1. Introduction

Most global weather and climate models solve the primitive equations for Earth’s atmosphere, assuming a thin atmosphere in the hydrostatic balance. To conserve energy and angular momentum, these models neglect the locally horizontal component of Earth’s rotation vector. This is called the traditional approximation (Eckart 1960), and the neglected Coriolis components are known as the nontraditional Coriolis terms (NCTs; which are terms involving 2Ω cosϑ, where Ω and ϑ denote planetary rotation rate and latitude). These approximations were believed to hold accurately for large-scale circulations in Earth’s atmosphere (e.g., Vallis 2017).

However, recent studies have emphasized that NCTs can significantly affect thermodynamic states and atmospheric circulations in the tropics (Hayashi and Itoh 2012; Igel and Biello 2020; Ong and Roundy 2019, 2020a,b). For example, NCTs affect thickness between pressure levels—a measure of effective atmospheric buoyancy—by about 10% of the thickness variability (Ong and Roundy 2020a). Ong and Roundy (2019) found that, in response to steady heat sources, NCTs can generate anomalous winds that are about 12% of the total wind response. Ong and Roundy (2020b, hereafter OR20b) further showed that NCTs produce an eastward propagation tendency for atmospheric waves in Earth’s tropics. Ruling out other effects of NCTs by testing linear models, OR20b identified that the underlying mechanism was the compressional beta effect (CBE), which had been only considered significant in the atmosphere of giant planets and stars (e.g., Gilman and Glatzmaier 1981; Glatzmaier et al. 2009; Verhoeven and Stellmach 2014).

These recent studies have advanced our understanding of NCTs. However, most of them were based on simple models with imposed flow field or prescribed moist physics and solved linear problems. Though Igel and Biello (2020) presented nonlinear cloud-resolving model (CRM) simulations, they focused on the stretching–tilting effect in our Eq. (1) (“Coriolis shear” in their words), which is distinct from the CBE. The present study uses a zonal vertical two-dimensional (2D) CRM to further explore possible effects of the CBE on explicitly simulated convectively coupled circulations. We remove the stretching–tilting effect by using the 2D domain. The 2D CRM experiments simulate gravity waves and self-aggregated convection (e.g., Held et al. 1993; Yang 2018) in radiative–convective equilibrium (RCE), and their propagation characteristics are examined given NCTs switched on and off.

The rest of this paper is organized as follows. Section 2 reviews the CBE and forms a hypothesis. Section 3 describes the methods of modeling and spectral analyses. Section 4 presents the results and tests the hypothesis. Section 5 summarizes this paper and discusses the implications for the Madden–Julian oscillation (MJO).

2. The CBE

OR20b introduced both NCTs and the vertically decreasing density to the wave theory on an equatorial beta plane (Matsuno 1966). The present study confines the domain to a zonal vertical plane at the equator. This choice of domain removes the traditional beta effect, which is the major source of west–east asymmetry of the inertio-gravity wave mode, isolating the CBE. Modified by the CBE, the dispersion relation of gravity waves can be approximated to the first order as follows. Start from rearranging Eq. (11) of OR20b and taking the quasi-hydrostatic approximation (see appendix A for details):

$c^2-\frac{2\mathrm{\Omega }}{m_H^{2}H}\hspace{0.17em}c-\frac{{\tilde{N}}^2}{m_H^2}=0,$(2)

where the variables are defined as follows: c is the phase speed, $\tilde{N}$ is the effective buoyancy frequency, and $m_H^2\equiv m^2+1/(4H^2)$, where m denotes vertical wavenumber. Solve Eq. (2) for c:

$c=\frac{\mathrm{\Omega }}{m_H^{2}H}\hspace{0.17em}\pm \hspace{0.17em}\frac{\tilde{N}}{m_H}\sqrt{1+{\left(\frac{\mathrm{\Omega }}{m_H^{2}H}\hspace{0.17em}\frac{m_H}{\tilde{N}}\right)}^2}.$(3)

Define $c_0\equiv c(\mathrm{\Omega }=0)=\pm \tilde{N}/m_H$ and rewrite Eq. (3):

$c=\frac{\mathrm{\Omega }}{m_H^{2}H}+c_0\sqrt{1+{\left(\frac{\mathrm{\Omega }}{m_H^{2}H{c}_0}\right)}^2}.$(4)

Using $\widehat{c}\equiv c/c_0$ and $\widehat{\mathrm{\Omega }}\equiv \mathrm{\Omega }/(m_H^{2}H{c}_0)$, nondimensionalize Eq. (4):

$\widehat{c}=\widehat{\mathrm{\Omega }}+\sqrt{1+{\widehat{\mathrm{\Omega }}}^2}.$(5)

Expand the square root in Eq. (5) around ${\widehat{\mathrm{\Omega }}}^2=0$:

$\widehat{c}=1+\widehat{\mathrm{\Omega }}+\frac{{\widehat{\mathrm{\Omega }}}^2}{2}+O({\widehat{\mathrm{\Omega }}}^4).$(6)

Define Δc ≡ c − c₀ and dimensionalize Eq. (6):

$\mathrm{\Delta }c=\frac{\mathrm{\Omega }}{m_H^{2}H}+\frac{1}{2c_0}{\left(\frac{\mathrm{\Omega }}{m_H^{2}H}\right)}^2+c_{0}O({\widehat{\mathrm{\Omega }}}^4).$(7)

The first term on the right-hand side is defined as the first-order CBE (FOCBE) parameter. A negative phase speed denotes westward propagation, and vice versa. Figure 1 depicts three pairs of c and c₀ in a wavenumber–frequency domain given the FOCBE parameter in the tropical troposphere derived from an observational dataset (see section 4 for estimation), and it shows that the FOCBE makes gravity waves propagate faster eastward and slower westward in a way that the dispersion lines rotate counterclockwise in the wavenumber–frequency domain. The higher-order terms are negligible since $\widehat{\mathrm{\Omega }}$ is small in general; for example, the three pairs of c₀ in Fig. 1 correspond to $\widehat{\mathrm{\Omega }}$ of 1/6, 1/18, and 1/54 (from slow to fast). To our hypothesis, regarding a wavenumber in the whole space–time spectrum of simulated precipitation or other variables, the CBE may shift eastward power to a higher frequency and westward power to a lower frequency. Thus, the corresponding power shift is counterclockwise. For gravity wave bands, the speed change is predicted by Eq. (7).

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Zonal temporal dispersion relations of gravity waves confined to a zonal vertical plane at the equator with the FOCBE parameter of Earth’s tropical troposphere of 0.265 m s⁻¹ (solid) and without the FOCBE (dash). The red, green, and blue lines correspond to $\widehat{\mathrm{\Omega }}$ of 1/6, 1/18, and 1/54, respectively.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0219.1

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3. Methods

4. Results

We first focus on the raw spectra without NCTs (top row of Fig. 2) of precipitation rate, precipitable water (PW), zonal velocity at the lowest level (USFC) and at 200 hPa (U200). They all show power peaks that follow the dispersion lines for convectively coupled gravity waves with a speed of about ±16 m s⁻¹ (26-m equivalent depth). In addition, they show weaker peaks corresponding to dry gravity waves with a speed of about ±42 m s⁻¹ (180-m equivalent depth) except in the precipitation spectrum (Fig. 2a). Here we take the power-weighted average of speed in each gravity wave band in the normalized U200 spectra (Figs. 3d,h,l). Moreover, the raw spectra (Fig. 2) show another pair of local power maxima at the lowest resolved frequency at around wavenumber ±12 except in the U200 spectra (Figs. 2d,h,l). These low-frequency spectral peaks correspond to slowly propagating long-lived convective systems with a characteristic wavelength of 2048 km, which conforms to the scale of the self-aggregated convective clusters in Yang’s (2018) 2D RCE simulation over 300 K. Appendix B shows the evolution of the self-aggregated convection filtered for its band within the dashed box in the top row of Fig. 2. With increasing system rotation rate (from top to bottom of Fig. 2), the spectra seem similar, but the gravity wave peaks shift with the dispersion lines (both dashed and solid) significantly in the X64 spectra (bottom row of Fig. 2).

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Raw spectra of (from left to right) the precipitation rate, precipitable water, and zonal velocity at the lowest level and at 200 hPa, respectively, from (from top to bottom) the X00, X08, and X64 simulations, respectively. The darker shading denotes the larger base-10 logarithm of the power; the shading interval is 0.3 arbitrary units. The dashed and solid lines denote 26- and 180-m equivalent depths, respectively.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0219.1

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As in Fig. 2, but depicting the ratio of the raw spectrum to the background spectrum. The sample length is set to 128 days. Each background spectrum is created by smoothing for 64 times in wavenumber and is different from one another.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0219.1

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We compare Fig. 2 with the raw observed power spectra of eastward propagating convection symmetric about the equator (Fig. 1 of Roundy 2020; see also Kiladis et al. 2009; WK99). When convectively coupled, the simulated gravity waves (dashed lines in Fig. 2) as well as the observed Kelvin waves (e.g., Fig. 4b of WK99) correspond to an equivalent depth of about 25 m. However, unlike the raw simulation spectra with multiple maxima in frequency (Fig. 2), the raw observational spectra do not show a peak in the Kelvin wave band. Instead, the Kelvin wave maximum emerges from the normalization by a smoothed background spectrum (Roundy 2020). Another notable difference lies in the horizontal scale of the low-frequency power maxima; the simulated convective clusters scale ∼2000 km while the MJO scales ∼20 000 km.

To test our hypothesis of counterclockwise power shift in the wavenumber–frequency domain, we first contrast the X64 spectra to the symmetrized X00 spectra (bottom row of Fig. 4; section 3b). First, the precipitation spectrum (Fig. 2a) shows bands where the power decreases toward a higher frequency eastward or a lower frequency westward, including the counterclockwise side of the dashed lines, the clockwise side of the solid lines, and a band roughly covering wavenumbers from 4 to 12 and a frequency of 0.125 cycle per day (cpd) or lower. In these bands, the spectral power increases in the X64 spectrum (Fig. 4i) as if the CBE shifts the power counterclockwise. Even the power minima in the precipitation spectrum associated with dry gravity waves (solid lines in Fig. 2a) shift toward a higher frequency eastward and a lower frequency westward (Fig. 4i). Similarly, in the USFC spectrum (Fig. 2c), such counterclockwise power shift due to the CBE occurs in bands including the counterclockwise side of the dashed and solid lines as well as a part of the self-aggregated convection band roughly within wavenumbers 2 and 14 along with its extension toward the dashed line. The power also increases in these bands in the X64 spectrum (Fig. 4k). On the other hand, the power generally decreases in these spectra (Figs. 4i,k) in the clockwise side of power maxima and counterclockwise side of power minima. Similar reasoning applies to the PW (Fig. 4j) and U200 (Fig. 4l) spectra except the upward side of the dispersion lines in the PW spectrum, which implies power changes in a nonshifting manner. With decreasing system rotation rate (from bottom to top of Fig. 4), the power ratio becomes noisy, but the above-mentioned power changes can mostly be found to have smaller magnitude.

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As in Fig. 2, but the shading denotes the power ratio of the X08, X16, and X64 to the symmetrized X00. The shading levels are spaced by a common ratio of $\sqrt[12]{2}$ except that the level 1 is omitted.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0219.1

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To quantitatively test the FOCBE theory, we first estimate the FOCBE parameter. The simulations and the observations yield different vertical-mode spectra of kinetic energy; the former peak at the third mode, but the latter peak at the first mode (Fig. 5). Unlike the observational red spectrum with a concentrated single maximum, the simulation spectra are characterized by not only the prominent third-mode peak but also weaker redness with energy dispersed into higher vertical modes (Fig. 5). Besides, the power slightly moves from the first two modes to higher when filtered for the time scale of simulated gravity waves (1.5–18 days in Fig. 2) and observed Kelvin waves (2.5–30 days in WK99). With these caveats, the most energetic vertical mode is taken to estimate the characteristic FOCBE parameter of the observation and each simulation (Table 1). The prominence of the third vertical mode in our simulations can lead to severe underestimation of the effect of NCTs according to the FOCBE theory; the theoretical gravity wave speed change is one-ninth of the first mode. Consequently, the X08 simulation best corresponds to Earth’s tropical troposphere, where the first vertical mode seems to dominate. Also shown in Table 1, we use the speed of a convectively coupled gravity wave (c₀ = 16 m s⁻¹) as the reference to nondimensionalize the FOCBE parameter [$\widehat{\mathrm{\Omega }}\equiv \mathrm{\Omega }/(m_H^{2}H{c}_0)$, see section 2]. The results suggest that the CBE is insignificant with respect to the gravity wave. However, given c₀ in the denominator of $\widehat{\mathrm{\Omega }}$, slower-propagating phenomena could be more prone to the CBE.

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Vertical-mode spectra of kinetic energy (power) derived from zonal velocity profiles in the Merged Sounding during DYNAMO (black) and the 2D RCE with SAM (blue), each unfiltered (solid) and filtered for the time scale of Kelvin or gravity waves (dashed; see text). The power is normalized by the average in the first eight modes.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0219.1

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Table 1

The characteristic FOCBE parameter.

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The X08 spectra show power changes in frequency lower than the convectively coupled gravity wave bands (top row of Fig. 4). In the low-frequency USFC spectrum (Fig. 4c), the power increases in the eastward band and decreases in the westward band. However, in the U200 spectrum (Fig. 4d), the power changes are the opposite. To explain this contrast, we compare their low-frequency raw spectra (Figs. 2c,d). The USFC spectrum (Fig. 2c) is red mostly in frequency. Hence, taking a constant wavenumber, the counterclockwise power shift would lead to an eastward power increase and a westward power decrease. Similar explanations can be applied to parts of the precipitation and PW spectra (Figs. 4a,b). On the other hand, the U200 spectrum (Fig. 2d) is red mostly in wavenumber, Hence, taking a constant frequency, the shift would result in an eastward power decrease and a westward power increase. Though noisy in X08, these features are robust; they incrementally amplify in X16 and X64 results.

Finally, we test the FOCBE theory quantitatively. To avoid complication due to the proximity of the two types of gravity wave peaks, here we measure the gravity wave speed from the normalized precipitation spectra (Fig. 3a column), where the dry gravity wave peaks are absent. The measured gravity wave speed (solid lines) captures the gravity wave peaks in the normalized spectra of the other variables (Fig. 3). As a caveat, the smoothed background spectra for the normalization also change with the system rotation rate. Nevertheless, our conclusion holds even when measuring the gravity wave speed from the raw spectra. We take the average of the positive and negative phase speed as the speed change (Δc).

The Δc results (Fig. 6) agree with their FOCBE parameter (Table 1). The X00 results are about zero and not shown in Fig. 6 because they include negative values, which cannot be converted to log scale. As shown in Fig. 6, the X01 results are almost undetectable because the asymmetry hardly stands out from the noise (see online supplemental material). The other simulation results are incremental to c (about 16 m s⁻¹) and discernable only by computers except that the X64 results yield Δc scaling 11% of c, which is noticeable by sight (bottom row of Figs. 2–4). Nevertheless, the robust agreement shown in Fig. 6 confirms that the dry-dynamical FOCBE theory can predict Δc due to the inclusion of NCTs even given the free interactions between circulation and moist physics in the 2D RCE.

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Scaling of the gravity wave speed change (boxes and whiskers) against the FOCBE parameter (horizontal axis; see Table 1). Each box-and-whiskers set denotes the quartiles and full range of the speed measurement results from each simulation. The blue star marks denote the first, second, and third vertical modes (from right to left) from the Merged Sounding during DYNAMO; the mark sizes are proportional to the unfiltered power. The axes are log scaled.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0219.1

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5. Summary and discussion

This study explores one of the effects of the nontraditional Coriolis terms (NCTs), specifically the compressional beta effect (CBE), on explicitly simulated convective system propagation in radiative–convective equilibrium (RCE) in a zonal vertical domain using the System for Atmospheric Modeling (SAM). The use of a zonal vertical domain eliminates the other effect of NCTs, the stretching–tilting effect. We restored NCTs to SAM to represent the effects of system rotation about a meridional axis. The system rotation rate is tested over a wide range. We hypothesize that the CBE makes the eastward propagation of convectively coupled circulations faster and the westward propagation slower. The simulated propagation characteristics are analyzed with space–time spectra. The raw spectra show local power maxima in bands associated with self-aggregated convection and convectively coupled gravity waves. Given the power spectra red in frequency, the CBE can qualitatively explain the power increase in the band of eastward propagating convective clusters and its decrease in the band of westward ones. Furthermore, this study measures the speed change of convectively coupled gravity waves from the precipitation spectra. The resulting speed change agrees with the theoretical speed change due to the FOCBE, which is estimated from the system rotation rate, the scale height, and the vertical wavenumber derived from the vertical-mode analysis. In this idealized framework, these changes due to the inclusion of NCTs are robust and meaningful while the computational cost for the implementation of NCTs in SAM increases by less than 0.01%. Also given that the restoration of NCTs to the dynamical core is simple, we recommend it for model development.

These results, for the first time, show that NCTs can affect convective system propagation because of the CBE, which was derived from dry dynamics. The dry-dynamical theory agrees with the moist simulation results over a wide range of the FOCBE parameter (Fig. 6). Such agreement allows us to infer changes in convective system propagation from the FOCBE parameter. Among our simulations, the one with an 8-times Earth’s rotation rate may be comparable to Earth’s tropical troposphere, because the first vertical mode dominates in the observations while the third mode is the strongest in our simulations.

The use of a zonal vertical 2D domain simplifies the interpretation of the results but limits their applicability to the real tropical atmosphere. The 2D domain disables the stretching and tilting of vorticity, leaving the CBE the only way for NCTs to affect the vorticity dynamics. However, the 2D domain also disables the traditional beta effect, which is the major source of west–east asymmetry of real equatorial waves. To better assess the relevance of NCTs in the real tropical atmosphere, further tests with general circulation models are needed. Though OR20b showed that NCTs cannot affect atmospheric wave propagation by the stretching–tilting effect in linear models, neither OR20b nor the present study can rule out this effect from affecting convective system propagation through 3D nonlinear processes.

For Earth’s tropical troposphere, the FOCBE theory implies a gravity wave speed change of one FOCBE parameter, 0.265 m s⁻¹, which scales only 2% of the gravity wave speed itself. Yet this change may have implications for the propagation speed of the MJO. Regarding the MJO speed, the FOCBE can be introduced to two of the diverse MJO theories. First, Yang and Ingersoll (2013) determined the MJO speed by the average of inertio-gravity wave speed (negative westward and vice versa), which may increase by one FOCBE parameter. Second, Kim and Zhang (2021) proposed that the MJO speed arises from the damped Kelvin wave speed, which may speed up by one FOCBE parameter. However, given the diversity of MJO theories (Zhang et al. 2020), further studies are needed.

Our results show that the CBE can produce a significant east–west asymmetry in the spectral power density of precipitation, PW, and winds. Given a power spectrum red in frequency, NCTs can lead to enhanced power of low-frequency variability of eastward propagating signals (e.g., Figs. 4a–c), which are associated with convective self-aggregation in our simulations and might become the MJO in the real atmosphere. Therefore, restoring NCTs in atmospheric models may improve the representation of the MJO, the variability of which is significantly underestimated by the current climate models (Le et al. 2021).

Data availability statement.

We thank the German Climate Computing Center (DKRZ) for hosting the standardized RCEMIP data, which are publicly available at https://hdl.handle.net/21.14101/d4beee8e-6996-453e-bbd1-ff53b6874c0e. The Merged Sounding data were obtained from the Atmospheric Radiation Measurement (ARM) Program sponsored by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research, Climate and Environmental Sciences Division via doi:10.5439/1034922 with free registration. The model, SAM, can be used only with permission from Marat Khairoutdinov (marat.khairoutdinov@stonybrook.edu).