Convectively Coupled Global Rossby Modes in an Idealized Moist GCM

Cameron G. MacDonald Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey

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Pablo Zurita-Gotor Departamento Física de la Tierra y Astrofísica, Universidad Complutense de Madrid, Madrid, Spain
Instituto de Geociencias UCM-CSIC, Madrid, Spain

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Isaac M. Held Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey

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Yi Ming Schiller Institute for Climate and Society, Boston College, Boston, Massachusetts

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Abstract

The westward-propagating convectively coupled equatorial wave (CCEW) variability produced by an idealized general circulation model (GCM) is investigated. The model is a zonally symmetric aquaplanet with a slab ocean. Water vapor in the model may condense and produce latent heating, but there is no parameterization of cloud processes, only a quasi-equilibrium convection scheme. The CCEWs produced by the model are found to be sensitive to the heat capacity of the slab and the strength of surface friction. In spectral space, the westward-propagating precipitation variability in the model is dominated by sharp peaks in spectral power at zonal wavenumbers 5 and 6. These precipitation peaks are situated along the dispersion curve of the Rossby–Haurwitz waves, suggesting a connection between the global Rossby modes and precipitation variability. Composites of these disturbances reveal global circulation patterns that extend into the midlatitudes. The moisture variance budget of these disturbances shows that moisture advection by the global Rossby modes maintains the accompanying moisture signal. This is interpreted as downgradient advection of the background moisture gradient of the intertropical convergence zone. The locations of the precipitation peaks are sensitive to Doppler shifting by the zonal winds; when this Doppler shift becomes too weak, the frequencies of the global Rossby modes become too high to effectively couple to convection. A linearized primitive equation model shows that the presence of vertical shear in the background zonal winds is vital for producing a forced response that resembles the modes produced by the GCM. The forced response of the linear model is optimally located to enhance the original circulation of the global mode.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Cameron G. MacDonald, cgm3@princeton.edu

Abstract

The westward-propagating convectively coupled equatorial wave (CCEW) variability produced by an idealized general circulation model (GCM) is investigated. The model is a zonally symmetric aquaplanet with a slab ocean. Water vapor in the model may condense and produce latent heating, but there is no parameterization of cloud processes, only a quasi-equilibrium convection scheme. The CCEWs produced by the model are found to be sensitive to the heat capacity of the slab and the strength of surface friction. In spectral space, the westward-propagating precipitation variability in the model is dominated by sharp peaks in spectral power at zonal wavenumbers 5 and 6. These precipitation peaks are situated along the dispersion curve of the Rossby–Haurwitz waves, suggesting a connection between the global Rossby modes and precipitation variability. Composites of these disturbances reveal global circulation patterns that extend into the midlatitudes. The moisture variance budget of these disturbances shows that moisture advection by the global Rossby modes maintains the accompanying moisture signal. This is interpreted as downgradient advection of the background moisture gradient of the intertropical convergence zone. The locations of the precipitation peaks are sensitive to Doppler shifting by the zonal winds; when this Doppler shift becomes too weak, the frequencies of the global Rossby modes become too high to effectively couple to convection. A linearized primitive equation model shows that the presence of vertical shear in the background zonal winds is vital for producing a forced response that resembles the modes produced by the GCM. The forced response of the linear model is optimally located to enhance the original circulation of the global mode.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Cameron G. MacDonald, cgm3@princeton.edu

1. Introduction

Convectively coupled equatorial waves (CCEWs) are a key component of the large-scale circulation of the tropical troposphere (Kiladis et al. 2009). The coherent propagation of these disturbances along the equator provides a key source of predictability in the atmosphere on synoptic and intraseasonal time scales, both in the tropics and around the entire globe. While progress is being made to improve the modeling of these disturbances (e.g., Bartana et al. 2023), comprehensive general circulation models (GCMs) still struggle to capture this wave variability, both as a component of the climatology of long integrations of these models (Lin et al. 2006; Hung et al. 2013; Ahn et al. 2020) and as a part of the subseasonal-to-seasonal prediction of the atmospheric state (Dias et al. 2023). The interaction of the large-scale circulation of these models with heavily parameterized moist convective processes is key to the genesis of these waves, and many of the challenges in producing high-fidelity simulations of tropical variability stem from the difficulties in representing these interactions (Lin et al. 2006). Here, we will consider the CCEWs produced by an idealized GCM which retains a relatively simple treatment of the essential interactions between convection and the large-scale circulation.

In spectral space, tropical convective heating is red in both frequency and wavenumber, having the greatest spectral power at large spatial scales and low frequencies (Salby and Garcia 1987; Wheeler and Kiladis 1999; Hendon and Wheeler 2008). Significant peaks above this red spectrum appear at wavenumbers and frequencies that lie along the dispersion curves of the linearized shallow water equations on the equatorial β plane with an appropriately chosen equivalent depth (Wheeler and Kiladis 1999). This equivalent depth is on the order of 25 m, far smaller than would be expected based on the dry static stability of the tropical troposphere, suggesting that interactions between convection and the large-scale circulation play a key role in producing these relatively slow waves in the tropical troposphere (Emanuel et al. 1994; Wheeler and Kiladis 1999; Kiladis et al. 2009). The shallow water equations on the equatorial β plane, first studied by Matsuno (1966), provide a foundation upon which many theories for the slow movement and instability of these CCEWs have been built, often through truncating the vertical structure of the atmosphere to a small number of internal modes (Neelin and Yu 1994; Mapes 2000; Majda and Shefter 2001; Kuang 2008).

GCMs of intermediate complexity provide a rung in the model hierarchy somewhere between shallow water models and more comprehensive GCMs (Held 2005; Maher et al. 2019). In particular, GCMs with an idealized treatment of the hydrological cycle have become popular tools for studying the influence of latent heating upon the general circulation of the atmosphere (Frierson et al. 2006, 2007; Merlis et al. 2013; Clark et al. 2018; MacDonald and Ming 2022; Garfinkel et al. 2022; Zurita-Gotor et al. 2023). The simplified boundary conditions and physical parameterizations in these models allow for easier interpretation of results, and their low computational burden allows for many runs to be performed to more thoroughly explore the parameter space of the model (O’Gorman and Schneider 2008). These models, when run with a rudimentary quasi-equilibrium convection scheme, but no parameterization of any cloud or ice processes, produce both red spectra of convection and CCEWs of various kinds as emergent components of the tropical circulation (Frierson 2007a; Shamir et al. 2021; MacDonald and Ming 2022; Garfinkel et al. 2022). This presents an arena in which to explore the sensitivities of CCEWs to physical processes that may not fit as cleanly into a simpler shallow water picture of the dynamics.

Frierson (2007a) was the first to investigate the equatorial wave variability in such an intermediate complexity GCM. He found that Kelvin waves were the most prominent form of variability; these waves were particularly sensitive to parameter choices in the model’s convection scheme. MacDonald and Ming (2022) have shown that a similar intermediate complexity GCM can produce MJO-like disturbances when augmented with a zonally asymmetric pattern of sea surface temperatures to mimic the presence of a warm pool on the equator.

In an attempt to better understand the CCEWs generated by the idealized moist GCM used in MacDonald and Ming (2022), in this paper we investigate the westward-propagating CCEWs produced by the same model and document their sensitivity to the surface characteristics of the model. We will show the rather surprising result that the most prominent westward variability is in fact not associated with equatorially trapped Rossby waves with equivalent depths on the order of 25 m but instead appears to be the result of coupling between the global Rossby modes of the atmosphere and moist convection in the tropics. These waves are a robust feature of the model across a vast swath of parameter space, appearing most strongly at westward wavenumbers 5 and 6 with periods greater than 10 days, and having spectral power highly concentrated close to the frequencies of the free external Rossby modes of the atmosphere.

Global Rossby modes (GRMs) of higher frequencies are a robustly observed feature of the real atmosphere; the most well known is the so-called 5-day wave which appears at zonal wavenumber 1 (Madden and Julian 1972; Madden 2007), but a large set of peaks have been identified elsewhere in the wavenumber–frequency spectrum (Eliasen and Machenhauer 1965; Salby 1984; Madden 2007). Sakazaki and Hamilton (2020) have recently identified an even more comprehensive set of GRMs and inertia–gravity waves in the ERA5 reanalysis. Some of these modes carry detectable signals in tropical precipitation, though there has historically been some vacillation as to the causality of the relationship between convective heating and the GRMs (Manzini and Hamilton 1993; Miyoshi and Hirooka 1999; Sakazaki 2021). King et al. (2015) have investigated the precipitation variability connected to the 5-day wave, finding that the 5-day wave mainly acted to modulate precipitation in convectively active zones in the tropical belt. Sakazaki (2021) examined precipitation signals associated with a wider set of symmetric and antisymmetric global modes. They proposed two different mechanisms for these precipitation anomalies; for modes of wavenumber 1 or 2, adiabatic compression from the passage of the wave offered a viable explanation for the observed precipitation patterns, while for modes of slightly smaller zonal scale, moisture convergence induced by the waves appeared to play a more important role. Zurita-Gotor and Held (2021) investigated the GRMs produced by both dry and moist idealized GCMs, finding peaks in the precipitation spectrum associated with the 5- and 10-day waves at wavenumber 1 in their moist GCM.

Evidently, precipitation and convective heating are robust components of the observed GRMs. However, in our idealized GCM, the precipitation signals associated with the GRMs at wavenumbers 5 and 6 dominate the overall low-frequency precipitation variability in the tropics. Through a combination of idealized GCM experiments and analysis using a linearized primitive equation model, we will illuminate a pathway for sufficiently slow GRMs to become convectively coupled and grow through their interaction with latent heating in the tropics. It will be shown that the background zonal winds—through both Doppler shifting of the GRMs and vertical shear effects—exert a strong influence on the character of the westward-propagating precipitation variability. In effect, our experiments suggest the potential for a markedly different regime of tropical wave variability in the presence of a stronger midlatitude jet relative to that of Earth. Additionally, we believe that the identification and characterization of these convectively coupled GRMs is important given their prominence in standard configurations of our idealized model; a thorough investigation into the nature of these waves is beneficial for the community who use this class of idealized GCM for studying tropical dynamics. Indeed, given—as will be shown—the global structure of the waves, those studying the midlatitudes within this modeling framework should also be aware of this mode of tropical variability.

This paper is organized in the following manner: The next section briefly introduces the idealized GCM and analysis techniques that we will employ. Section 3 documents the tropical wave variability in the model, highlighting the prominent peaks in precipitation associated with the GRMs, as well as their associated moisture variance budgets. A discussion of these results is provided in section 4, with a particular focus on the influence of vertical shear on these modes. Finally, conclusions are given in section 5.

2. Methods

a. Model formulation

We employ an idealized moist GCM, the specifics of which are described extensively in Clark et al. (2018) and MacDonald and Ming (2022). This model has evolved from the idealized GCM designed by Frierson et al. (2006, 2007) and is similar to those used in many contemporary studies (Merlis et al. 2013; Jucker and Gerber 2017; Vallis et al. 2018; Garfinkel et al. 2022). In short, the model consists of a spectral dynamical core in which specific humidity is an active tracer that produces latent heating upon condensation and thus influences the circulation of the atmosphere. Moist convection is parameterized using a simplified Betts–Miller scheme which relaxes the atmospheric temperature profile to a moist pseudoadiabat (Frierson 2007b); there is no parameterization of cloud or ice processes and any condensed moisture is rained out immediately. Parameters for the convection scheme are the same as the control run of MacDonald and Ming (2022). Large-scale condensation may also occur when the atmosphere becomes supersaturated; however, with common parameter choices, precipitation in the tropics is dominated by the parameterized component (Frierson 2007b). The model has a full-physics radiation scheme which allows for water vapor–radiation interactions, as opposed to the two-stream gray radiative transfer used in Frierson et al. (2006, 2007). All of our runs are performed in a perpetual equinox setting, so there is no seasonal cycle; however, the model still has a diurnal cycle.

The core experiments we perform in this study perturb parameters in the oceanic mixed layer and surface flux parameterizations. The bottom boundary of the model is a mixed layer ocean so that the model is energetically closed (Frierson et al. 2006; Merlis et al. 2013; MacDonald and Ming 2022). The ocean has no dynamics, only a spatially uniform heat capacity specified in the form of the depth of the mixed layer. The temperature of the mixed layer responds to the turbulent (sensible and latent) and net radiative (shortwave and longwave) fluxes at the surface. An additional prescribed heating of the ocean can also be imposed in the model, expressed as the convergence of an oceanic heat flux. Our core set of experiments does not impose any oceanic heat fluxes, but additional experiments with fixed meridional heat transport in the ocean will be discussed in section 4.

Surface turbulent fluxes of momentum, moisture, and temperature are parameterized via bulk aerodynamic formulas (Frierson et al. 2006). The winds used in the bulk formulas are bolstered by a convective gustiness factor that is set to 1 m s−1. In neutral conditions, the transfer coefficients are functions of a set of roughness lengths (z0,m, z0,q, z0,t) for momentum, moisture, and temperature, respectively (Garratt 1994). In nonneutral conditions, the drag coefficients are set by universal similarity functions which are functions of za/L, with za the height of the lowest model level and L the Obukhov length. Separate universal functions are used for the unstable case, in which the surface Richardson number Ria is negative, and the stable case, in which Ria > 0. When the Richardson number exceeds a critical value Ric, the transfer coefficients effectively become zero and there is negligible turbulent transfer of momentum, heat, and moisture. We have set Ric = 1 in all of our experiments.1

b. Experiments

We perform a series of model experiments to test the sensitivity of equatorial waves in the model to changes in the surface parameterizations. Two parameters serve to create our core set of experiments. The first of these is the momentum roughness length z0,m. We have selected three values for z0,m: a weak value of 1 × 10−5 m similar to that used in the original model of Frierson et al. (2006), a control value of 0.005 m used in Clark et al. (2018) and MacDonald and Ming (2022), and a strong value of 0.1 m. The weak and control values bracket the range of roughness lengths generally used in this class of model; the strong value is likely too large for a realistic ocean surface and is more suitable for certain land surfaces. There are, as will be shown, interesting transitions in model behavior when increasing z0,m to this value and so we include this choice as an additional core experiment. Across all of our core experiments, the roughness lengths for moisture and temperature are set to 1 × 10−5 m.

The second parameter that we vary is the depth of the slab ocean. We choose two values: a shallow mixed layer depth of 1 m and a deep mixed layer of 20 m. In the deep case, surface temperature variations on the time scale of CCEWs are very weak and these runs may be thought of as similar to a fixed SST run from the viewpoint of the CCEWs. For the shallow slab, temperature variations on the order of 0.1 K can be observed in conjunction with traveling CCEWs. Importantly, the choice of 1 m is not so shallow as to let the diurnal cycle dominate; for example, further decreasing the depth to 0.1 m leads to extremely strong day–night contrasts. These strong day–night contrasts lead to the formation of a very stable boundary layer during the nighttime, effectively eliminating turbulent surface fluxes.

We index this set of simulations by their parameters as follows: First, the depth of the mixed layer is denoted by D1 for the 1-m mixed layer depth and D20 for the 20-m depth. The three choices of momentum roughness length are then identified as W (weak) for z0,m = 1 × 10−5 m, C (control) for z0,m = 0.005 m, and S (strong) for z0,m = 0.1 m. For example, the experiment with a 20-m mixed layer and control friction will be referred to as D20C.

c. Analysis techniques

We employ a set of analysis techniques with the intention of characterizing the spectral properties of the tropical variability in our model. First and foremost among these methods is the computation of wavenumber–frequency diagrams of spectral power, following Wheeler and Kiladis (1999). We partition the time series of a given variable into 192-day segments, overlapping by 128 days. After removing the mean and linear trend from each segment, fast Fourier transforms are taken in the zonal and time dimensions, and the resulting spectral power (the squared modulus of the spectrum) is averaged over all the segments. A background spectrum is calculated by repeatedly applying a 1-2-1 filter to the spectrum along both the wavenumber and frequency dimensions; as in Wheeler and Kiladis (1999), we apply the filter 40 times in the frequency dimension and 10 times in the wavenumber dimension. Significant spectral peaks above the background are seen by taking the ratio of the power spectrum to the background, with a chi-squared test determining the minimum normalized power that is deemed significant. Cross-spectra between various model fields will also be used to characterize the global coherence of circulation patterns by computing the squared coherence and phase lag between the variables (Hayashi 1982; Hendon and Wheeler 2008).

When a more precise definition of the background spectrum is required, we use a nonlinear smoothing algorithm described by Zurita-Gotor and Held (2021). Beginning with wavenumber–frequency spectra averaged over 192-day segments, for each wavenumber and latitude we iteratively apply the smoothing operator
Dk+1(f)=min{D^k(f),N(f)},
where f is the frequency and N(f) = logP(f)/logB0(f), in which P(f) = |S(f)|2 is the power of the spectrum S(f) and logB0(f) is a cubic polynomial fit to logP(f). The hat indicates the application of a 1-2-1 filter in frequency to Dk(f). In the first iteration, we take D0(f) = N(f). As in Zurita-Gotor and Held (2021), we iterate the procedure 500 times before computing the final background spectrum as B(f)=B0D500. For any wavenumber, the peak frequency f0 is determined as the frequency at which the globally averaged P/B attains its maximum.

d. Linearized primitive equation model

To better understand the response of the atmosphere to tropical heating, especially in the presence of both meridional and vertical shear in the basic-state zonal winds, we will also employ a linearized primitive equation model, following Kasahara and da Silva Dias (1986). The model solves for either the free modes or the forced response of the atmosphere to a prescribed heating via projection onto a set of vertical and meridional basis functions. Solution of the vertical structure problem follows Kasahara (1984) and decomposes the full primitive equations into a countable set of shallow water systems with different equivalent depths [computed as the eigenvalues of the vertical structure problem considered by Kasahara (1984)]. Additional details on the formulation of the linear model are provided in the appendix.

3. Results

a. Basic characteristics

We start by documenting some essential changes to the mean state that occur when the surface roughness and mixed layer depth are altered. Figure 1a shows the zonal and time mean profiles of zonal wind at the surface (dashed lines) and in the upper troposphere near the level of the jet maximum (solid lines). Consistent with previous studies which investigated the effect of surface friction on the general circulation, reducing z0,m leads to a poleward shift of the jet (Chen et al. 2007; Polichtchouk and Shepherd 2016), and with weak friction, the jet widens significantly. There is a corresponding poleward shift in the midlatitude surface westerlies, suggesting a greater separation of the subtropical and eddy-driven components of the jet. In the tropics, reduced friction leads to an acceleration of the surface easterlies. The zonal-mean zonal winds are less sensitive to the choice of mixed layer depth, though a deeper mixed layer does lead to a slightly stronger and more equatorward jet, most easily seen in the weak friction case.

Fig. 1.
Fig. 1.

(a) Meridional profiles of the zonal-mean zonal winds for the six core experiments. Solid lines show the winds in the upper troposphere at 200 hPa, and the dashed lines show winds at the lowest model level. (b) Meridional profiles of zonal mean precipitation and evaporation for the core experiments. Solid lines show the precipitation and dashed lines the evaporation. In both panels, the profiles have been symmetrized about the equator. (c) Meridional profiles of zonal mean surface temperature for the six core experiments.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Figure 1b displays some basic changes to the hydrological cycle as the surface properties are varied. All of our experiments have a single intertropical convergence zone (ITCZ) located on the equator. Evidently, stronger surface friction is associated with a weaker precipitation maximum at the equator. Furthermore, a deeper mixed layer appears to enhance this sensitivity; the D20C and D20S cases produce less equatorial precipitation than the corresponding experiments with shallower mixed layers. Midlatitude precipitation also exhibits some sensitivity to the choice of surface friction; the storm tracks shift poleward with the jet and become less active as the friction is weakened. The latitudinal distribution of surface evaporation remains consistent across the whole set of experiments but weakens slightly with decreasing friction. Figure 1c shows the zonal mean surface temperatures for the core experiments; due to the lack of oceanic energy transport and the perpetual equinoctial insolation, there is a strong meridional temperature gradient. The experiments with stronger surface friction tend to have slightly warmer mean tropical climates.

In Fig. 2, we show representative Hovmöller plots of equatorial precipitation for the six core experiments. It is clear that varying the surface properties of the model greatly modify the dominant tropical wave variability. Figure 2a shows precipitation for the D1W case, in which the dominant mode of precipitation variability is a westward-propagating signal primarily taking on a wavenumber 5 pattern and having a phase speed of around 7 m s−1. As the strength of surface friction is increased (Figs. 2b,c), the prominence of this westward signal becomes diminished, and more eastward variability is present. In the D1S case shown in Fig. 2c, the most prominent variability is instead a slow-moving eastward mode, most visible between days 75 and 125, though westward-propagating and almost stationary patterns may also be seen in the time series.

Fig. 2.
Fig. 2.

Hovmöller plots of equatorially averaged (10°S–10°N) precipitation for the six core experiments. The time segments are taken from year 15 of each of the simulations. Note the various transitions in the dominant forms of wave variability as the parameters of the model are altered.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Further transitions in wave variability occur when the depth of the mixed layer is increased. In the D20W case (shown in Fig. 2d), one can see the imprint of eastward-propagating planetary-scale Kelvin waves with phase speeds of around 25 m s−1, which are able to make several circumnavigations of the globe as coherent structures, as was found by Frierson (2007a). These Kelvin waves are interlaced with the same wavenumber 5 westward signal observed in the D1W experiment. With the deeper mixed layer, increasing the friction leads to the emergence of a quasi-stationary wavenumber 6 mode as the dominant form of precipitation variability, which is most visible between days 125 and 200 in the D20S case shown in Fig. 2f. From this point forward, we will limit our focus to the properties of the westward-propagating precipitation variability produced by the model.

b. Spectral characteristics

A complementary spectral view of the symmetric component of precipitation variability is shown in Fig. 3. The presence of planetary-scale Kelvin waves is apparent for the experiments with the larger mixed layer depth. In the two weak friction cases (Figs. 3a,d), the strong westward modes seen in the Hovmöller plots of Fig. 2 appear as strong, sharp peaks in spectral power at westward wavenumber 5, having periods of between 10 and 20 days. These peaks are roughly situated on the dispersion curve of the equatorial Rossby waves with equivalent depths characteristic of CCEWs. For the experiments with stronger friction, we note the emergence of a second sharp peak at westward wavenumber 6 with a far slower period on the order of 100 days. These peaks are associated with the near stationary mode of variability identified in Fig. 2f. These two peaks at wavenumbers 5 and 6 are quite distinct from one another despite their close proximity in spectral space. Observations of tropical convection generally show significant spectral power along the dispersion curves of equatorial Rossby waves of the lowest meridional index, but this signal is generally more smeared out across a range of wavenumbers (Wheeler and Kiladis 1999; Hendon and Wheeler 2008; Kiladis et al. 2009). The organization of the spectral power of precipitation into a discrete set of strong peaks, some of which are located away from the dispersion curves of equatorial Rossby waves, is a surprising departure from observed spectra.

Fig. 3.
Fig. 3.

Space–time spectral diagrams of the spectral power of the symmetric component equatorial precipitation (the spectral power is averaged over 10°S–10°N before computing the background spectrum). Contours show the power below the significance threshold, which is taken to be 1.2 based on a chi-squared test. Shaded areas indicate significant spectral power above the background. Spectral power exceeding 2.6 is colored purple. The dashed lines show dispersion curves for equatorial Kelvin and Rossby waves (with meridional index 1) with equivalent depths of 25, 50, and 100 m. An additional Rossby wave curve is shown with an equivalent depth of 13 km and Doppler shifted by 20 m s−1, meant as an approximation to the Rossby–Haurwitz waves.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Greater context may be gained into the nature of these strong peaks at wavenumbers 5 and 6 by looking at the symmetric component of the power spectrum of upper-tropospheric zonal wind. These spectra are shown in Fig. 4 for our core experiments. It is clear that there exist significant spectral peaks in the zonal wind at westward wavenumbers 5 and 6 at the same frequencies where peaks in precipitation are found. These peaks roughly reside on the dispersion curve of the gravest equatorial Rossby wave with an equivalent depth of 13 km that has been Doppler shifted by 20 m s−1, used as an approximation for the frequencies of the Rossby–Haurwitz waves with the lowest symmetric meridional index (Hendon and Wheeler 2008; Potter et al. 2014). The peaks at westward wavenumbers 1 and 2 correspond to the well-studied 5- and 4-day waves, respectively (Salby 1984). Close examination of the precipitation spectra in Fig. 3 suggests that some weaker peaks, which do not exceed the significance criteria, may be seen at low wavenumbers along the dispersion curve of the Rossby–Haurwitz waves. The conspicuous location of the precipitation peaks seen at westward wavenumbers 5 and 6 in Fig. 3 suggests a clear association between GRMs and precipitation variability in the tropics. We will examine more closely the nature of this association in section 4, and in particular whether there exists some form of convective coupling between convection and the GRMs which serves to mutually reinforce the precipitation and circulation patterns displayed here.

Fig. 4.
Fig. 4.

As in Fig. 3, but for the symmetric component of the upper-tropospheric (200 hPa) zonal wind. Spectra are again averaged over 10°S–10°N.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Having noted the compelling alignment of the precipitation peaks with the dispersion curve of the Rossby–Haurwitz waves, we now characterize the global structure of these signals. To confirm that these modes have a coherent structure extending across the globe, in Fig. 5 we display the squared coherence of the surface pressure at both the equator and the Northern Hemisphere midlatitudes with the remainder of the globe as a function of frequency at wavenumbers 1–6 for the D20C case. It is clear that the frequencies associated with the GRMs display a strong band of coherence extending from the southern to the northern midlatitudes, indicating that these waves have a coherent global circulation pattern. For planetary wavenumbers 1–3, there are some additional, slower frequencies at which we see significant global coherence. These peaks may be associated with additional GRMs of higher meridional index (both symmetric and antisymmetric) present in the model (Zurita-Gotor and Held 2021), but we do not attempt to characterize these any further.

Fig. 5.
Fig. 5.

Squared coherence as a function of latitude and frequency for the D20C experiment. (a)–(f) Plots of squared coherence between surface pressure at 2°N and all other latitudes for wavenumbers 1–6. (g)–(l) Plots of squared coherence surface pressure at 40°N and all other latitudes for wavenumbers 1–6. Note that negative frequencies correspond to westward propagation.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

c. Structure of global Rossby modes

To make clear the association of these signals in convection with the GRMs, we now construct composites of the structure of the modes. To generate an index upon which to regress model output, we perform a principal component analysis of the 300-hPa streamfunction filtered to a single wavenumber (either 5 or 6) and bandpass filter around the peak frequency of the modes as determined from the spectral analysis performed above. For the k = 5 mode, the band is chosen as 10–25 days, whereas for the k = 6 mode we use the 20–100-day band, in each case taking only westward-propagating waves. The resulting empirical orthogonal functions (EOFs) are the horizontal spatial patterns of the streamfunction and the corresponding principal components (PCs) their amplitude. For both the wavenumber 5 and 6 modes, the first pair of EOFs is in quadrature with one another and together explains more than 60% of the filtered streamfunction variance. We use the first PC as our index for regression.

Figures 6a and 6b show the regressed 850-hPa geopotential height patterns for the modes in the D20C experiment for wavenumbers 5 and 6, respectively. The global, virtually untilted structure of the mode is apparent for both wavenumbers, similar to the theoretical structures for GRMs shown, for example, by Sakazaki (2021). The corresponding regression maps at the 200-hPa level are shown in Figs. 6c and 6d. Outside the deep tropics, where the regressed values are small, the geopotential height anomalies are largely in phase throughout the depth of the troposphere; the wind patterns (the vectors in Fig. 6) also look similar between the two levels, suggesting that they have a considerable barotropic (vertically averaged) component. Furthermore, it is notable that the meridional structure of the k = 6 mode is confined slightly closer to the equator than the k = 5 pattern; this is consistent with the fact that in the limit of small Lamb’s parameter, barotropic Rossby–Haurwitz waves on the sphere become trapped more closely to the equator as k increases (Boyd 1985); similar increasing meridional confinement with zonal wavenumber may be seen for the Hough modes with a finite equivalent depth (Madden 2007). The meridional structure of the meridional winds has a single robust zero crossing, located on the equator, further justifying the connection with the GRMs of the lowest symmetric meridional index.

Fig. 6.
Fig. 6.

(a) Regression patterns of 850-hPa geopotential height (shading) and horizontal winds (vectors) for the k = 5 mode of the D20C experiment. A reference wind vector is provided in the top-right corner of the panel. (b) As in (a), but for the k = 6 mode of the same experiment. (c) As in (a), but for regression patterns at the 200-hPa level. (d) As in (c), but for the k = 6 mode.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Figure 7 shows the amplitude of the regressed geopotential height patterns for the k = 5 and k = 6 modes. The amplitude is strongest in the midlatitudes and in the upper troposphere, reaching a maximum at around 200 hPa and decreasing above that level. The modes thus do not have the vertical structure of the Lamb wave (increasing exponentially in a log pressure coordinate). The 5-day wave produced by the model, on the other hand, does have a vertical structure resembling the Lamb wave (not shown), as was also found for the 5-day wave in the dry simulations of Zurita-Gotor and Held (2021). Given the strong background zonal flow and the relatively small phase speed of the low-frequency modes at westward wavenumbers 5 and 6, it should be expected that the vertical structures of the modes bear a greater resemblance to the mean flow than to the Lamb wave (Held et al. 1985).

Fig. 7.
Fig. 7.

(a) Amplitude of the regressed geopotential pattern as a function of latitude and pressure for the k = 5 mode in the D20C experiment. The amplitude is computed as 2[z2], where z′ is the regressed geopotential anomaly and the square brackets indicate a zonal average. (b) As in (a) but for the k = 6 mode. Note the logarithmic scale of the vertical axis.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Regressed patterns of precipitation and column-integrated water vapor are shown in Fig. 8, along with the vertically averaged horizontal winds. In the tropics, the precipitation is strongest off the equator, with the pattern attaining its maximum amplitude at around 10°N/S. These maxima and minima in the regressed precipitation are well correlated with column water vapor anomalies. There also exist weaker, extratropical peaks and troughs of precipitation that do not have commensurate anomalies in column-integrated water vapor; we speculate that these are due to the modulation of midlatitude storms by the circulation pattern of the GRM. In the tropics, regions of enhanced (reduced) precipitation are associated with poleward (equatorward) barotropic flow, though the poleward flow tends to slightly lead (in a westward sense) the regressed column water vapor and precipitation. This relation points to advection by the barotropic circulation of the GRMs as a potential mechanism for the maintenance of the moisture signal accompanying the modes. Precipitation in the deep tropics has a phase shift relative to the off-equatorial peaks, with this shift being more pronounced for the k = 5 mode than for the k = 6 mode, for which the equatorial precipitation is more nearly in phase with the off-equatorial maxima.

Fig. 8.
Fig. 8.

Horizontal structures of the (a) k = 5 and (b) k = 6 modes of the D20C experiment. Regressed precipitation anomalies are shown in the shading, and regressed column water vapor anomalies are shown by the black contours, with the zero contour omitted (contour interval 0.4 mm). The vectors show the regressed vertically averaged wind vectors. Reference vectors are provided in the top-right corner of each panel.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

d. Moisture budget

Having characterized the structure of these low-frequency GRMs, we now examine the column-integrated moisture budget associated with the disturbances. This budget may be written as
qt=qvP+E,
where ⟨q⟩ is the column-integrated water vapor, v is the horizontal wind vector, P is the precipitation, and E is the latent heat flux out of the surface. The angle brackets indicate a vertical pressure integral from the surface to the top of the atmosphere. Given the strongly barotropic nature of the GRMs, it is useful to separate out the barotropic component of the winds, so that we have
v=V+v˜,
where V = gv⟩/ps is the vertically averaged wind vector and v˜ is the residual baroclinic wind. The vertically averaged component captures the barotropic circulation of the GRMs, while the residual wind accounts for circulations with vertical motion. Assuming that the barotropic divergence ∇ ⋅ V is small, the moisture budget can be rewritten as
qt=Vqqv˜P+E.
In practice, we regress each individual term in the moisture budget against the first PC of the 300-hPa streamfunction and evaluate the column-integrated moisture variance budget by multiplying Eq. (4) by the regressed ⟨q⟩ pattern and averaging in the zonal direction, so that we have
t[q2]2=[q(Vq)][q(qv˜)][q(PE)],
where [⋅] indicates a zonal average and (⋅)′ denotes a regressed quantity. We are thus able to separate contributions to the production of column moisture variance from barotropic and baroclinic circulations, and each of these advection terms can be further decomposed into contributions from zonal and meridional circulations. Such variance budgets have been used in various forms as diagnostic tools for a variety of tropical circulations (Andersen and Kuang 2012; Wing and Emanuel 2014; Adames 2017; MacDonald and Ming 2022; Yao et al. 2022). Given that the modes we are investigating appear so strongly at a single wavenumber, this approach is particularly well suited as the variance can be formed via a simple zonal average.

Figure 9 shows the resulting moisture variance budgets as functions of latitude for both the wavenumber 5 and 6 modes in the D20C experiment. For both modes, the strongest contributor to the maintenance of the moisture variance is the meridional advection of moisture by the barotropic circulation. Slightly equatorward of the maximum moisture variance, there is also a significant positive contribution from the residual baroclinic circulation; this baroclinic contribution is larger relative to the barotropic production for k = 5 than for k = 6. In both cases, this production of variance is primarily balanced by the destruction of variance by precipitation.

Fig. 9.
Fig. 9.

(a) Moisture variance budget for the k = 5 mode of the D20C experiment. The solid lines show the individual contributions to the production of moisture variance. The dashed black line denotes the moisture variance using the right axis. (b) As in (a), but for the k = 6 mode of the D20C experiment.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

In Fig. 10a, we show the variance contribution from meridional barotropic moisture advection as a function of latitude. Also shown are the contributions to this term from barotropic advection of the background moisture gradient by the regressed circulation of the global modes, i.e., the component Vyq¯, where the overbar denotes a basic-state quantity. It is clear that the meridional structure of the full meridional advection term is very similar to that of the advection background moisture gradient; in fact for both wavenumbers 5 and 6, the variance contribution from Vyq¯ is about 50% larger than the full term, showing that advection of the background moisture gradient is the main source of moisture variance for the modes. Figure 10b shows the zonal phases as a function of latitude for ⟨q⟩′, (Vyq⟩)′ and Vyq¯, for both wavenumbers 5 and 6. It is clear that the phases of the full barotropic meridional advection and the contribution from the advection of the background moisture gradient are quite similar, again suggesting that Vyq¯ is the most important portion of the meridional moisture advection. The picture is then that the barotropic circulation of the GRMs can efficiently converge moisture into a column via advection down the strong off-equator meridional moisture gradients in the tropics.

Fig. 10.
Fig. 10.

(a) Moisture variance contribution from total meridional advection by the barotropic circulation (solid lines) and the component due to advection of the background meridional moisture gradient (dashed lines). The k = 5 (k = 6) cases are shown in blue (red). (b) Zonal phase of the regressed column water vapor (solid lines), the total meridional barotropic moisture advection (dashed lines), and the component due to advection of the background meridional moisture gradient (dash–dotted lines). Phases have been shifted so that the regressed ⟨q′⟩ have a phase of zero at the equator in each case.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

The GRMs we have simulated also cause significant perturbations to radiative fluxes at the top of the atmosphere. Outgoing longwave radiation (OLR) anomalies on the order of ±15 W m2 are associated with the signature of the global mode, with negative OLR anomalies associated with enhanced precipitation and column water vapor (not shown). Therefore, there is the potential for anomalous radiative heating to act as a source of column moist static energy variance that is not considered in the column moisture variance budget. To evaluate the importance of this effect, we have performed a variant of the D20C experiment where this radiative feedback is explicitly eliminated. This is done by providing the radiation scheme with a climatological specific humidity field from the original D20C experiment in place of the actual specific humidity field. The symmetric component of the spectrum of equatorial precipitation is nearly indistinguishable from that of the original D20C experiment and thus is not shown for brevity. It is therefore apparent that the radiative perturbations accompanying the GRMs are not fundamental to their existence in the model.

4. Discussion

a. Location of precipitation peaks

We have shown that barotropic moisture advection by GRMs can induce a pattern of moisture convergence in the tropics that is largely balanced by a compensating pattern of precipitation. The resulting spectral peaks of precipitation tend to only appear at wavenumbers 5 and 6, yet we have seen that significant peaks in the spectrum of zonal wind exist along the Rossby–Haurwitz dispersion curve for wavenumbers 1–4 as well, these having only very weak peaks in precipitation that do not contribute significantly to the overall tropical variability.

Some of the runs we have performed show precipitation peaks only at wavenumber 5, while others display distinct peaks at wavenumber 6 as well. The GRMs are Doppler shifted by the zonal mean winds, causing the modes to propagate at slower westward phase speeds than would occur in a resting atmosphere. Figure 11 shows how the approximate dispersion curves for Rossby–Haurwitz waves are influenced by Doppler shifting by the zonal-mean zonal wind; as the strength of the Doppler shifting is increased, the frequency of the Rossby–Haurwitz wave at any given westward wavenumber is decreased. For eastward Doppler shifts of around 20 m s−1, the dispersion curves reside in the spectral region where we have observed the precipitation peaks in our experiments. Of course, the actual frequencies of the peaks will also be influenced by the meridional and vertical shear of the basic-state zonal winds, and therefore, the specific choice of equivalent depth and Doppler shift are necessarily empirical, but the alignment of the spectral peaks along the approximate dispersion curves (Fig. 4) suggests that a single bulk Doppler shift is a reasonable paradigm with which to understand the locations of the peak frequencies.

Fig. 11.
Fig. 11.

Doppler shifting of the Rossby–Haurwitz wave. As before, the dispersion curves of the Rossby–Haurwitz wave are approximated by the dispersion curve of the gravest symmetric equatorial Rossby wave with an equivalent depth of 13 km. The dispersion curve with no Doppler shifting (the intrinsic frequency) is shown as a solid black line. Further dispersion curves with eastward Doppler shifts of 10 m s−1 (blue), 20 m s−1 (red), and 30 m s−1 (orange) are also shown. Dotted lines show dispersion curves with equivalent depths of 25, 50, and 150 m, representative of observed convectively coupled equatorial Rossby waves. Magenta triangles indicate the frequencies of free modes with external vertical structures in our linearized primitive equation model, using basic-state winds taken from the D20C experiment.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

We have seen in Fig. 3 that the experiments with the weakest surface friction produce peaks only at wavenumber 5, while the runs with stronger surface friction generally produce peaks at both wavenumbers 5 and 6. From Fig. 1, we can see that the weak friction runs have a significantly wider and more barotropic jet. A sufficient Doppler shifting from this stronger zonal mean flow could eliminate the possibility of a westward-propagating mode at wavenumber 6, though the details likely depend on the meridional and vertical structure of the zonal winds.

As a more definitive test of the sensitivity of these GRMs to Doppler shifting by the basic state, we have run an additional series of experiments with a prescribed oceanic heat flux. The form of the heat flux is that used in Bischoff and Schneider (2016) and is given by
F=Q0(cosφ)1(1φ2Δ2)exp[φ2/(2Δ2)],
where Q0 is the amplitude of the heat flux, φ is the latitude, and Δ is a parameter which sets the meridional scale of the heating; as in Bischoff and Schneider (2016), we use Δ = 16°. This heat flux is zonally and hemispherically symmetric. Its purpose, with a positive amplitude, is to mimic the observed shape of the zonal mean oceanic energy transport. As such, it modulates the energy transport required of the atmosphere. A positive (negative) value of Q0 will produce an additional poleward (equatorward) energy transport in the tropics and thus decrease (increase) the energy transport required of the atmospheric circulation (Merlis et al. 2013). The corresponding change in the strength of the Hadley cell will affect the speed of the subtropical jet (STJ) and thereby influence the Doppler shifting of the GRMs.

Figure 12 shows the zonal-mean zonal winds for a set of experiments with the above oceanic heat flux added to the D20C experiment with Q0 = {−50, −25, 25, 50} W m−2, as well as the original D20C experiment with no heat flux added. A positive (negative) value of Q0 weakens (strengthens) the STJ and causes a poleward (equatorward) shift in its latitude. The choice of Q0 also has considerable influence on the upper-tropospheric winds in the tropics, where negative values of Q0 lead to weak superrotation. The surface westerlies in the midlatitudes shift to align with the jet but do not change in magnitude.

Fig. 12.
Fig. 12.

Profiles of the zonal-mean zonal winds in experiments with various amplitudes of the forcing given in Eq. (6) added to the D20C experiment. As in Fig. 1, the solid lines show the 200-hPa zonal winds and the dashed lines show the zonal winds at the lowest model level.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

In Figs. 13a–e, we show the precipitation and 200-hPa zonal wind spectra for this set of experiments. As the strength of the STJ is varied, the spectral peaks in the zonal wind are shifted in frequency. In accordance with the Doppler shifting of the Rossby–Haurwitz wave by the basic-state winds, when the STJ is made stronger (weaker), the peaks at a given wavenumber move to lower (higher) frequencies. Equivalently, when the STJ strengthens (weakens), the range of wavenumbers which support low-frequency modes moves to smaller (larger) westward wavenumbers. For the case with the strongest STJ, this leads to the precipitation peaks now residing at westward wavenumbers 4 and 5, with the k = 5 mode now having a much smaller westward phase speed. In the case with the weakest STJ, the wavenumber 5 and 6 modes have frequencies closer to 0.2 cpd, seemingly too high to have an associated strong peak in the precipitation power spectrum.

Fig. 13.
Fig. 13.

Space–time spectral diagrams of the spectral power of the symmetric components of 200-hPa zonal wind (contours) and precipitation (shading) averaged over the tropics (10°S–10°N) for the experiments with oceanic heat flux added. Contours for the zonal wind spectra start at 1.2 and the greatest contour is drawn at 4.4. For the precipitation contours, power greater than 2.6 is colored purple. The dotted lines are dispersion curves for the equatorial Rossby waves as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

In performing these prescribed ocean heat flux experiments, we have shown that the locations of the GRMs are influenced by Doppler shifting by the basic-state zonal winds and that our idealized model must support GRMs at sufficiently low frequencies in order to produce strong, coherent westward wave variability. Presumably, there should also be a lower bound on the frequencies that can support these GRMs: For sufficiently slow (or eastward) propagation speeds, the discrete set of GRMs will give way to a continuum of modes as critical layers are formed where the zonal flow matches the phase speed of the wave (Kasahara 1980). This effect may be important for understanding why our experiments with stronger surface friction (which have weaker easterlies in the tropics) favor generating the very slow mode at wavenumber 6.

b. Convective coupling of the global Rossby modes

The existence of strong westward-propagating precipitation variability is conditioned on the presence of low-frequency GRMs, so it seems reasonable that the large-scale circulation is playing a key role in promoting the growth of these waves. The spectral peaks in precipitation are located along the dispersion curve of the Rossby–Haurwitz waves, suggesting that convection is not significantly modifying the phase speed of the GRMs. The insensitivity of the GRMs to interactive radiation further points to a growth mechanism rooted in large-scale dynamics. It is then helpful to consider the response of the local convective dynamics to the moisture advection induced by the passage of a GRM. In the most basic treatment, the moisture budget [Eq. (4)] can be written as
ddtq=P+F,
where F′ represents the forcing of the column moisture by the GRM, which is viewed as external to the local convective dynamics. Our treatment here bears some similarities to the method of Lindzen (2003). Taking precipitation to be related to column moisture by a convective relaxation time τc, the convective response to a purely oscillatory forcing with local frequency σ and amplitude F0 is given by
P0F0=11iστc.
The convective relaxation time may be computed for each mode as the ratio [P′⟨q⟩′]/[P2]; averaging from 30°S to 30°N yields bulk estimates of τc = 42 h and τc = 36 h for the k = 5 and k = 6 modes, respectively. The amplitude of the normalized precipitation response depends on the frequency of the forcing as 1/1+σ2τc2, so that higher-frequency waves elicit a weaker convective signal for the same forcing. With the computed estimates of τc, a forcing with a frequency of around 0.2 cpd (characteristic of the 5-day wave, for example) would elicit a precipitation response about 4 times smaller than the magnitude of the forcing.

The phase lag between the forcing and precipitation response also depends on the frequency of the forcing. Figure 14 shows the resulting phase between the forcing and precipitation as a function of frequency using the τc values computed above. For very low frequencies, the precipitation response is essentially in phase with the forcing and approaches a phase lag of 90° as the frequency becomes large. Also shown in the figure are the measured phase differences between the regressed meridional barotropic moisture advection −(Vyq⟩)′ and the regressed column water vapor ⟨q⟩′. The phase lags are in good agreement with those predicted by the simple theory presented above, solidifying barotropic advection as a key process in setting the precipitation associated with the global modes.

Fig. 14.
Fig. 14.

Phase difference between the moisture forcing and precipitation response. Solid lines show the phase of the response computed using Eq. (8). The blue and red vertical lines denote the frequencies of the k = 5 and k = 6 modes, respectively. The triangles show the phase between the meridional barotropic moisture advection and the column water vapor.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

This argument is rather simple but serves to illustrate the basic idea that convection responds more strongly to a slow forcing. The actual response function for the amplitude of the precipitation may drop off far more sharply as a function of frequency than what we have described, or be influenced by further processes that we have not considered in this most basic treatment. For example, the strength of the peaks in the idealized GCM shows some sensitivity to the mixed layer depth, with the wavenumber 6 mode favored when the heat capacity of the ocean is larger. Further consideration of ocean coupling, such as that in Sobel and Gildor (2003), could potentially be able to capture this effect.

c. The influence of vertical shear

The GRMs that are produced in our idealized GCM have a distinctive structure. Outside of the deep tropics, geopotential height anomalies have the same sign throughout the depth of the troposphere and exhibit minimal meridional tilt. In an atmosphere in uniform rotation, the circulation response to an imposed heating in the tropics is primarily baroclinic. That is, for a heat source with a vertical structure that is essentially a half-sine wave extending through the depth of the troposphere, the geopotential height and horizontal winds of the forced response will have opposite signs in the upper and lower troposphere (Geisler and Stevens 1982; Kasahara 1984; Salby and Garcia 1987). However, the presence of vertical shear offers a pathway for the imposed heating to produce a more barotropic response through a coupling of the external mode to the internal modes of the chosen vertical discretization (Lim and Chang 1986; Kasahara and da Silva Dias 1986).

We shall use a linearized model similar to that of Kasahara and da Silva Dias (1986) to understand the role of vertical shear in setting the meridional and vertical structure of the GRMs in our idealized GCM. This is done, as in preceding studies (Geisler and Stevens 1982; Kasahara 1984; Kasahara and da Silva Dias 1986), by computing the forced response to an imposed tropical heat source. Our heat source is meant to roughly mimic the convective heating associated with the GRMs in our GCM experiments, i.e., it has a zonal wavenumber 5 pattern and has a frequency of −0.064 cpd. The exact meridional and vertical structure of the heating is described in the appendix. We solve the linear model for basic states both with and without vertical shear. In the case without vertical shear, the basic-state zonal winds match the 500-hPa climatology of the D20C experiment at every vertical level, following Kasahara and da Silva Dias (1986). When vertical shear is included, we enforce a linear vertical shear profile such that the basic-state zonal winds match the climatology of the model at the surface (1000 hPa) and at 200 hPa. The specifics of the construction of this shear profile are discussed in the appendix.

Figures 15a and 15c show the forced response in the absence of vertical shear in the lower and upper troposphere, respectively. Consistent with the results of Kasahara and da Silva Dias (1986), the response is largely baroclinic, with the upper-level pattern in the tropics being of the opposite sign to that of the lower-level pattern. The amplitude of the response outside of the tropics is quite small. Figures 15b and 15d show the response in the presence of vertical shear for an identical heating. It is clear that vertical shear has a huge influence on both the structure and amplitude of the response. Most importantly for our purposes, the extratropical response has strengthened significantly and has the same sign at upper and lower levels at any given longitude. The amplitude of both the upper- and lower-level response has also greatly increased and extends much further poleward.

Fig. 15.
Fig. 15.

(a) Forced response of the linearized primitive equation model to an imposed wavenumber 5 heat source in the absence of vertical shear. Shading shows the response of the 870-hPa geopotential height, and the vectors show the response of the horizontal winds at the same level. A reference wind vector is provided in the top-right corner of the panel. The orange triangles situated on the equator mark the crests of the prescribed heating distribution. (b) As in (a), but the response is for an atmosphere with vertical shear. (c) As in (a), but for the response at 227 hPa without vertical shear. (d) As in (c), but for an atmosphere with vertical shear.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

In Fig. 16, we show the amplitude of the geopotential height response to the imposed heating in the linearized primitive equation model. For the case without vertical shear (Fig. 16a), the dominant baroclinic structure of the response is apparent in the tropics. With the addition of vertical shear (Fig. 16b), the structure of the amplitude now resembles that shown for the simulated mode in Fig. 7a, with maximum amplitude in the upper troposphere around 35°N/S. The linear model also produces a relatively strong response near the surface in the deep tropics, not seen for the mode in the idealized GCM, where the lower-tropospheric winds are far more aligned with those of the upper troposphere. Except for spatially uniform momentum and thermal damping, we have not included any effects of friction in our linearized treatment, which may influence the strength of the tropical response. The basic state of the linear model is also assumed to be in gradient wind balance (Kasahara and da Silva Dias 1986), whereas the GCM, which has a meridional overturning circulation, would be expected to have considerably weaker tropical temperature gradients (Held and Hou 1980; Plumb and Hou 1992), which could also affect the structure of the forced response.

Fig. 16.
Fig. 16.

(a) Amplitude of the geopotential height response in the linearized model, computed as 2[z2], as a function of latitude and pressure for the case with no vertical shear. (b) As in (a), but for the case with vertical shear. (c) Total eddy energy, computed following Kasahara and da Silva Dias (1986), as a function of the forcing frequency for the cases with and without vertical shear. The red (blue) dashed vertical line denotes the frequency of the free external mode with (without) vertical shear.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

Figure 16c shows the total eddy energy (Kasahara 1984; Kasahara and da Silva Dias 1986) of the forced response for a wide range of forcing frequencies. For the case with vertical shear, the amplitude achieves a peak close to the free external mode frequency; the peak is rather broad and extends out to frequencies around −0.1 cpd. The spectral peaks in zonal wind we have seen in our idealized GCM also extend out to similar frequencies (cf. Fig. 4), suggesting that the forced response to heating may exert some control on the width of spectral peaks in the model, in addition to the wave–wave interactions and frictional damping discussed by Zurita-Gotor and Held (2021). The barotropic case exhibits no peak in amplitude close to the frequency of the free external mode, highlighting the importance of vertical shear.

Further computations with the linearized primitive equation model (not shown) have suggested that the barotropic structure and lack of meridional tilt are rather special properties of the chosen forcing frequencies. That is, when the prescribed heating has a wavenumber and frequency that do not closely correspond to the location of a free GRM in the linear model, the resulting forced response will in general have pronounced vertical and meridional tilts throughout the troposphere. Computing the free modes of the linear model confirms that these peak locations are associated with the external modes—appropriately modified by the sheared basic-state winds—of the atmosphere (see Fig. 11). There is excellent agreement between the frequencies of the computed free external modes and the frequencies of the surface pressure peaks in our idealized GCM across all of our experiments.

d. Potential for positive feedback

The presence of strong peaks in the spectra of upper-tropospheric zonal wind along the Rossby–Haurwitz dispersion curve at wavenumbers 1–4 suggests that the model can generate GRMs without there needing to be a coherent convective signal accompanying the modes. Previous studies have shown that dry GCMs can generate a similar spectrum of GRMs at these wavenumbers (Potter et al. 2014; Zurita-Gotor and Held 2021). Zurita-Gotor and Held (2021) have argued that the forcing of the GRMs in these dry models is due to the breaking of baroclinic waves in the midlatitudes. The strong, low-frequency peaks that we have focused on in this work are seemingly absent from dry models (Potter et al. 2014, their Fig. 11), suggesting that wave breaking is less efficient at exciting these modes. This seems consistent with the picture of an inverse energy cascade put forth by Zurita-Gotor and Held (2021), as less energy would reside in these larger wavenumbers. While wave breaking could certainly still act as an initial excitation mechanism for the low-frequency GRMs, we now consider instead the potential for the convective heating accompanying a GRM to couple to its circulation and thereby provide a positive feedback to promote further growth of the modes.

First, we will carefully characterize the strength of the GRMs in our model runs. Zurita-Gotor and Held (2021) showed that in their dry model, where GRMs were forced by wave breaking, the strength of the peaks scaled as a power law with the associated background spectrum. Figure 17 shows the strength of the spectral peaks of surface pressure for the six core experiments and four experiments with prescribed ocean energy transport as a function of the background spectrum computed using the algorithm described in section 2c. Consistent with Zurita-Gotor and Held (2021, their Fig. 10a), we find that the strength of the peaks that exhibit no significant precipitation signal has a power-law scaling with their background spectra. On the other hand, the peaks with a strong associated precipitation signal (marked by square brackets in Fig. 17) tend to lie significantly above the dry scaling law.2 In the experiments with ocean energy transport, it is clear that the peaks at westward wavenumbers 1–4 continue to follow roughly the same power law as our core experiments. However, the peaks at wavenumbers 5 and 6 are clearly sensitive to the ocean energy transport; when the oceanic energy flux is equatorward (poleward), the peaks lie further above (closer to) the power-law scaling of the lower wavenumber peaks. Evidently, some factor other than wave breaking is setting the strengths of these peaks. If we think of these modes as being truly convectively coupled, then the moisture budget presented in section 3d points to the background meridional moisture gradient—which should become steeper with stronger equatorward ocean energy flux—as a reasonable candidate to be setting the strength of the peaks.

Fig. 17.
Fig. 17.

Scaling of the peak powers of the GRMs with the computed background spectrum (cf. section 2c). The numbers indicate the zonal wavenumber, with peaks with a strong associated precipitation signal having square brackets around the wavenumber. Note the log scaling of the axes.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

The moisture variance budgets analyzed in section 3d suggest that precipitation is largely balanced by moisture advection by the barotropic circulation, with the dominant contribution coming from poleward advection down the meridional moisture gradient of the equatorial ITCZ. Based on the phase lags shown in Fig. 14, for the wavenumber 5 mode we should expect the convective heating to lag (in a westward sense) the poleward flow with a phase lag of around 45°. Now, consider the forced response (in the presence of vertical shear) shown in Fig. 15d. It is clear that the pattern of the geopotential height slightly lags the prescribed convective heating (marked by the orange triangles), producing a response with poleward flow leading the heating by around 45°. Thus, the circulation response to the convective heating is essentially exactly in phase with the barotropic circulation that initially gave rise to the heating. Hence, the heating accompanying these modes seems to offer a viable pathway for their amplification via coupling to convection, which is sketched schematically in Fig. 18. Our approach has shown that the forcing of the GRMs by moisture advection is self-consistent in the sense of Stevens and Lindzen (1978) and Lindzen (2003). That is, when meridional advection is treated as a forcing of the moisture equation, the resulting circulation response produces meridional advection that is precisely in phase with the forcing. This approach, in which we have prescribed a forcing with a real frequency in both the convective (section 4b) and large-scale (section 4c) dynamics, is considerably more flexible than the more traditional search for the most unstable mode of a linear problem—in particular, we were able to include the influence of both vertical and meridional shear in a straightforward manner.

Fig. 18.
Fig. 18.

Schematic of the potential pathway for positive feedback. The original circulation is shown as the black ovals, representing geopotential height contours. The red arrow indicates poleward barotropic flow that acts as the dominant source of moisture. The location of the resulting positive convective heating perturbation is marked by an orange triangle. The circulation response to this heating is shown in the orange contours. The response is in phase with the original circulation and can promote the growth of the disturbance.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

We would also like to understand why this pathway for wave growth is apparently unavailable to GRMs of lower wavenumber and higher frequency. The basic mechanism whereby barotropic moisture advection induces a precipitation response—albeit with a weaker amplitude—should be active in these modes. The resultant heating would, however, have a greater phase lag, closer to around 75° (Fig. 14), if τc remains on the order of 1 day. Figure 19 shows the forced response for a zonal wavenumber 2 heat source with a frequency of −0.25 cpd, i.e., the wavenumber and frequency of the 4-day wave in the idealized model. The geopotential response is now essentially in phase with the convective heating. Taking into account the larger phase lag of the convection relative to the original circulation, it again appears that the forced response is close to being optimally situated to enhance the barotropic circulation of the mode. Based on this, it seems that the lack of convective coupling for the faster GRMs is more likely rooted in the response of convection to the passage of the modes. That is, even though the phase lags are aligned for cooperative interaction, the heating produced in response to the presence of the GRM is too small to overcome whatever processes—be it friction or wave–wave interactions—are damping the modes.

Fig. 19.
Fig. 19.

As in Fig. 15, but for a prescribed heating with a frequency of −0.25 cpd and a zonal wavenumber 2 pattern in an atmosphere with vertical shear.

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

e. Absence of equatorial Rossby waves

It is notable that across all of our simulations, there is a distinct absence of significant spectral power associated with equatorially trapped Rossby waves with equivalent depths on the order of 25 m. Even with the addition of a strong poleward oceanic heat transport, where the precipitation distribution is less concentrated at the equator and the GRMs have considerably higher frequencies, a significant precipitation signal associated with equatorial Rossby waves is not forthcoming (e.g., Fig. 13e). In observational analyses of convection, there is typically a strong band of spectral power aligned with the dispersion curve of the gravest equatorial Rossby wave (Wheeler and Kiladis 1999; Hendon and Wheeler 2008; Kiladis et al. 2009). The accompanying dynamical signatures of these waves differ significantly from the global structures we have produced in our model (Kiladis and Wheeler 1995; Wheeler et al. 2000).

This absence of equatorial Rossby wave variability presents a challenge for simple theoretical models of these waves. Such simple models generally leverage instability mechanisms such as wind-induced surface heat exchange (WISHE) to motivate the instability of these waves and therefore their emergence above the background spectrum of convection (Fuchs-Stone et al. 2019; Chen 2022). The processes necessary to these instability mechanisms should be well represented in our model, so a key question becomes what processes are suppressing this instability. There is, of course, the possibility that further exploration of the parameter space of the model could produce regimes in which bonafide convectively coupled equatorial Rossby waves exist as a dominant mode of tropical variability.

5. Conclusions

We have shown that the dominant source of westward-propagating precipitation variability in an idealized moist GCM appears to be associated not with equatorial Rossby waves with small equivalent depths as observed for Earth, but instead due to a convective coupling of the global Rossby modes (GRMs) of the atmosphere at westward wavenumbers 5 and 6. The existence of these modes is persistent across a range of climates simulated by varying the roughness and heat capacity of the model’s lower boundary. It was found that the maintenance of the moisture signal associated with the modes was mainly engendered by the downgradient advection of moisture by the barotropic circulation pattern of the GRM itself.

It appears that the sector of spectral space in which convectively coupled GRMs can exist is rather limited. The strong convective coupling was found to disappear for GRMs with periods less than 10 days; there should also exist an upper bound (in an eastward sense) on the phase speeds of the waves when critical layers form and the meridional spectrum of eigenfunctions becomes continuous. The locations of the spectral peaks were strongly influenced by Doppler shifting induced by the basic-state zonal winds of the model and were consistent with the simpler picture of a uniform Doppler shift of the Rossby–Haurwitz waves. A sufficient weakening of the STJ (and hence the Doppler shift) through the inclusion of prescribed poleward oceanic energy transport was shown to greatly weaken the convective signal of the GRMs and displace their spectral peaks to higher frequencies. Experiments with a broader set of prescribed ocean energy transports showed that the spectral peaks of precipitation respond to this energy transport in a manner consistent with Doppler shifting by the basic-state zonal winds. In its standard configuration, our model, given its perpetual equinox solar forcing and lack of oceanic transport, tends to have a very strong jet and so the periods of the free external modes in the model—in particular at wavenumbers 5 and 6—are considerably longer than those that have been identified for Earth (Sakazaki 2021).

It was further shown, through the use of a linearized primitive equation model, that vertical shear in the basic-state zonal winds played a vital role in setting the structure of the circulation response to an imposed heat source, consistent with the results of Kasahara and da Silva Dias (1986). Specific to our purposes, we found that the presence of vertical shear led to a forced circulation with a global extent that was largely barotropic and had no meridional tilt, similar to the structure of the GRMs in our idealized GCM. Without any vertical shear, the forced response was primarily baroclinic and confined to the tropics. It was additionally shown that vertical shear is a necessary condition for an amplitude peak to occur close to the frequency of the free external mode at a given wavenumber. These linear computations were then used to show that a wavenumber 5 heat source propagating at the phase speed of the k = 5 GRM could cooperatively interact with the barotropic circulation of the GRM that gave rise to the heating in the first place. This interaction is conditioned on having a sufficiently strong convective response to the GRM, and the relatively weaker response for higher-frequency forcing provides the most likely candidate for explaining why this convective coupling is only observed for sufficiently slow GRMs. The overall picture that emerges from this work is that in this idealized GCM, the character of the westward-propagating CCEWs is largely controlled by the zonal winds, both through Doppler shifting and through the presence of vertical shear. Indeed, changes to the westward-propagating CCEWs induced by altering the surface friction in the model seem to be largely explainable via the influence these changes have on the basic-state zonal winds.

It would be interesting to systematically investigate the existence of these modes across a wider set of GCMs of varying degrees of complexity. Such experiments could give a better sense of how important the specifics of the model’s convection scheme are to the growth of these GRMs. One could additionally hope to gain some insight into how modifications to the lower boundary conditions—beyond those to mixed layer depth and surface friction considered here—can act to damp these modes from the model. Shamir et al. (2021) have investigated the wave variability in a very similar idealized model which produces a signal of enhanced spectral power along the Rossby–Haurwitz dispersion curve and have shown (their Fig. 4) that the spectral peaks associated with the GRMs are diminished when realistic topography, land–sea contrast, and oceanic heat fluxes are included in the model, suggesting that these additional complexities may act to weaken this particular component of tropical variability. This seems consistent with our result that the imposition of poleward oceanic heat transport—which should bring the model’s climate into a state closer to that of Earth—tends to lessen the prominence of these global modes in the model. Of course, we have only run the model with the simplified Betts–Miller scheme of Frierson (2007b), so it could also be the case that some aspect of this specific treatment of moist convection greatly favors the growth of these modes. However, a cursory examination of the symmetric precipitation spectra produced by more comprehensive models in a similar zonally symmetric aquaplanet setting (Nasuno et al. 2008; Medeiros et al. 2016; Rios-Berrios et al. 2022) suggests that some of these models have a propensity to generate convectively coupled GRMs. The results of Rios-Berrios et al. (2022, their Fig. 7) suggest that, like other CCEWs, the existence of convectively coupled GRMs is very sensitive to the particular treatment of convection in a given model.

1

It should be noted that the model has another distinct parameter which is also referred to as the critical Richardson number in its boundary layer diffusivity scheme. This other parameter determines the height of the planetary boundary layer in the model. Details are provided in Frierson et al. (2006).

2

As an aside, we also note that peaks associated with the 5-day waves in the two weak friction experiments also lie far above the dry scaling law. Zurita-Gotor and Held (2021) have shown that the strength of this peak is sensitive to surface friction in this class of model.

Acknowledgments.

We thank Chaim Garfinkel, George Kiladis, and an anonymous reviewer for their comments which greatly improved the manuscript. We acknowledge GFDL computing resources made available for this research. This report was prepared by CGM under Awards NA18OAR4320123 and NA23OAR4320198 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the author(s) and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration, or the U.S. Department of Commerce. PZG acknowledges funding by Grant PID2022-136316NB-I00 by MCIN/AEI/10.13039/501100011033. This work was completed as part of NSF Grant AGS 2246700.

Data availability statement.

The numerical experiments performed in this study were carried out using GFDL’s Flexible Modeling System (FMS). FMS can be accessed at https://www.gfdl.noaa.gov/fms/. A very similar modeling suite is accessible through the Isca project at https://execlim.github.io/IscaWebsite/.

APPENDIX

Details of the Linearized Primitive Equation Model

In our linearized primitive equation model, we enforce a three-dimensional heating profile of the form
Q=CpAF(p)H(φ)ei(kλσt),
where Cp is the isobaric specific heat capacity, A=105Ks1 is the amplitude of the heating, k is the zonal wavenumber of the heating, and σ is the (real) frequency. Both k and σ are treated as given. Following Kasahara (1984), the vertical structure of the heating is given by
F(p)=32pspspt[1(2pptpspt1)2]H(ppt),
where ps = 1000 hPa, pt = 200 hPa, and H(⋅) is the Heaviside function. This profile constructs a parabola with zeros at the levels ps and pt, and a maximum in the midtroposphere. The meridional profile of the heating is
H(φ)=12[I(μ,μ+)+I(μ,μ)],
where μ = sinφ and μ± = sin(±15°). The function I is given by
I(μ,μ*)=βexp[sgn(μμ*)β(μμ*)]2eβ(1μ*)eβ(1+μ*),
where β = 10 is a parameter that controls the decay rate of the heating with the sine of latitude. The profile is designed to have maxima located at ±15° (roughly where we see maxima in precipitation for the modes produced in the idealized GCM) and decay exponentially poleward from there.
The essence of the forced problem is to discretize the atmosphere into a finite number of vertical and meridional modes and project both the shear profile and the heating onto these modes. The nondimensionalized linear response W˜=(u˜,υ˜,z˜)T may then be written as
W˜=aWaΦa(η)Θ(φ)ei(kλσt),
where Wa are the amplitudes of the modes and the subscript a denotes a sum of both the vertical and meridional modes. The vertical structure functions Φ(η) and their corresponding equivalent depths are determined following Kasahara (1984); as in that paper, we retain the first six modes of the vertical decomposition and use the same vertical profile of the reference temperature. The meridional structure functions Θ(φ) = [U(φ), −iV(φ), Z(φ)]T are the Hough vector functions, which form a complete and orthonormal basis. These are determined following the method of Swarztrauber and Kasahara (1985) as sums over a truncated series of vector spherical harmonics. For each shallow water system, we use a total of 48 meridional modes, taking 16 rotational modes and 32 inertia–gravity modes. Thus, we have a total of N = 288 modes to solve for in the linear problem.

Under these assumptions, the forced response problem may be reduced to a matrix inverse problem for the vector (W1, W2, …, WN)T of the amplitudes of the modal decomposition (Kasahara 1984). The matrix in question will have off-diagonal elements due to the coupling of the vertical and meridional modes in the presence of shear (Kasahara 1980; Kasahara and da Silva Dias 1986). The diagonal elements will also depend on the characteristic frequencies of the modal decomposition, determined as part of the meridional structure problem (Swarztrauber and Kasahara 1985). A full description of the problem is given in Kasahara and da Silva Dias (1986). The free problem is closely related and amounts to finding the eigenvalues and eigenvectors of the same matrix. The eigenvectors give the amplitudes Wa in Eq. (A5) to compute the three-dimensional structure of each free mode.

We have employed two different profiles of the basic-state zonal wind in order to demonstrate the importance of vertical shear. In addition to the barotropic case used by Kasahara and da Silva Dias (1986), we have constructed a profile with a linear vertical shear that varies in the meridional direction to match the zonal wind climatology of the idealized model at the surface and 200 hPa. Taking the surface and 200-hPa zonal winds to be u¯s(φ) and u¯200(φ), respectively, the full zonal wind profile is given by
u¯(φ,η)=u¯s(φ)+58(u¯su¯200)(η1),
where η = 2p/ps − 1 is a nondimensional pressure coordinate that takes values in the interval [−1, 1]. This expression for u¯(φ,η) may be written in the form
u¯(φ,η)=(38u¯s+58u¯200)+[58(u¯su¯200)η].
Here, each term in brackets is a product of a function of φ and a function of η and may each be individually expanded in terms of the vertical and meridional basis functions and treated using the methods described by Kasahara and da Silva Dias (1986) to compute the associated interaction coefficients. The basic-state zonal wind profile produced using this method is compared to the climatological zonal winds from the D20C model run in Fig. A1.
Fig. A1.
Fig. A1.

(a) Climatological zonal-mean zonal winds from the D20C experiment. (b) Approximation of the zonal-mean zonal wind profile using Eq. (A7).

Citation: Journal of the Atmospheric Sciences 82, 2; 10.1175/JAS-D-24-0081.1

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