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\begin{document}

\title{The nontraditional Coriolis terms and convective system propagation}



\author[1,1,1,1]{Hing Ong}%
\author[2,2,2,2]{Da Yang}%
\affil[1]{University of California, Davis}%
\affil[2]{University of California}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
\let\center\flushleft
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\selectlanguage{english}
\begin{abstract}
This study explores effects of the nontraditional Coriolis terms (NCTs)
on convective system propagation in radiative-convective equilibrium
(RCE). NCTs are restored to the System for Atmospheric Modeling (SAM) to
explicitly simulate the temporal evolution of convective systems in a
zonal vertical domain rotating about a meridional axis. The system
rotation rate is tested over a wide range. The results are transformed
into space-time spectra to analyze the overall propagation
characteristics. The raw spectra show local power maxima in bands
associated with self-aggregated convection and convectively coupled
gravity waves. Changes in the spectra due to the inclusion of NCTs can
mostly be explained by the compressional beta effect (CBE), which speeds
up the eastward propagation and slows down the westward propagation of
zonal vertical circulation. For example, given the power spectra red in
frequency, the power increases in the band of eastward propagating
convective clusters and decreases in the band of westward ones.
Furthermore, the speed changes of convectively coupled gravity waves are
measured from the spectra. The magnitude of the speed changes increases
with the system rotation rate, and this increase agrees with the
theoretical speed change due to the CBE. These results suggest that the
dry CBE theory can explain the effect of NCTs on the propagation of
convective systems. This study recommends the restoration of NCTs to
model dynamical cores because NCTs meaningfully correct the propagation
speed of convectively coupled circulations and computationally costs
little.  SIGNIFICANCE STATEMENT The rotation of Earth turns eastward
motion upward and upward motion westward, and vice versa. This effect is
called the nontraditional Coriolis effect and is omitted in most of the
current atmospheric models for predicting weather and climate. Using an
idealized model with cloud physics, this study suggests that the
inclusion of the nontraditional Coriolis effect speeds up eastward
moving rainy systems and slows down westward moving ones. The speed
change agrees with a theory without cloud physics. This study encourages
restoring the nontraditional Coriolis effect to the atmospheric models
since it increases the accuracy of tropical large-scale weather
prediction while the cost is low.%
\end{abstract}%



\sloppy


The nontraditional Coriolis terms and convective system propagation

Hing Ong \textsuperscript{a} and Da Yang \textsuperscript{a,b}

\textsuperscript{a} University of California, Davis, Davis, CA

\textsuperscript{b} Lawrence Berkeley National Laboratory, Berkeley, CA

\emph{Corresponding author} : Hing Ong, hxong@ucdavis.edu

ABSTRACT

This study explores effects of the nontraditional Coriolis terms (NCTs)
on convective system propagation in radiative-convective equilibrium
(RCE). NCTs are restored to the System for Atmospheric Modeling (SAM) to
explicitly simulate the temporal evolution of convective systems in a
zonal vertical domain rotating about a meridional axis. The system
rotation rate is tested over a wide range. The results are transformed
into space-time spectra to analyze the overall propagation
characteristics. The raw spectra show local power maxima in bands
associated with self-aggregated convection and convectively coupled
gravity waves. Changes in the spectra due to the inclusion of NCTs can
mostly be explained by the compressional beta effect (CBE), which speeds
up the eastward propagation and slows down the westward propagation of
zonal vertical circulation. For example, given the power spectra red in
frequency, the power increases in the band of eastward propagating
convective clusters and decreases in the band of westward ones.
Furthermore, the speed changes of convectively coupled gravity waves are
measured from the spectra. The magnitude of the speed changes increases
with the system rotation rate, and this increase agrees with the
theoretical speed change due to the CBE. These results suggest that the
dry CBE theory can explain the effect of NCTs on the propagation of
convective systems. This study recommends the restoration of NCTs to
model dynamical cores because NCTs meaningfully correct the propagation
speed of convectively coupled circulations and computationally costs
little.

SIGNIFICANCE STATEMENT

The rotation of Earth turns eastward motion upward and upward motion
westward, and vice versa. This effect is called the nontraditional
Coriolis effect and is omitted in most of the current atmospheric models
for predicting weather and climate. Using an idealized model with cloud
physics, this study suggests that the inclusion of the nontraditional
Coriolis effect speeds up eastward moving rainy systems and slows down
westward moving ones. The speed change agrees with a theory without
cloud physics. This study encourages restoring the nontraditional
Coriolis effect to the atmospheric models since it increases the
accuracy of tropical large-scale weather prediction while the cost is
low.

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 the 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 \(2\Omega\cos\vartheta\),
where\(\Omega\) and \(\vartheta\) 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
(M. Hayashi and Itoh 2012; Igel and Biello 2020; Ong and Roundy 2019,
2020a, 2020b). 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.
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).

While the thickness theory (Ong and Roundy 2020a) is inherent to all the
above-mentioned theories about NCTs, the CBE theory (OR20b) is
dynamically distinct from the others by vorticity thinking. In the
three-dimensional (3D) vorticity equation with the anelastic
approximation on an equatorial beta plane (see Appendix A for
derivations), NCTs (\(2\Omega\) given \(\vartheta=0\)) appear
in two terms:

\(\frac{\partial\mathbf{\omega}}{\partial t}+\left[\left(\mathbf{v}\bullet\mathbf{\nabla}\right)\mathbf{\omega}+\selectlanguage{greek}\text{βv}\selectlanguage{english}\mathbf{k}-\left(\mathbf{\omega}\bullet\mathbf{\nabla}\right)\mathbf{v}-\beta y\frac{\partial\mathbf{v}}{\partial z}+\beta y\frac{w}{H}\mathbf{k}-\mathbf{\nabla}\times b\mathbf{k}\right]=2\Omega\frac{\partial\mathbf{v}}{\partial y}-2\Omega\frac{w}{H}\mathbf{j}\). (1)

The variables are defined as follows: \(\mathbf{\omega}\), 3D relative
vorticity; \(\mathbf{v}\), 3D velocity; \(w\), vertical
velocity; \(H\), scale height; \(v\),
meridional velocity; \(\beta\), meridional derivative of the
traditional Coriolis parameter; and \(b\), buoyancy. The
left-hand side shows terms without NCTs, including local tendency of
vorticity and others (bracketed). The right-hand side shows terms with
NCTs, including the stretching-tilting term (stretching of and tilting
from meridional planetary vorticity) and the CBE term. The
stretching-tilting term needs the meridional dimension while the CBE
does not. The stretching-tilting term has been associated with zonal
wind response to heat sources (M. Hayashi and Itoh 2012; Igel and Biello
2020; Ong and Roundy 2019).

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 Compressional Beta Effect (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 equation (11) of OR20b and taking the quasi-hydrostatic
approximation (see Appendix A for details):

\(c^{2}-\frac{2\Omega}{{m_{H}}^{2}H}c-\frac{{\tilde{N}}^{2}}{{m_{H}}^{2}}=0\), (2)

where the variables are defined as follows: \(c\), phase
speed;\(\tilde{N}\), effective buoyancy
frequency;\({m_{H}}^{2}\equiv m^{2}+\frac{1}{\left(4H^{2}\right)}\), where\(m\) denotes vertical
wavenumber. Solve Eq. (2) for \(c\):

\(c=\frac{\Omega}{{m_{H}}^{2}H}\pm\frac{\tilde{N}}{m_{H}}\sqrt{1+\left(\frac{\Omega}{{m_{H}}^{2}H}\frac{m_{H}}{\tilde{N}}\right)^{2}}\). (3)

Define\(c_{0}\equiv c\left(\Omega=0\right)=\pm\frac{\tilde{N}}{m_{H}}\)and rewrite Eq. (3):

\(c=\frac{\Omega}{{m_{H}}^{2}H}+c_{0}\sqrt{1+\left(\frac{\Omega}{{m_{H}}^{2}Hc_{0}}\right)^{2}}\). (4)

Using \(\hat{c}\equiv\frac{c}{c_{0}}\) and\(\hat{\Omega}\equiv\frac{\Omega}{\left({m_{H}}^{2}Hc_{0}\right)}\), nondimensionalize Eq.
(4):

\(\hat{c}=\hat{\Omega}+\sqrt{1+{\hat{\Omega}}^{2}}\). (5)

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

\(\hat{c}=1+\hat{\Omega}+\frac{{\hat{\Omega}}^{2}}{2}+O\left({\hat{\Omega}}^{4}\right)\). (6)

Define \(c\equiv c-c_{0}\) and dimensionalize Eq. (6):

\(c=\frac{\Omega}{{m_{H}}^{2}H}+\frac{1}{2c_{0}}\left(\frac{\Omega}{{m_{H}}^{2}H}\right)^{2}+c_{0}O\left({\hat{\Omega}}^{4}\right)\). (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
\(c_{0}\) 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
\(\hat{\Omega}\) is small in general; for example, the three pairs of
\(c_{0}\) in Figure 1 correspond to \(\hat{\Omega}\) 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).

3. Methods

a. Model Description

This study uses the System for Atmospheric Modeling (SAM) (Khairoutdinov
and Randall 2003) version 6.11.2 and complies with the protocol of RCE
Model Intercomparison Project (RCEMIP; Wing et al. 2018, 2020). We
restored NCTs to SAM as released in version 6.11.7. The implementation
of NCTs was validated by the compressional Rossby wave benchmark
(OR20b). It increased the central processing unit (CPU) time by less
than 0.01\% of the total cost of full-physics simulations. The whole
process from restoration, validation, to releasing was simple. Thus, the
cost for NCTs was low. This is consistent with Skamarock et al. (2021),
who reported that the restoration of NCTs increased the cost of the
global atmospheric dynamical core by less than 5\%. For spherical
coordinates, a little extra care is needed to retain the angular
momentum conservation (e.g., White and Bromley 1995). Since SAM uses
Cartesian coordinates, the restoration is straightforward.

Deviations from the RCEMIP protocol (Wing et al. 2018) are explained as
follows. First, six simulations are conducted with the system rotation
rate set to 0, 1, 4, 8, 16, and 64 times the Earth's rotation rate
(named X00, X01, X04, X08, X16, and X64). Second, the meridional
dimension is eliminated (2D). Third, the size of the zonal and temporal
dimensions is 24,576 km and 1,632 Earth's days. Fourth, the model top is
lowered from 33 km to 28 km. Last, mean zonal velocity is nudged toward
zero by linear relaxation at a timescale of 86,400 s at each level to
prevent development of mean winds that may affect convective system
propagation.

All other procedures and settings follow the elongated channel SAM
simulation over sea surface temperature of 300 K in RCEMIP (labeled
SAM\_CRM/RCE\_large300). The initial vertical profile is derived from
the corresponding small domain simulation data in RCEMIP (labeled
SAM\_CRM/RCE\_small300). In theory, the X00 spectrum should be west-east
symmetric.

b. Spectral Analyses

To analyze the overall propagation characteristics of convective systems
in the 2D RCE simulations, the data except the first 96 days are
transformed into space-time spectra following the sampling procedure by
Wheeler and Kiladis (1999, hereafter WK99) based on the method by Y.
Hayashi (1971, 1982). The sample length is 64 days. This procedure
yields raw spectral power in the wavenumber-frequency domain. To focus
on the east-west asymmetry, we contrast the raw spectra to reference
spectra made from the X00 spectra by taking the average of spectral
power density symmetric about wavenumber zero. We symmetrize the
reference for two reasons. First, the X00 simulation does not
distinguish west from east, so zonally flipped X00 spectra are equally
valid as the raw X00 spectra. Second, a symmetric reference spectrum
would help highlight NCT-induced zonal asymmetry.

To measure the gravity wave speed, we take the power-weighted average of
phase speed in a gravity wave band. Using the normalization procedure by
WK99, we take contiguous bands with normalized power greater than unity
following gravity wave dispersion lines as gravity wave bands. To
quantify the uncertainty, we test the sample lengths of 128, 192, 256,
384, 512, and 768 days, and the smoothing passes in wavenumber of 16,
32, 48, and 64 times. The smoothing passes in frequency are set to the
sample length divided by 8.

To estimate the vertical wavenumber relevant to the FOCBE parameter, the
data are transformed into vertical normal modes assuming a rigid
tropopause (Fulton and Schubert 1985) with the approximated set of
analytic basis functions omitting small vertical variations of scale
height and static stability (Kiladis et al. 2009) as follows:

\(\sqrt{\rho\left(z\right)}u\left(x,z,t\right)=\selectlanguage{greek}\sum_{M=0}^{\infty}{U_{M}\left(x,t\right)\cos\left(\frac{\text{Mπz}}{D}\right)}\selectlanguage{english},\ z\in\left[0,D\right]\), (8)

where the variables are defined as follows: \(\rho\),
density; \(u\), zonal velocity; \(D\),
tropopause height; and \(U_{M}\), amplitude of the
\(M\)th vertical mode. The vertical wavenumber
\(m\) equals to\(\selectlanguage{greek}\selectlanguage{english}\frac{\text{Mπ}}{D}\). The power
\({U_{M}}^{2}\) is proportional to the vertically integrated kinetic
energy of the vertical mode. Given the level of the minimum temperature
in the mean profile, \(D\) is located at 15.5 km. The data
are linearly interpolated to vertical levels equally spaced by
\(z=\) 500 m from \(\frac{z}{2}\)
to\(D-\frac{z}{2}\), and then \(\sqrt{\rho}u\) is calculated. Then,
the domain is extended to \(2D-\frac{z}{2}\), and the upper half of the
new domain is filled with the mirror image of \(\sqrt{\rho}u\) from
the lower half. For each profile, a fast Fourier transform is performed.
The resulting coefficients can be filtered for a target timescale with
the Lanczos filter. This procedure yields the vertical-mode spectrum.

For comparison, the same vertical-mode analysis is applied to an
observational dataset. The dataset used is the Merged Sounding
(Giangrande et al. 2011) over Gan Island, Maldives from 1 October 2011
to 9 February 2012 during the Dynamics of the MJO (DYNAMO) field
campaign (Yoneyama et al. 2013). The above-mentioned procedure is
followed, but \(D\) and \(z\) are 17.0 km
and 200 m.

4. Results

We first focus on the raw spectra without NCTs (top row of Figure 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 \selectlanguage{ngerman}±16 m s\textsuperscript{--1} (26-m equivalent
depth). In addition, they show weaker peaks corresponding to dry gravity
waves with a speed of about ±42 m s\textsuperscript{--1} (180-m
equivalent depth) except in the precipitation spectrum (Figure 2a). Here
we take the power-weighted average of speed in each gravity wave band in
the normalized U200 spectra (Figure 3dhl). Moreover, the raw spectra
(Figure 2) show another pair of local power maxima at the lowest
resolved frequency at around wavenumber ±12 except in the U200 spectra
(Figure 2dhl). These low-frequency spectral peaks correspond to slowly
propagating long-lived convective systems with a characteristic
wavelength of 2,048 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
Figure 2. With increasing system rotation rate (from top to bottom of
Figure 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 Figure 2).

We compare Figure 2 with the raw observed power spectra of eastward
propagating convection symmetric about the equator (Figure 1 of Roundy
2020; see also Kiladis et al. 2009; WK99). When convectively coupled,
the simulated gravity waves (dashed lines in Figure 2) as well as the
observed Kelvin waves (e.g., Figure 4b of WK99) correspond to an
equivalent depth of about 25 m. However, unlike the raw simulation
spectra with multiple maxima in frequency (Figure 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 \textasciitilde{} 2,000 km while the
MJO scales \textasciitilde{} 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 Figure 4; Section 3b). First, the
precipitation spectrum (Figure 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 wavenumber from 4
to 12 and frequency of 0.125 cycle per day (cpd) or lower. In these
bands, the spectral power increases in the X64 spectrum (Figure 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 Figure 2a) shift toward a higher frequency eastward and a lower
frequency westward (Figure 4i). Similarly, in the USFC spectrum (Figure
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
wavenumber 2 and 14 along with its extension toward the dashed line. The
power also increases in these bands in the X64 spectrum (Figure 4k). On
the other hand, the power generally decreases in these spectra (Figure
4i and 4k) in the clockwise side of power maxima and counterclockwise
side of power minima. Similar reasoning applies to the PW (Figure 4j)
and U200 (Figure 4l) spectra except the upward side of the dispersion
lines in the PW spectrum, which implies power changes in a non-shifting
manner. With decreasing system rotation rate (from bottom to top of
Figure 4), the power ratio becomes noisy, but the above-mentioned power
changes can mostly be found to have smaller magnitude.

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 (Figure 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 (Figure 5). Besides, the power slightly moves from
the first two modes to higher when filtered for the timescale of
simulated gravity waves (1.5 to 18 days in Figure 2) and observed Kelvin
waves (2.5 to 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 nineth of the first mode.
Consequently, the X08 simulation best corresponds to Earth's tropical
troposphere, where the first vertical mode seems to dominate.

The X08 spectra show power changes in frequency lower than the
convectively coupled gravity wave bands (top row of Figure 4). In the
low-frequency USFC spectrum (Figure 4c), the power increases in the
eastward band and decreases in the westward band. However, in the U200
spectrum (Figure 4d), the power changes are the opposite. To explain
this contrast, we compare their low-frequency raw spectra (Figure 2cd).
The USFC spectrum (Figure 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
(Figure 4ab). On the other hand, the U200 spectrum (Figure 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
(Figure 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 (Figure 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
(\(c\)).

The \(c\) results (Figure 6) agree with their FOCBE
parameter (Table 1). The X00 results are about zero and not shown in
Figure 6 because they include negative values, which cannot be converted
to log scale. The X01 results are uncertain because of a lack of the
resolution in the wavenumber-frequency domain. The other simulation
results are incremental to \(c\) (about 16 m
s\textsuperscript{--1}) and discernable only by computers except that
the X64 results yield \(c\) scaling 12\% of
\(c\), which is noticeable by sight (bottom row of Figures
2, 3, and 4). Nevertheless, the robust agreement shown in Figure 6
confirms that the dry-dynamical FOCBE theory can predict
\(c\) due to the inclusion of NCTs even given the free
interactions between circulation and moist physics in the 2D RCE.

5. Summaries and Discussions

This study explores effects of the nontraditional Coriolis terms (NCTs)
on explicitly simulated convective system propagation in
radiative-convective equilibrium (RCE) in a zonal vertical domain using
the System for Atmospheric Modeling (SAM). 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 NCT-induced compressional beta effect 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. 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 (Figure 6). Such
agreement allows us to infer changes in convective system propagation
from the FOCBE parameter. Among our simulations, the one with eight-time
Earth's rotation rate best corresponds to Earth's tropical troposphere,
because the first vertical mode dominates in the observations while the
third mode dominates 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.

For Earth's tropical troposphere, the FOCBE theory implies a gravity
wave speed change of one FOCBE parameter, 0.265 m
s\textsuperscript{--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 NCTs 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., Figure
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).

Acknowledgments.

This research is supported by a Packard Fellowship awarded to D.Y. We
thank Brandon Wolding and two anonymous reviewers for their useful
comments.

Data Availability Statement.

We thank the German Climate Computing Center (DKRZ) for hosting the
standardized RCEMIP data, which is 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).

APPENDIX

Appendix A: Derivations

To derive Eq. (1), start from the 3D momentum equation on an equatorial
beta plane:

\(\frac{\partial\mathbf{v}}{\partial t}+\left(\mathbf{v}\bullet\mathbf{\nabla}\right)\mathbf{v}+\beta y\mathbf{k}\times\mathbf{v}+\mathbf{\nabla}\varphi-b\mathbf{k}=-2\Omega\mathbf{j}\times\mathbf{v}\),

where \(\varphi\) denotes the pressure-like variable defined by
Lipps and Hemler (1982). Taking the curl and applying\(\mathbf{\nabla}\bullet\mathbf{v}=\frac{w}{H}\)
(i.e., the anelastic approximation) yields Eq. (1).

To derive Eq. (2), start from equation (11) of OR20b:

\(-k^{2}\left({\tilde{N}}^{2}-\epsilon\omega^{2}\right)-k\selectlanguage{greek}\selectlanguage{english}\frac{2\text{Ωω}}{H}+\omega^{2}\left(m^{2}+\frac{1}{4H^{2}}\right)=0\),

where \(k\) and \(\omega\) denote wavenumber and
frequency. Setting\(\epsilon\) to zero (i.e., the
quasi-hydrostatic approximation) and applying \(c\equiv\frac{\omega}{k}\) yields
Eq. (2).

Appendix B: Filtering for the Self-Aggregated Convection

To confirm the association of the self-aggregated convection with the
spectral band in the dashed box in the top row of Figure 2, the PW is
filtered for this band. The Hovm\selectlanguage{ngerman}öller diagrams of the filtered PW
(Figure A1) show many convective clusters (solid contours) and dry
regions (dashed contours). These systems grow (or split), migrate, and
decay (or merge) in a complex manner with generally 12 convective
clusters maintained in total. Such metabolism occurs in periods other
than the 96 days shown in Figure A1.

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TABLES

\textbf{Table 1} . The Characteristic FOCBE Parameter\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Case name & \(\Omega\)11Normalized by the Earth's rotation rate
of 7.292 \selectlanguage{ngerman}× 10\textsuperscript{--5} s\textsuperscript{--1}. &
\(M\) & \(D\) (km) & \(H\)
(km)22Virtual temperature times gas constant divided by gravity
acceleration averaged in the troposphere with the data equally spaced in
height. & FOCBE (m s\textsuperscript{--1})\tabularnewline
\midrule
\endhead
DYNAMO & 1 & 1 & 17.0 & 7.0 & 0.265\tabularnewline
X00 & 0 & 3 & 15.5 & 7.0 & 0\tabularnewline
X01 & 1 & 3 & 15.5 & 7.0 & 0.028\tabularnewline
X04 & 4 & 3 & 15.5 & 7.0 & 0.111\tabularnewline
X08 & 8 & 3 & 15.5 & 7.0 & 0.222\tabularnewline
X16 & 16 & 3 & 15.5 & 7.0 & 0.445\tabularnewline
X64 & 64 & 3 & 15.5 & 7.0 & 1.779\tabularnewline
\bottomrule
\end{longtable}

FIGURES\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure 1} . 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\textsuperscript{--1} (solid) and without the FOCBE (dash). The red,
green, and blue lines correspond to \(\hat{\Omega}\) of 1/6, 1/18, and
1/54.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 2} . Raw spectra of (from left to right) the
precipitation rate, precipitable water, zonal velocity at the lowest
level and at 200 hPa from (from top to bottom) the X00, X08, and X64
simulations. 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-m and 180-m equivalent depth.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 3} . Like Figure 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.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure 4} . Like Figure 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.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure 5} . 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 timescale of Kelvin or gravity waves (dashed, see
text). The power is normalized by the average in the first eight modes.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

\textbf{Figure 6} . Scaling of the gravity wave speed change (boxes and
whiskers) against the FOCBE parameter (horizontal axis, see Table 1).
Each box-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.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}

\textbf{Figure A1} . Hovm\selectlanguage{ngerman}öller diagrams of the PW in the 2D RCE (a) with
NCTs (X16) and (b) without NCTs from the raw data (shading, mm) and data
filtered for the self-aggregated convection (contours, interval: 5 mm).
The solid and dashed contours denote positive and negative values, and
the zero contour is omitted.

\selectlanguage{english}
\FloatBarrier
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