planetary-scale signals is similar to that of the remaining non-dispersive signals in Figure 15a, the interpretation is that the former are near-neutral "modes," whose emergence requires some level of midlatitude forcing. Meanwhile, comparison of Figures 14e and 15b show that the net contribution of the 1 E n  remote eddy source K1 E P is roughly a factor of three smaller than in CNTL-D, while the 3 E n  tropical heating contribution K3 E H is more or less the same. Insight into the reduced  Figure 15c, which shows that the accompanying Rossby wave train in this case is slightly weaker in amplitude, in addition to being less trapped in the key subtropical belt 15 E -20 E . The reason for this reduced trapping is not clear, but may be tied to the faster propagation speed of the waves at roughly 17 versus 11 m 1 s E  , which ultimately sets the speed of the disturbance's associated Rossby wave source.
To explain the increase in Kelvin wave propagation speed, Figure 16 compares the vertical profiles of the climatological zonal wind near the equator in these two cases (as well as in IPAC-D), where different offsets have been added to allow comparison of the mean flow in the wave's moving frame of reference (see the figure caption for details). The comparison shows the waves to have a well-defined "steering level" at around 325 hPa or roughly 9.1 km, corresponding to an intrinsic wave propagation speed of around 17 m 1 s E  . This estimate, though empirical, is considered to be reliable for two reasons, both of which point to the 3 E n  mode as being critical to the wave's propagation, despite the energetic dominance of the 1 E n  mode. The first is that the implied steering level of the waves, as indicated by the green line in Figure 16, is almost perfectly coincident with the peak altitude of the net tropical heating source K E H (whose dominant contribution is from the 3 E n  mode; see Figure 7), while lying several kilometers below that of the remote eddy source K E P . The second is that the intrinsic propagation speed of the waves is very close to that of the n  3 mode (i.e., 16.9 vs. 17.5 m 1 s E  , respectively), where the latter was derived in a completely independent fashion based on a dry linear model calculation. Given this correspondence, it is suggested that the 3 E n  mode may actually play a primary role in setting the propagation speed of the waves, as opposed to the 1 E n  mode, in agreement with previous modeling work by Tulich et al. (2007) and Tulich and Mapes (2008).

Effects of Midlatitude Eddy Damping Under the IPAC Basic State
Turning finally to the effects of eddy damping under the easterly sheared IPAC basic state (IPAC-D), Figure 17a reveals an eastward-moving tropical wave spectrum that is now dominated not only by the signals of high-frequency Kelvin waves, but also those of the model's lower-frequency MJO. This result shows that external forcing from the midlatitudes is not essential for either of these two distinct modes of variability, in accordance with MK16. At the same time, however, the dearth of power at frequencies and zonal wavenumbers between these two modes is evidence once again that such forcing is nevertheless critical for driving the intermediate band of slow-moving dispersive Kelvin-like signals, in addition to amplifying the signals of the simulated MJO (compare Figure 17a vs. Figure 9a). Additional evidence of an important supporting role of the midlatitudes in forcing the MJO can be found in a modeling study by Hall et al. (2017).
Considering the net modal energetics of the simulated MJO, Figure 17b shows that the disturbance is primarily driven by the 1 E n  tropical heating source K1 E H , with an important secondary contribution from the 1 E n  remote eddy source. Inspection of Figure 17c shows that the mechanism of this internal eddy feedback is essentially the same as discussed previously, but where the associated flanking Rossby gyres have shallower northwest-southeast tilts, implying predominantly poleward propagation of Rossby wave energy.
Overall, the pattern looks very similar to that obtained in idealized simulations of the remote response to a prescribed MJO-like heat source, described in Monteiro et al. (2014, see their Figure 2c). Results confirming that these Rossby gyres indeed act to strengthen the model's MJO analog, in addition to its higher-frequency Kelvin waves, are contained in Figure 18, which shows how the rain signals of both wave types are reduced in response to weakening of the subtropical jet, through a reduced contribution of the remote eddy source K1 E P (see the inset in Figure 18b).  in panel (b) shows how this weakening, in addition that of the higher-frequency Kelvin waves, is associated with reduction in the 1 E n  remote eddy source K1 E P , following the same plotting convention as in Figure 12e.

Summary and Concluding Remarks
This study employed a global model with superparameterized physics to address the problem of moist tropical waves and their dependence on the basic state, with an emphasis on two forms of eastward-moving tropical wave variability: Kelvin waves and the MJO. The primary goal was to shed light on the observed regionality of these two wave types (as quantified in Figure 1), which has yet to be fully explained. Results from a series of aquaplanet simulations and analyses support the hypothesis that regional variations of the background zonal flow E u are of leading importance, owing to their mediating influence on at least two different mechanistic pathways, in addition to Doppler-shifting effects. Brief sketches of these two affected pathways, encapsulating the main findings of this study, are discussed below.
The first pathway involves the forcing of equatorially trapped Kelvin-mode circulations by eastward and equatorward-propagating Rossby-type eddies impinging on the tropics from higher latitudes. The forcing is thought to arise as the eddies encounter a critical latitude where their zonal phase speed matches the background zonal flow (i.e., 0 E U c   ). Despite being confined to the subtropical upper troposphere, the primary effect of the eddy forcing is to excite and maintain deep overturning Kelvin-mode circulations in the tropical troposphere that are manifested in spectral space by eastward-propagating signals in tropical rainfall. The spectral structure of these signals is similar to that of their parent eddies, whose dispersion is well explained by the linear Rossby wave theory. Because eddies are absorbed where 0 E U c   , the spectrum of forced rain signals is seen to depend crucially on E u, and especially the strength of the subtropical jet. This spectrum of forced variability includes not only slower-moving, dispersive signals inherited from the storm tracks, but also faster-moving, non-dispersive signals that are regarded as traditional free Kelvin waves. Rather than being set by the forcing, however, the phase speed of these free waves appears to be set by the "dry" speed of the 3 E n  vertical mode of the troposphere, modified by Doppler shifting at a steering level in the upper troposphere at around 9 km.
The second pathway lies in the lower free troposphere. Intraseasonal fluctuations in Kelvin-mode zonal wind modulate the activity of higher-frequency equatorial Rossby-type eddies (referred to here as "shear-induced eddy modulation" or SIEM), in such a way as to promote the slow eastward propagation of moisture anomalies near the equator. While the importance of this modulation toward MJO propagation is generally accepted, here its efficacy was seen to be directly tied to the strength of the background cyclonic shear on the flanks of the simulated ITCZ. This mean-flow dependence of the SIEM mechanism appears to lie at the heart of the explanation for why the MJO's convective signals in nature tend to be confined mainly to the tropical Indo-Pacific (where low-level westerlies are the norm and the associated flanking belts of mean cyclonic shear are generally larger than elsewhere). A further reason is evidently tied to the modulating effects of the background zonal flow at upper levels. In particular, results showed how the presence of strong upper-level mean easterlies in the tropics, together with a strong subtropical jet (as is typical of the Indo-Pacific sector, especially during the solstice seasons) tends to be optimal for the MJO, not only by insulating the disturbance from the effects of relatively fast-moving eddies in the extratropics (that otherwise tend to excite higher-frequency Kelvin waves), but also by fostering positive eddy-momentum feedbacks involving the disturbance's associated flanking Rossby gyres.
Many of the above findings are novel, with very little in the way of theoretical guidance. It thus remains unclear as to how profound (or secondary) these upper-level flanking Rossby gyres are to the dynamics of either Kelvin waves or the MJO. Another important question concerns the underlying causes for the mean-flow dependence of the SIEM mechanism, which has only recently been hinted at through careful processing of model reanalysis data (Wang et al., 2019). Finally, there is a need to understand how the SIEM mechanism is affected not only by the background zonal flow structure, but also by the distributions of time-mean convective heating and moisture, all of which are shaped by the time-mean SST field. Investigation of this issue might ultimately help to explain why the MJO tends to be strongest during boreal winter, despite the fact that low-level mean westerlies over the tropical Indo-Pacific tend to be strongest during boreal summer (Zhang & Dong, 2004).
Future work should try to reduce this study's main limitations. The approach of using a global model with axisymmetric forcing and lower boundary conditions to study the regionality of moist tropical waves, while convenient for isolating the effects of different local basic states, leaves open questions about the effects of  radians.
The above strategy is quite different from that originally implemented in the SP-WRF, where E  was chosen to match the direction of the large-scale horizontal wind at low levels, following the suggestion of Grabowski (2004). To assess the impact of this revision, the model was used to perform a series of real-world seasonal climate integrations, following the protocol outlined in Section 5 of T15. Results in Figures A1 and A2 show that the impact in terms of the simulated time-mean climate is generally quite small. The only significant difference is a slight improvement in the correlation between the simulated versus observed spatial patterns of the E u-wind and E v-wind components, both near the surface and at 200 hPa (see Figure A2). This insensitivity is also seen when comparing the simulated versus observed space-time spectra of tropical rain, depicted in Figure A3. The model spectra appear broadly similar, with only modest differences pointing to either slightly weaker Kelvin and tropical depression-type disturbances, or slightly stronger westward-moving inertia-gravity waves, leading to marginally better agreement with observations. The reason(s) for these differences, in addition to those seen in Figure A2, is not clear but may be due to changes in the parameterized CMT, via changes in the statistical sampling of the two large-scale horizontal wind components. Regardless, such improvements are welcome and provide justification for the revised approach, in addition to its intended purpose of ensuring all directions are treated equally in a statistical sense.
The conclusion that the model performance is largely insensitive to the choice of CRM orientation is somewhat different from that of T15, which examined the effects of aligning the CRMs everywhere perpendicular, as opposed to parallel, to the large-scale horizontal flow at low levels, except in regions of strong convection. The focus in that study, however, was on the simulated time-mean pattern of surface rain, where regional differences exceeding 3 mm 1 dy E  (in an absolute sense) were reported to be statistically significant at the 80% confidence interval, which is lower than the 90% interval used here. Another important distinction is that the model performance documented in Figures A1-A3 is significantly better than that seen in T15's Figures 10, 12a and 14, respectively. The reason is due mainly to the correction of a coding error involving the specified surface albedo, which was inadvertently set to zero over all non-glaciated land points, in addition to the use of a different radiation scheme and some tuning/modification to the TKEbased turbulent mixing scheme.  , 1997). Model results in panels (b) and (c) are for the flow-parallel versus random orientation strategies, respectively. Based on a Student's t-test, the null hypothesis that the two simulations are statistically identical cannot be rejected anywhere using a 90% confidence level or above. decreases to 7 for the dominant synoptic-scale eddies at 4 E k  and 5, before dropping off sharply from 5 to 0 at 7 E k  and 8, respectively. Overall, these results are supportive of the use of linear Rossby wave theory, in the form of Equation 6, to explain the propagation and dispersion of the model's storm track disturbances, with the steering level of the waves implied a posteriori to lie at around 300 hPa. Figure A2. Taylor diagram for SP-WRF seasonal climate integrations performed using either the original flow-parallel or newly devised random CRM orientation strategies described in the text (red vs. green symbols, respectively). Symbols with annotation denote the surface rainfall (R), ocean-masked precipitable water (PW), outgoing longwave radiation (OLR), and zonal/meridional wind components at both 10 m and 200 hPa (U10/V10 and U200/V200, respectively). Results were obtained using the same observational datasets as described in Section 5 of T15.  is a theoretical parameter that determines the degree to which (dry) linear equatorial waves of a given equivalent depth 0 E h (or alternatively, Kelvin/gravity wave speed * E c) are trapped near the equator (Gill, 1980). To arrive at an appropriate value of * E  for the SP-WRF's simulated moist eastward-moving tropical wave disturbances, the model output from the control simulation was first spectrally filtered to retain eastward-moving zonal wavenumbers in the range 1 E k  -14 and periods in the range 2-120 days. Next, a principal component (PC/EOF) analysis was applied to the combined fields of daily averaged precipitation and the divergent component of the 200-hPa zonal wind 200 E U  , both normalized to have a standard deviation of unity, where the structural dimension of the analysis Figure A3. Similar to Figure A1 but for the average global space-time spectrum of tropical rain. Results were obtained using the same methods (and observational data) as in Figure 1, except that no regional tapering was applied and the time window was reduced from 96 to 30 days, due to the smaller data record. Shading with contours denote where the signal-to-noise ratio E  1.05, with intervals of 0.05. Solid curves denote the dispersion relations of various dry equatorial wave modes with equivalent depth of 25 m.  Figure C1a, where the percentage of the total filtered variance explained by the PC is around 38%. Note that this percentage increases to 85% when the analysis is restricted to 200 E U  , with very little change in the associated eigenstructure (results not shown). As indicated by the black dotted curve in Figure C1a, the regressed structure of 200 E U  is well captured by that theoretically expected for dry Kelvin waves with a trapping scale 0 9 E   , corresponding to a dry Kelvin/gravity wave speed * 45

Journal of Advances in Modeling Earth Systems
The implication is that eddy momentum forcing in the subtropics can indeed potentially act to excite the simulated moist Kelvin waves, despite the relatively narrow structure of their associated precipitation anomalies. Comparison of the gray solid and dotted curves in Figure C1a shows that the latter have an estimated trapping scale * 3 E   , which is presumably set by the width of the model's time-mean tropical rain band (see Figure 2b).
To assess the realism of the above picture, a similar PC/EOF analysis was applied to TRMM rainfall data combined with ERA5 (Hersbach et al., 2020) estimates of 200 E U  for the period 1998-2017. Preparation of the data involved removing the first three harmonics of the seasonal cycle, as well as horizontal coarse graining to achieve a spatial resolution comparable to that of the SP-WRF (i.e., 2.5 E  vs. 2.8 E ). Also, due to the broader diversity of tropical wave types in the real world as compared to the control simulation, the latitude range of the analysis was restricted to within 10 E   of the equator, while the spectral filter was revised to match that devised for isolating moist Kelvin wave signals by Kiladis et al. (2009;see their Figure 1). The leading PC structures obtained under these modifications, depicted in Figure C1b, are very similar to those obtained for the model, but where the implied values of * E  are roughly 2 E  larger in both cases. Meanwhile, repeating the analysis for MJO-filtered anomalies, using the same filter as in Wheeler and Kiladis (1999), yields an even broader zonal wind structure ( * 17 E   ), while the precipitation structure appears more like that in Figure C1a, but with substantial negative side lobes (results not shown). In summary, it appears that the potential for extratropical forcing of eastward-moving moist tropical variability, via the projection pathway of Hoskins and Yang (2000), is substantial in both the real world and the aquaplanet setup considered herein.

Appendix D: Composite Structures of the Model's MJO Analog
As discussed in Section 4, the tropical rain spectrum produced in the standard IPAC case (Figure 9a) is dominated by a pronounced MJO-like spectral peak at 2 E k  , closely matching the corresponding observed spectrum in Figure 1a. To further assess the realism of this simulated disturbance, a composite analysis was performed in physical space using the same basic methodology as for the Kelvin wave composite in Figure 5, but with the object-defining filter specified to retain only zonal wavenumbers in the range 1 E k  -5 and periods in the range 30-120 days. The resulting horizontal flow and streamfunction anomaly patterns at 200 and 850 hPa are plotted in Figures D1a and D1b, respectively. The familiar quadrapole gyre patterns are readily apparent in both cases, with the centers of the dominant gyres located at around 25 E  latitude in the upper panel, compared to around 10 E  in the lower one, broadly similar to what is observed in the context of the real-world MJO (e.g., see Figure 2 in Kiladis et al., 2005). Taking the latter observational study as a benchmark, however, it is evident that the simulated equatorial zonal wind anomalies at 850 hPa are too strong by roughly a factor of 2, an error that is interpreted to stem from an overly strong Kelvin-mode circulation component in the model. This interpretation is based on the fact that the amplitude of the simulated streamfunction anomalies is comparable to what is actually observed. Despite this discrepancy, it is clear that the simulated disturbance is more than just a pure Kelvin wave, since the low-level westerlies trailing its convection center at the equator are substantially stronger than the low-level easterlies out ahead, as part of an associated "westerly wind burst" (WWB)-a tell-tale feature of the MJO. The accompanying "swallowtail" pattern in the simulated rainfall anomalies is also reminiscent of that seen in observations (Adames & Wallace, 2014b;Zhang & Ling, 2012). Looking at longitude-height cross-sections of the disturbance at the equator, Figures D1c and D1d reveal several additional similarities to the observed MJO, including tilted structures in temperature E T, specific humidity v E q, and apparent heating E Q anomalies that imply a gradual deepening of moisture and convection as the precipitating phase of the disturbance arrives at a fixed point from the west. Particularly noteworthy in this regard is the fact that fluctuations in v E q in the middle troposphere are roughly an order of magnitude larger (in moist static energy units) than those in E T, which is unlike what is seen in convectively coupled Kelvin waves, where fluctuations in v E q and E T are more comparable (Straub & Kiladis, 2003a). In summary, it appears that the MJO-like mode of variability produced in the model can indeed be considered as analog of the real MJO.

Data Availability Statement
All data for the SP-WRF simulations and analyses described in this study are available at https://downloads. psl.noaa.gov/Projects/FAIR_paper_data/20210302_01/.