# Lorenz Energy Cycle: Another Way to Understand the Atmospheric Circulation
on Tidally Locked Terrestrial Planets

Shuang Wang Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of
Atmospheric and Oceanic Sciences, School of Physics, Peking University,
Beijing 100871, China  Jun Yang Laboratory for Climate and Ocean-Atmosphere
Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking
University, Beijing 100871, China Jun Yang<EMAIL_ADDRESS>
###### Abstract

In this study, we employ and modify the Lorenz energy cycle (LEC) framework as
another way to understand the atmospheric circulation on tidally locked
terrestrial planets. It well describes the atmospheric general circulation in
the perspective of energy transformation, involved with several dynamical
processes. We find that on rapidly rotating, tidally locked terrestrial
planets, mean potential energy (PM) and eddy potential energy (PE) are
comparable to those on Earth, as they have similar steep meridional
temperature gradients. Mean kinetic energy (KM) and eddy kinetic energy (KE)
are larger than those on Earth, related to stronger winds. The two conversion
paths, P${}_{\rm M}\rightarrow$P${}_{\rm E}\rightarrow$KE and P${}_{\rm
M}\rightarrow$K${}_{\rm M}\rightarrow$KE, are both efficient. The former is
associated with strong baroclinic instabilities, and the latter is associated
with Hadley cells. On slowly rotating, tidally locked terrestrial planets,
weak temperature gradients in the free atmosphere and strong nightside
temperature inversion make PM and PE are much smaller than those on Earth.
Meanwhile, large day–night surface temperature contrast and small rotation
rate make the overturning circulation extend to the globe, so that the main
conversion path is P${}_{\rm M}\rightarrow$K${}_{\rm M}\rightarrow$KE. This
study shows that the LEC analyses improve the understanding of the atmospheric
circulation on tidally locked terrestrial planets.

##  1 Introduction

The substellar point of 1:1 tidally locked (or synchronously rotating)
terrestrial planet is fixed with time. Such a state can drive different
atmospheric circulation compared to Earth, which has been simulated by general
circulation models (GCMs) in previous studies (e.g., Joshi et al., 1997;
Merlis & Schneider, 2010; Edson et al., 2011; Leconte et al., 2013;
Wordsworth, 2015; Koll & Abbot, 2016; Noda et al., 2017; Haqq-Misra et al.,
2018; Pierrehumbert & Hammond, 2019; Hammond & Lewis, 2021; Sergeev et al.,
2022; Turbet et al., 2022; Wang & Yang, 2022). These simulations showed that
the atmospheric circulation on tidally locked terrestrial planets is dominated
by a global-scale overturning circulation, consisting of winds with upwelling
in the substellar region, horizontally flowing from the dayside to the
nightside in the upper troposphere, downwelling in the region away from the
substellar point, and flowing back from the nightside to the dayside near the
surface. Besides, there are a westerly jet over the equator (equatorial
superrotation) and planet-sized wavenumber-1 stationary Rossby and Kelvin
waves.

For understanding the atmospheric circulation on tidally locked terrestrial
planets, various methods have been used. Momentum budgets were widely used to
explore the interactions between zonal jets and planetary waves (e.g., Showman
& Polvani, 2010, 2011; Perez-Becker & Showman, 2013; Tsai et al., 2014;
Hammond & Pierrehumbert, 2018; Mendonça, 2019; Debras et al., 2020; Hammond et
al., 2020; Wang & Yang, 2021). It well demonstrated that up-gradient momentum
transports by the stationary waves maintain the equatorial superrotation
against friction. This method was also beneficial for the prediction of the
equatorial jet speed (Hammond et al., 2020). Besides, a Helmholtz
decomposition technique, which was suggested by Hammond & Lewis (2021), was
also used in several recent studies (Ding & Wordsworth, 2021; Sergeev et al.,
2022; Turbet et al., 2022). This technique separated the total circulation
into divergent (overturning circulation) and rotational components (zonal jets
and stationary waves). It was helpful to classify the dynamical regimes and to
quantify the transports of energy and tracers. For example, Hammond & Lewis
(2021) found that the global overturning circulation could dominate the
day–night heat transport even when the zonal jet is strong.

Another method is describing the atmospheric circulation from the perspective
of energy and energy transformation. Atmospheric heat engine framework depicts
the Earth’s atmospheric circulation as an ideal Carnot’s heat engine (Peixóto
& Oort, 1992). Koll & Abbot (2016) applied this framework on the tidally
locked terrestrial planet: the atmosphere absorbs heat on the dayside at a
dayside surface temperature of $T_{d}$ and emits it to space at a planet’s
equilibrium temperature of $T_{eq}$, which allows the atmosphere to maintain
the global overturning circulation against friction in the boundary layer,
following a Carnot’s efficiency of $\eta=(T_{d}-T_{eq})/T_{d}$. By using this
framework, they estimated the surface wind speed and developed an upper limit
on the strength of the overturning circulation.

Figure 1: (a) Framework of the Lorenz energy cycle (LEC) on Earth: boxes
represent the energy reservoirs; arrows represent generation (G),
transformation (C), and dissipation (D) of energy. (b) Generation of PM:
uneven heating and cooling tilt the zonal-mean isotherms (black curves)
departed from the global-mean isotherms (grey dashed lines), which generates
PM. (c) Heat transport by baroclinic eddies, converting PM to PE: the zonal-
mean isotherm (grey dashed line) is warped to the isotherm (black curve) by
baroclinic eddies, which decreases zonal-mean meridional difference, i.e., PM,
but leads zonal variance, i.e., PE. (d) Cross-isobaric motions, converting
potential energy and kinetic energy with each other. (e) Wave–mean flow
interactions, converting KE and KM with each other: under a positive
$\beta$-plane, arbitrary waves with group velocity directed away from the
source region in the y-direction ($c_{gy}$) can transport momentum back, along
with conversion from KE to KM; shear instabilities of zonal jets generate
eddies, along with conversion from KM to KE.

In this study, the Lorenz energy cycle (LEC, Lorenz, 1955) is employed as
another way to understand the atmospheric circulation on tidally locked
terrestrial planets. It describes the general circulation from the perspective
of energy transformation, and has been widely applied to Earth. For example,
the LEC has been used to understand the atmospheric circulation and dynamical
processes, such as waves, zonal jets, and their interactions (e.g., Peixóto &
Oort, 1974, 1992; Ulbrich & Speth, 1991; Duan & Wu, 2005). Moreover, the
ability to simulate LEC calculated from observational data is a useful
diagnostic for climate models (e.g., Hernández-Deckers & von Storch, 2010;
Marques et al., 2011). Comparison of the LEC calculated from various
reanalysis data is also beneficial to evaluate these data (e.g., Ulbrich &
Speth, 1991; Li et al., 2007; Marques et al., 2009, 2010; Kim & Kim, 2013).
Recently, the LEC has been applied to show the variability of the energy cycle
in response to climate change over the last 40 years (e.g., Kim & Choi, 2017;
Pan et al., 2017). The same framework also has been used to predict the future
variability of the energy cycle in the Coupled Model Intercomparison Project
(e.g., Michaelides, 2021; Kanno & Iwasaki, 2022). Briefly, the net incoming
solar radiation, latent heat release in the tropics, and net infrared cooling
in mid- and high-latitudes together generate mean potential energy (PM) in
Earth’s atmosphere. The growing baroclinic eddies convert PM to eddy available
potential energy (PE), and then convert PE to eddy kinetic energy (KE). A
portion of KE is converted to the mean kinetic energy (KM) through wave–mean
flow interactions. The bulk of KM and KE is ultimately dissipated through
small-scale turbulence and surface friction (Figure 1). The main pathway of
enengy conversion on Earth follows
P${}_{\rm{M}}\rightarrow$P${}_{\rm{E}}\rightarrow$K${}_{\rm{E}}\rightarrow$KM.
In addition, the LEC has also been used to understand the oceanic circulation
on Earth (e.g., Peixóto & Oort, 1992; Olbers et al., 2012; von Storch et al.,
2012) and the atmospheric circulation on other planets such as Venus and Titan
(e.g., Del Genio et al., 1993; Yamamoto & Takahashi, 2006; Lee & Richardson,
2010).

A key point of this study is that we apply the LEC to tidally locked
terrestrial planets. We evaluate the LEC on Earth-like tidally locked planets
and compare it with that on Earth. We find that PM is very small on slowly
rotating tidally locked planets, and the main path of the energy conversion is
P${}_{\rm{M}}\rightarrow$K${}_{\rm{M}}\rightarrow$KE. We also find that the
LEC on rapidly rotating planets is like a combination of those on Earth and on
slowly rotating planets. The structure of this paper is as follows. Section 2
describes the methodology and data. Section 3 shows the thermal structure,
atmospheric circulation, and LECs on three different types of planets. Section
4 is the summary and discussions.

##  2 Methodology and data

###  2.1 LEC in tidally locked coordinates

Figure 2: (a) Frame of standard (ST) coordinates. The substellar point is
located at latitude/longitude
$\left(\phi,\lambda\right)=\left(0^{\circ},180^{\circ}\right)$. Three black
arrows are unit vectors of the local Cartesian coordinates
$(\mathbf{\hat{e}_{i}},\mathbf{\hat{e}_{j}},\mathbf{\hat{e}_{k}})$. Red vector
is the rotational angular velocity, and its projections onto the local axes
are shown at the top-right corner. (b) Frame of tidally locked (TL)
coordinates. The substellar point is located at tidally locked latitude
$\phi_{TL}=90^{\circ}$. The projections of the rotational angular velocity are
shown at the top-right corner, and the solid line represents the terminator,
and dots represent the substellar points. The frames of the tidally locked
latitude and longitude lines are shown at the bottom-right corner. (c) Annual-
mean surface temperature (contours) and the near-surface winds (vectors) on
Earth. (d) Same as (c) but on a tidally locked terrstrial planet with an orbit
period of 60 Earth days. (e) Same as (d) but shown in the tidally locked
coordinates. SP and AP are the substellar point and the antistellar point,
respectively.

LEC is more regarded as a fundamental property of the Eulerian mean system
rather than of the real atmosphere, because the Eulerian means are employed to
define the mean energy, and the departures from the Eulerian means are
employed to define the eddy energy (Chapter 10.4 in Holton & Hakim, 2013).
Thus, it is important to employ a suitable Eulerian mean. For Earth, zonal
means are always employed, because solar radiation, temperature, and winds are
almost zonally homogeneous for long-term averages (Figure 2(c)). For tidally
locked terrestrial planets, especially slowly rotating ones, the Eulerian mean
should match the monotonically decreasing stellar radiation and temperature
along an arbitrary direction from dayside to nightside (Figure 2(d)). So for
slowly rotating tidally locked terrestrial planets, the tidally locked
coordinates are employed in this study, similar to that used in previous
studies (e.g., Koll & Abbot, 2015, 2016; Ding & Wordsworth, 2020; Ding &
Pierrehumbert, 2020; Hammond & Lewis, 2021; Sergeev et al., 2022; Turbet et
al., 2022; Wang & Yang, 2022). In the tidally locked coordinates, the nominal
“North/South Pole” is the substellar/antistellar point, the tidally locked
latitude lines are a series of concentric circles around the substellar point,
and the tidally locked longitude lines are the great circles linking the
substellar and antistellar points (Figure 2(b)). The transformation relations
between the standard and the tidally locked coordinates are given in Appendix
A. By the transformation, the overturning circulation (Figure 2(d)) become a
zonal-mean component, i.e., homogeneous along the tidally locked longitudes
(Figure 2(e)). That is, the Eulerian mean on a tidally locked planet could be
defined as the zonal averages in the tidally locked coordinates.

Note that a zonal-mean zonal jet in the standard coordinates would be
transformed into an eddy component in the tidally locked coordinates. For
example, a uniform zonal-mean zonal jet in the standard coordinates $U_{0}$,
as shown in Figure 3(a), would be transformed into corresponding winds in the
tidally locked coordinates following the relations of

$\displaystyle u_{TL}$ $\displaystyle=$
$\displaystyle\frac{\cos\lambda_{TL}\tan\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}}U_{0},$
(1) $\displaystyle v_{TL}$ $\displaystyle=$
$\displaystyle-\frac{\sin\lambda_{TL}\sec\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}}U_{0},$
(2)

where $u_{TL}$ is tidally locked zonal wind, $v_{TL}$ is tidally locked
meridional wind, $\lambda_{TL}$ is tidally locked longitude, and $\phi_{TL}$
is tidally locked latitude (see Equation (A8) in Appendix A). Figure 3(b)
shows the transformed winds from Equations (1) and (2), and suggests that the
zonal-mean zonal jet becomes an eddy component.

Figure 3: A hypothetical uniform zonal-mean zonal jet in the standard
coordinates (a) is transformed into the winds in the tidally locked
coordinates (b) using Equations (1) and (2). Blue dashed lines represent the
equator (EQ) in the standard coordinates; red dashed lines represent the
terminators (TM); black dot in panel (a) represents the substellar point (SP);
black triangle in panel (b) represents the North Pole (NP) in the standard
coordinates.

A critical step is deriving the governing equations in the tidally locked
coordinates. We start this step from the primitive equations, including a
momentum vector equation,

$\frac{D\mathbf{v}}{Dt}+2\bm{\Omega}\times\mathbf{v}=-\frac{1}{\rho}\nabla
p+\mathbf{g}+\mathbf{F},$ (3)

a mass continuity equation,

$\frac{D\rho}{Dt}+\rho\nabla\cdot\mathbf{v}=0,$ (4)

and a thermodynamic equation,

$\frac{DT}{Dt}+\frac{p}{\rho
c_{v}}\nabla\cdot\mathbf{v}=\frac{\dot{Q}}{c_{v}},$ (5)

where $D/Dt=\partial/\partial t+\mathbf{v}\cdot\nabla$,
$\mathbf{v}\cdot\nabla$ is advection operator, $\mathbf{v}$ is velocity of
flows, $\bm{\Omega}$ is planetary rotation rate, $\rho$ is air density, $p$ is
pressure, $\mathbf{g}$ is effective gravity vector, $\mathbf{F}$ is friction
force, $T$ is air temperature, $c_{v}$ is specific heat at constant volume,
and $\dot{Q}$ is heating rate (Chapter 1 in Vallis, 2019). Mathematically, the
transformation of coordinates does not change the values and forms of scalars
and scalar operators, i.e., temperature, density, divergence of velocity,
heating rate, and advection operator (Chapter 2 in Kundu et al., 2016). Thus,
the two scalar equations, Equations (4) and (5), will not change. However, the
transformation of coordinates will change the projections of vectors onto new
axes, i.e., wind velocity, rotation rate, and pressure gradient, so that the
projections of Equation (3) will change.

Considering that the directions of unit vectors of axes change as these
vectors move with the atmosphere, it would introduce an effective rotation
rate, so that Equation (3) in the tidally locked coordinates is written as

$\frac{Du_{TL}}{Dt}\mathbf{\hat{e}_{i}}+\frac{Dv_{TL}}{Dt}\mathbf{\hat{e}_{j}}+\frac{Dw_{TL}}{Dt}\mathbf{\hat{e}_{k}}=-\bm{\Omega}_{flow}\times\mathbf{v}-2\bm{\Omega}\times\mathbf{v}-\frac{1}{\rho}\nabla
p-g\mathbf{\hat{e}_{k}}+\mathbf{F},$ (6)

where $u_{TL}$, $v_{TL}$, and $w_{TL}$ are the tidally locked zonal,
meridional, and vertical winds, respectively. $\mathbf{\hat{e}_{i}}$,
$\mathbf{\hat{e}_{j}}$, and $\mathbf{\hat{e}_{k}}$ are the unit vectors of the
tidally locked axes, respectively. $\bm{\Omega}_{flow}$ is the effective
rotation rate (Equation 2.31 in Vallis, 2019), with formula of

${\color[rgb]{0,0,0}\bm{\Omega}_{flow}=-\frac{v_{TL}}{r}\mathbf{\hat{e}_{i}}+\frac{u_{TL}}{r}\mathbf{\hat{e}_{j}}+\frac{u_{TL}\tan\phi_{TL}}{r}\mathbf{\hat{e}_{k}}},$
(7)

$r$ is the radial distance from the center of the planet, and $\phi_{TL}$ is
the tidally locked latitude. From Figure 2(b), $\bm{\Omega}$ can be written as

${\color[rgb]{0,0,0}\bm{\Omega}=\mit\Omega\sin\lambda_{TL}\mathbf{\hat{e}_{i}}+\mit\Omega\cos\lambda_{TL}\sin\phi_{TL}\mathbf{\hat{e}_{j}}-\mit\Omega\cos\lambda_{TL}\cos\phi_{TL}\mathbf{\hat{e}_{k}}},$
(8)

where $\mit\Omega$ represents the magnitude of planetary rotation rate, and
$\lambda_{TL}$ is the tidally locked longitude. Combining Equations (6)–(8)
yields

${\color[rgb]{0,0,0}\frac{Du_{TL}}{Dt}=-\frac{u_{TL}w_{TL}}{r}+\frac{\tan\phi_{TL}}{r}u_{TL}v_{TL}-\frac{1}{\rho
r\cos\phi_{TL}}\frac{\partial p}{\partial\lambda_{TL}}-2\mit\Omega
v_{TL}\cos\lambda_{TL}\cos\phi_{TL}-\rm 2\mit\Omega
w_{TL}\cos\lambda_{TL}\sin\phi_{TL}+F_{\lambda}},$ (9)
${\color[rgb]{0,0,0}\frac{Dv_{TL}}{Dt}=-\frac{v_{TL}w_{TL}}{r}-\frac{\tan\phi_{TL}}{r}u_{TL}^{2}-\frac{1}{\rho
r}\frac{\partial p}{\partial\phi_{TL}}+2\mit\Omega
u_{TL}\cos\lambda_{TL}\cos\phi_{TL}+\rm 2\mit\Omega
w_{TL}\sin\lambda_{TL}+F_{\phi}},$ (10)
${\color[rgb]{0,0,0}\frac{Dw_{TL}}{Dt}=\frac{u_{TL}^{2}+v_{TL}^{2}}{r}-\frac{1}{\rho}\frac{\partial
p}{\partial r}-g+2\mit\Omega u_{TL}\cos\lambda_{TL}\sin\phi_{TL}-\rm
2\mit\Omega v_{TL}\sin\lambda_{TL}+F_{r}},$ (11)

where $F_{\lambda}$, $F_{\phi}$, and $F_{r}$ are the projections of the
friction force on axes. On a typical terrestrial planet, the thickness of the
atmosphere are usually ignored compared to its horizontal scale, and so is the
vertical motion ($\omega$). Thus, the shallow atmosphere approximation
($r=a+z\approx a$, $a$ is planetary radius, and $z$ is height above surface;
$\partial/\partial r\approx\partial/\partial z$;
$\left|w_{TL}\right|\ll\left|u_{TL}\right|,\left|v_{TL}\right|$) is employed
to simplify Equations (9)–(11), shown as

${\color[rgb]{0,0,0}\frac{Du_{TL}}{Dt}=f_{TL}v_{TL}+\frac{\tan\phi_{TL}}{a}u_{TL}v_{TL}-\frac{1}{\rho
a\cos\phi_{TL}}\frac{\partial p}{\partial\lambda_{TL}}+F_{\lambda}},$ (12)
${\color[rgb]{0,0,0}\frac{Dv_{TL}}{Dt}=-f_{TL}u_{TL}-\frac{\tan\phi_{TL}}{a}u_{TL}^{2}-\frac{1}{\rho
a}\frac{\partial p}{\partial\phi_{TL}}+F_{\phi}},$ (13) $\frac{\partial
p}{\partial z}=-\rho g,$ (14)

where $f_{TL}\equiv-2\mit\Omega\cos\lambda_{TL}\cos\phi_{TL}$ is the Coriolis
parameter in the tidally locked coordinates. The three momentum equations have
the same forms as those in the standard spherical coordinates (Equation 2.41
in Vallis, 2019). Furthermore, Equations (12)–(14) along with the unchanged
Equations (4) and (5) suggest that the projected forms of the primitive
equations in the two coordinates are the same except that the formulas of the
Coriolis parameter and the velocities are different. Therefore, the processes
to obtain the LEC in the standard coordinates are valid in the tidally locked
coordinates.

Table 1: General descriptions of symbols used in the LEC. Note that zonal mean is defined as the average along the standard longitudes in the standard coordinates, but along the tidally locked longitudes in the tidally locked coordinates. So does the deviation from the zonal mean. Symbols | Description
---|---
$X$ | Arbitrary quantity
$\bar{X}$ | Temporal-mean of $X$
$\left[X\right]$ | Zonal-mean of $X$
$\tilde{X}$ | Global-mean of $X$
$X^{{}^{\prime}}$ | Deviation from the temporal-mean of $X$, equal to $X-\bar{X}$
$X^{*}$ | Deviation from the zonal-mean of $X$, equal to $X-\left[X\right]$
$X^{{}^{\prime\prime}}$ | Deviation from the global-mean of $X$, equal to $X-\tilde{X}$
$C\left(X_{1},X_{2}\right)$ | Conversion rate from $X_{1}$ to $X_{2}$
$G\left(X\right)$ | Generation rate of $X$
$D\left(X\right)$ | Dissipation rate of $X$
$\lambda$ | Longitude
$\phi$ | Latitude
$Z$ | Geopotential height
$u$ | Zonal wind
$v$ | Meridional wind
$\omega$ | Vertical pressure velocity
$a$ | Planetary radius (solid part)
$g$ | Gravity
$T$ | Air temperature
$\theta$ | Air potential temperature
$R$ | Gas constant for dry air
$c_{p}$ | Specific heat at constant pressure
$\kappa$ | $R/c_{p}$
$\gamma$ | Stability factor, equal to $-\frac{\kappa\theta}{pT}\left(\frac{\partial\tilde{\theta}}{\partial p}\right)^{-1}$
$dm$ | Mass element, equal to $a^{2}\cos\phi d\lambda d\phi dp/g$

Following section 14.3 in Peixóto & Oort (1992), we combine Equations (4),
(5), and (12)–(14), and then integrate over the whole atmosphere to disregard
all boundary terms. This process yields

$\frac{\partial\rm P_{M}}{\partial t}=-{\rm C\left(P_{M},P_{E}\right)}-{\rm
C\left(P_{M},K_{M}\right)}+\rm{G\left(P_{M}\right)},$ (15) $\frac{\partial\rm
P_{E}}{\partial t}={\rm C\left(P_{M},P_{E}\right)}-{\rm
C\left(P_{E},K_{E}\right)}+\rm{G\left(P_{E}\right)},$ (16) $\frac{\partial\rm
K_{M}}{\partial t}={\rm C\left(K_{E},K_{M}\right)}+{\rm
C\left(P_{M},K_{M}\right)}-\rm{D\left(K_{M}\right)},$ (17) $\frac{\partial\rm
K_{E}}{\partial t}=-{\rm C\left(K_{E},K_{M}\right)}+{\rm
C\left(P_{E},K_{E}\right)}-\rm{D\left(K_{E}\right)},$ (18)

where PM is mean potential energy, PE is eddy potential energy, KM is mean
kinetic energy, KE is eddy kinetic energy, $\rm C\left(X_{1},X_{2}\right)$ is
conversion rate from X1 to X2, $\rm G\left(X\right)$ is generation rate of X,
and $\rm D\left(X\right)$ is dissipation rate of X (Figure 1). The general
descriptions of symbols we used are shown in Table 1. The detailed
descriptions of these terms are shown in Appendix B.

###  2.2 Data

Daily-mean data for the atmosphere on Earth is from the National Center for
Environmental Prediction and the Department of Energy reanalysis datasets
(NCEP R2). These datasets are produced by an advanced data assimilation method
combining numerical models and observational data. The spatial resolution is
$2.5^{\circ}\times 2.5^{\circ}$ in latitude and longitude with 17 levels from
surface to 10 hPa. The period covered is the whole year of 1979. The data is
grouped by 12 months and the LEC is calculated in each month and then averaged
to get the annual-mean LEC. This evaluation is based on the fact that there
are strong seasonal cycles and that synoptic eddies have lifetimes of mostly
several days and less than a month.

Daily-mean data for the atmosphere on tidally locked terrestrial planets is
from simulations by the Exoplanet Community Atmosphere Model (ExoCAM, Wolf &
Toon, 2014, 2015; Wolf et al., 2017, 2022). One simulation is for a rapidly
rotating, tidally locked planet with a rotation period (= orbital period) of 5
Earth days, and the other simulation is for a slowly rotating, tidally locked
planet with a rotation period (= orbital period) of 60 Earth days. In the two
simulations, the solar constant is 1360 W m-2, planets are Earth-sized
aquaplanets with terrestrial atmospheres (1 bar N2, 400 ppmv CO2, and flexible
water vapor) and the same gravity as Earth. The spatial resolution is
$4^{\circ}\times 5^{\circ}$ in latitude and longitude with 40 pressure levels
from surface to 10 Pa. The period covered is the last 300 model days. We
directly calculate the LEC over all 300 days instead of month by month, based
on that there is no seasonal or diurnal cycle on the 1:1 tidally locked
planets.

In our calculations, all GCMs’ data over 100 hPa are excluded. This is because
temperatures and winds are strongly model-dependent near the top of the
atmosphere (Sergeev et al., 2022; Turbet et al., 2022). For consistency, the
NCEP R2 data over 100 hPa are also excluded. These treatments do not affect
the main conclusions in this work.

##  3 Results

###  3.1 Thermal structure and atmospheric circulation

Figure 4: Climatic elements on three planets. (a)–(c) Temporal- and zonal-mean
temperature (color shading) and zonal-mean zonal winds (contour lines) on
Earth, on the rapidly rotating tidally locked terrestrial planet, and on the
slowly rotating tidally locked terrestrial planet; (d)–(f) same as (a)–(c) but
for deviations of temporal- and zonal-mean geopotential height from the global
means (color shading), and mass stream functions (contour lines, in units of
$10^{10}$ kg s-1). Note that the ranges of color bars in panels (d), (e), and
(f) are different. Panels (a), (b), (d) and (e) are shown in the standard
coordinates, and (c) and (f) are shown in the tidally locked coordinates. SP
and AP are the substellar point and the antistellar point, respectively.
Figure 5: Deviations of temporal-mean temperature from the global means at 500
hPa on (a) Earth, (b) one rapidly rotating tidally locked terrestrial planet,
and (c) one slowly rotating tidally locked terrestrial planet. The temporal-
and global-mean temperatures are shown in the top of each panel. The black
dots represent substellar points.

For Earth, the rapidly rotating tidally locked planet, and the slowly rotating
tidally locked planet, the solar constant is 1360 W m-2. It corresponds to a
global mean of 340 W m-2 received stellar radiation at the top of the
atmosphere, but the absorbed values are different for the three planets,
suggesting different efficiencies of energy input to the planets. On Earth,
the absorbed stellar radiation is 238 W m-2, including 80 W m-2 absorbed by
the atmosphere and 158 W m-2 absorbed by the surface. The global-mean surface
temperature is 288 K. The absorbed stellar radiation on the tidally locked
planets is smaller than that on Earth, i.e. 187 W m-2 on the rapidly rotating
planet (57 W m-2 absorbed by the atmosphere and 130 W m-2 absorbed by the
surface) and 175 W m-2 on the slowly rotating planet (57 W m-2 absorbed by the
atmosphere and 118 W m-2 absorbed by the surface), due to the reflection by
thick clouds near the substellar location (Yang et al., 2013). Naturally, the
global-mean surface temperatures on the two tidally locked planets are lower
than that on Earth, i.e. 257 K on the rapidly rotating planet and 247 K on the
slowly rotating planet.

The air temperature, geopotential height, and atmospheric circulation on the
three planets are shown in Figure 4. In the free atmosphere, these climatic
elements are zonally homogeneous on the rapidly rotating tidally locked
terrestrial planet and axisymmetric on the slowly rotating tidally locked
terrestrial planet, so that the former is shown in the standard coordinates
and the latter is shown in the tidally locked coordinates.

The temperature structure on Earth and on the rapidly rotating tidally locked
terrestrial planet are analogous, as they show steep meridional (south–north)
temperature gradients in the middle latitudes (Figures 4(a) and (b)). While on
the slowly rotating tidally locked terrestrial planet, the global free
atmosphere is in a weak temperature gradient (WTG) regime (Pierrehumbert,
2010; Pierrehumbert & Hammond, 2019), for which the air temperature is almost
horizontally homogeneous everywhere (Figure 4(c)). This is because weak
Coriolis effect cannot maintain a large pressure or temperature gradient. The
WTGs can be more obviously seen in the horizontal temperature structures. For
example, the equator–pole temperature difference at 500 hPa is about 40 K on
Earth and about 60 K on the rapidly rotating tidally locked terrestrial planet
(Figures 5(a) and (b)). However, the temperature difference on the slowly
rotating tidally locked terrestrial planet is no more than 5 K (Figure 5(c)).
The WTGs are destroyed only very close to the surface, due to the effect of
surface friction. Note that in the WTG regime, the difference of geopotential
height is also small, e.g., no more than 500 m on the slowly rotating planet,
while this value is about 3000 m on Earth and on the rapidly rotating planet
(Figures 4(d)–(f)).

Figure 4(c) shows a strong temperature inversion on the slowly rotating
tidally locked planet, which is mainly on the nightside and extends to the
dayside. It is caused by the uneven distribution in the stellar radiation and
the effective energy transport from dayside to nightside in the free
atmosphere (Joshi et al., 2020). The strong inversion can make the atmosphere
be stable and inhibit the growth of eddies, which can influence the LEC and
will be shown in the subsequent sections.

On Earth, the annual- and zonal-mean mass stream functions clearly show the
Hadley cells in the tropics and the Ferrel cells in the middle latitudes
(Figure 4(d)). On the rapidly rotating tidally locked terrestrial planet, the
Hadley cells expand and become dominant while the Ferrel cells almost
disappear (Figure 4(e)). This is due to that the planetary rotation rate is
1/5 of that on Earth. The strength of Hadley cells on this rapidly rotating
planet is about $25\times 10^{10}$ kg s-1, and is somewhat larger than the
value of $10\times 10^{10}$ kg s-1 on Earth. Differently, the circulation on
the slowly rotating tidally locked terrestrial planet is an isotropic
overturning circulation around the substellar point (Figure 4(f)). It extends
to the globe due to the very small rotation rate, which is 1/60 of that on
Earth. The strength of the global overturning circulation is about $120\times
10^{10}$ kg s-1, and is much stronger than the Hadley cells on Earth and on
the rapidly rotating planet. This is due to that the day–night surface
temperature contrast is about 100 K, much larger than the equator–pole surface
temperature difference on Earth, about 50 K (Figures 2(c) and (d)).

###  3.2 LEC on Earth

Figure 6: The global-mean vertical integrals of energy components (in boxes)
and conversion rates (close to arrows) on Earth. Units are $10^{5}$ J m-2 for
energy and W m-2 for conversion rates. Arrows without specific values
represent generation rates and dissipation rates, same as those in Figure
1(a). Directions of arrows represent the directions of energy conversion, and
the main conversion paths are highlighted by blue color. Values are calculated
in the standard coordinates.

The global-mean vertical integrals of the LEC on Earth are shown in Figure 6.
The total available energy stored in the atmosphere is about $62.0\times
10^{5}$ J m-2, including $44.1\times 10^{5}$ J m-2 of PM, $5.0\times 10^{5}$ J
m-2 of PE, $6.9\times 10^{5}$ J m-2 of KM, and $6.0\times 10^{5}$ J m-2 of KE.
The main conversion path is P${}_{\rm M}\rightarrow$P${}_{\rm
E}\rightarrow$K${}_{\rm E}\rightarrow$KM. PM is converted to PE at a rate of
1.70 W m-2 through heat transport by baroclinic eddies. PE is converted to KE
at a rate of 2.37 W m-2 through cross-isobaric motions in baroclinic eddies.
Some portion of KE is converted to KM at a rate of 0.48 W m-2 through
wave–mean flow interactions. In addition, PM is ultimately converted to KM at
a relatively inefficient rate of 0.24 W m-2 through cross-isobaric motions in
the Hadley and Ferrel cells. Our results are consistent with previous
estimations (e.g., Peixóto & Oort, 1974; Li et al., 2007; Kim & Kim, 2013). We
also recalculate the LEC using the data simulated by ExoCAM, and obtain
analogous results to those based on reanalysis data (figures not shown).

Note that the four arrows in Figure 6 without specific values represent the
generation rates of PM and PE and the dissipation rates of KM and KE,
respectively. It is difficult to calculate directly from the reanalysis data,
so we omit their values here. However, their values can be estimated by
assuming an equilibrium energy cycle. That is, PM is converted to PE and KM
with 1.70 and 0.24 W m-2, respectively, and therefore a generation rate of PM
with 1.94 W m-2 is required. Similarly, the generation rate of PE is 0.67 W
m-2, the dissipation rate of KM is 0.72 W m-2, and the dissipation rate of KE
is 1.89 W m-2. We omit the corresponding results on the other two planets, and
discuss them in Summary and Discussions.

Figure 7: Latitude-altitude cross-sections of the LEC on Earth. (a)–(d) Mean
potential energy (PM), eddy potential energy (PE), mean kinetic energy (KM),
and eddy kinetic energy (KE), respectively; (e)–(h) conversion rates of mean
potential energy to eddy potential energy, eddy potential energy to eddy
kinetic energy, eddy kinetic energy to mean kinetic energy, and mean potential
energy to mean kinetic energy, respectively. Global-mean vertical integrals of
each panel are corresponding values in Figure 6. Panels are shown in the
standard coordinates. The range of color bar in panel (a) is different from it
in panels (b)–(d).

The majority of PM lies in the range of 1000–300 hPa over the polar regions,
especially over the Antarctic continent, where the temperature departure from
the global mean is largest (Figures 4(a) and 7(a)). There is also a secondary
maximum near the tropopause in the tropics, as the temperature there also
deviates significantly from the global mean, although the meridional
temperature gradient is weak. PE is mainly in 30–90∘S/N (Figure 7(b)),
resulting from temperature anomalies induced by baroclinic eddies, forced
orographic waves, and land–sea surface temperature contrast (Li et al., 2007).
Most of KM is in the troposphere over the middle latitudes of 30–60∘S/N, as it
reflects the subtropical and mid-latitude jets (Figures 4(a) and 7(c)). In
addition, there is an extra maximum extending into the stratosphere in the
southern hemisphere, as it reflects the stratospheric jet (over 100 hPa, not
entirely shown). KE is mainly centered in the troposphere over 50–60∘S/N
(Figure 7(d)), associated with the storm tracks over the Pacific, Atlantic,
and Southern oceans (Trenberth, 1991; Harnik & Chang, 2003).

The structure of PE is analogous to that of PM except for a slightly shift to
the equator, since PE is conversed from PM through baroclinic eddies so that
is affected by PM. Likewise, the structure of KE is analogous to that of KM.
Besides, KE is also affected by PE as it is conversed from PE. Thus, the
maxima of KE are between the maxima of PE and KM, suggesting KE is a result of
a balance between PE and KM.

The conversion from PM to PE occurs mainly in the middle troposphere over the
middle latitudes of 30–60∘S/N, where the baroclinic eddies are strong (Figure
7(e)). The growing baroclinic eddies transport heat poleward and reduce the
south–north temperature gradients, but lead additional east–west temperature
variance, i.e., reduce PM but generate PE. Inside these eddies, the cross-
isobaric motion converts a small portion of PE to KE (Figure 7(f)). PE is
converted to KE mainly near the surface over mid-to-high latitudes, especially
over the Antarctic, and is associated with the heat-driven rising and sinking
motions (Li et al., 2007). In the middle troposphere over the middle
latitudes, eddies are generate and then propagate out of this region, but
bring the momentum back to accelerate the jet, and these wave–mean flow
interactions convert KE to KM (Figure 7(g)). PM is converted to KM in the
tropics and subtropics by the motion along the pressure gradient in the Hadley
cells nearly following angular momentum conservation, but KM is converted back
to PM in the middle latitudes by the motion against the pressure gradient in
the Ferrel cells (Figures 4(d) and 7(h)). The combined action of the
conversion between PM and KM makes the global-mean conversion from PM to KM be
relatively inefficient, and even be negative sometimes (e.g., Figure 14.8 in
Peixóto & Oort, 1992).

The LEC perspective is consistent with the momentum budget. The maxima of KM
over 40–60∘S/N coincide with the regions where KE is converted to KM but KM is
converted to PM. It indicates that the mid-latitude jets are eddy-driven—the
poleward momentum transport by eddies maintains the mid-latitude jets.
Likewise, the maxima of KM in the subtropics coincide with the regions where
both PM and KE are converted to KM, indicating that the poleward angular
momentum transport by meridional circulation and eddies together maintain the
subtropical jets.

###  3.3 LEC on rapidly rotating, tidally locked terrestrial planet

Figure 8: Same as Figure 6 but on the rapidly rotating tidally locked
terrestrial planet. Values are calculated in the standard coordinates (a) and
in the tidally locked coordinates (b).

Figure 8(a) displays the global-mean vertical integrals of the LEC on the
rapidly rotating tidally locked terrestrial planet. In our estimations, PM is
about $37.9\times 10^{5}$ J m-2, PE is about $6.9\times 10^{5}$ J m-2, KM is
about $66.3\times 10^{5}$ J m-2, and KE is about $12.1\times 10^{5}$ J m-2.
There are two main conversion paths, P${}_{\rm M}\rightarrow$P${}_{\rm
E}\rightarrow$KE and P${}_{\rm M}\rightarrow$K${}_{\rm M}\rightarrow$KE. On
the one hand, PM is converted to PE at a rate of 0.45 W m-2, and then into KE
at a rate of 2.34 W m-2. On the other hand, PM is converted to KM at a rate of
1.90 W m-2, and then into KE at a rate of 0.83 W m-2.

Figure 9: Same as Figure 7 but on the rapidly rotating tidally locked
terrestrial planet. Panels are shown in the standard coordinates. The maximum
in panel (a) is about $600\times 10^{5}$ J m-2 bar-1; the maximum in panel (h)
is about $40$ W m-2 bar-1, and the minimum is about $-30$ W m-2 bar-1.

The stellar flux received by the planet has an equator–pole contrast, which
can generate PM in the atmosphere. The majority of PM is in the middle
troposphere over the polar regions, and a portion is in the tropics (Figure
9(a)). The structure is similar to that on Earth, because they both have
similar south–north temperature gradient (Figures 4(a), 4(b), 5(a), and 5(b)).
PE is much smaller than PM and mostly over the polar regions, with
contributions from the extratropical stationary Rossby waves (Figure 9(b)).
The cold lobes of the Rossby waves are visible in the horizontal temperature
structure (Figure 5(b)). KM is located in the mid-to-upper troposphere over
the middle latitudes, coinciding with the two westerly jets (Figures 4(b) and
9(c)). Moreover, KM over the equator is also non-negligible due to the
equatorial superrotation. KE is concentrated in the polar regions, associated
with the extratropical stationary Rossby waves (Figure 9(d)).

On the rapidly rotating tidally locked planets, although the received stellar
radiation and the underlying surface are invariant, the atmosphere still has
large variability, i.e., transient eddies, which can be seen in the
instantaneous pressure field, temperature, winds, water vapor, and clouds
(e.g., Merlis & Schneider, 2010; Pierrehumbert & Hammond, 2019; Song & Yang,
2021). The variability also contributes a portion of PE and KE.

Figure 10: Longitude-altitude cross-sections of (a) eddy meridional velocity,
(b) eddy temperature, and (c) eddy geopotential height at 70∘N on the rapidly
rotating tidally locked terrestrial planet. Panels are instantaneous
quantities departed from their zonal means.

In the middle troposphere over the polar regions, PM is converted to PE by
eddies (Figure 9(e)). These eddies are conjectured to arise from a form of
baroclinic instability (Pierrehumbert & Hammond, 2019). Figure 10 shows the
instantaneous longitude-altitude structure of these eddies with a slightly
westward tilt, which is similar to the unstable baroclinic mode on Earth, but
on a larger scale due to the smaller planetary rotation rate (Holton & Hakim,
2013). Meanwhile, in the upper troposphere over the equator, the conversion
from PM to PE is caused by the equatorial wave activity. However, a fraction
of PE is converted back to PM in the range of 500–300 hPa over the middle
latitudes and near the surface, which is due to the up-gradient heat transport
by stationary waves. The cross-isobaric motion converts PE to KE in the
tropics and middle latitudes, but converts KE back to PE in the subtropics and
polar regions (Figure 9(f)). In general, the net conversion is from PE to KE.
In the free troposphere, KE is converted to KM over the tropics, but KM is
converted to KE over the middle latitudes, which is caused by the wave–mean
flow interactions (Figure 9(g)). That is, the stationary Rossby and Kelvin
waves form as a Matsuno-Gill mode that transports westerly momentum from the
middle latitudes to the equator, accelerating the equatorial jet but damping
the jets in the middle latitudes (e.g., Matsuno, 1966; Gill, 1980; Showman &
Polvani, 2011). However, the net conversion is from KM to KE. The conversion
from PM to KM is dominant from the tropics to the middle latitudes through the
expanded Hadley cells, while the conversion from KM back to PM only occurs at
high latitudes through the weak and narrow Ferrel cells (Figures 4(e) and
9(h)). Thus, the net conversion is from PM to KM.

The maxima of KM over 30–60∘S/N coincide with the regions where PM is
converted to KM. It suggests that the jets there are maintained by poleward
angular momentum transport through the Hadley cells rather than by momentum
transport through waves. Zonal jets constrained by the angular momentum
conservation would obey

$\left[u\right]=\Omega a\frac{\sin^{2}\phi}{\cos\phi},$ (19)

where $\Omega$ is the planetary rotation rate, $a$ is the planetary radius,
$\phi$ is the latitude, and wind speed over the equator is assumed to be zero
(Equation 11.4 in Vallis, 2019). It gives an estimate of the zonal-mean wind
speed at 50∘S/N to be about 80 m s-1, in agreement with the simulation. Thus,
the jets are more like subtropical jets, because their driving mechanism is
similar to that of the subtropical jets on Earth.

Comparing to Earth, a main difference of the LEC is that KM is much larger,
i.e., about ten times larger, which is related to the larger wind speeds. For
example, a zonal jet has a maximum wind speed of about 80 m s-1 on the rapidly
rotating tidally locked planet, while this value is only 30 m s-1 on Earth
(Figures 4(a) and (b)). This is due to that wider Hadley cells move the air
further away from the equator, allowing larger wind speeds, although the
planet rotates at 1/5 the rate of Earth (Equation (19)). The stronger winds
are also consistent with the larger meridional temperature gradient in the
high latitudes of the planet (Figure 5(b)). Another big difference from Earth
is that the net conversion from PM to KM becomes efficient. This is due to
that the Hadley cells become stronger and wider, but the Ferrel cells become
weaker, as a result of the smaller rotation rate. This situation is similar to
Venus, where the Hadley cells extend throughout the atmosphere and make the
conversion from PM to KM be most efficient (Lee & Richardson, 2010). In
addition, KM is eventually converted to KE rather than KE being converted to
KM.

###  3.4 LEC on slowly rotating, tidally locked terrestrial planet

Figure 11: Same as Figure 6 but on the slowly rotating tidally locked
terrestrial planet. Values are calculated in the tidally locked coordinates
(a) and in the standard coordinates (b).

In this section, the LEC on the slowly rotating tidally locked planet is
calculated using the tidally locked formulas. The global-mean vertical
integrals are shown in Figure 11(a). Overall, the total available energy is
about $12.7\times 10^{5}$ J m-2, including $1.9\times 10^{5}$ J m-2 of PM,
$0.9\times 10^{5}$ J m-2 of PE, $3.2\times 10^{5}$ J m-2 of KM, and $6.7\times
10^{5}$ J m-2 of KE. The main conversion path is P${}_{\rm
M}\rightarrow$K${}_{\rm M}\rightarrow$KE: PM is converted to KM at a rate of
2.25 W m-2 through cross-isobaric motions in global overturning circulation,
and a portion of KM is converted to KE at a rate of 0.51 W m-2 through
interactions between the overturning circulation and eddies. While the other
path, P${}_{\rm M}\rightarrow$P${}_{\rm E}\rightarrow$KE, which occurs in
baroclinic eddies and stationary planetary waves, is relatively inefficient,
as that the conversion rate from PM to PE is only 0.07 W m-2 and the
conversion rate from PE to KE is only 0.14 W m-2.

Figure 12: Same as Figure 7 but on the slowly rotating tidally locked
terrestrial planet. Panels are shown in the tidally locked coordinates. SP and
AP are the substellar point and the antistellar point, respectively. The
maximum in panel (a) is about $70\times 10^{5}$ J m-2 bar-1; the maximum in
panel (c) is about $70\times 10^{5}$ J m-2 bar-1; the maximum in panel (h) is
about $150$ W m-2 bar-1.

As a result of the WTGs in the free atmosphere (Figures 4(c) and 5(c)), PM is
nearly zero except very close to the surface around the substellar point
(Figure 12(a)). Likewise, PE is nearly zero throughout the atmosphere (Figure
12(b)). The horizontal flow in the upper branch of the global overturning
circulation, known as zonal-mean meridional winds in the tidally locked
coordinates (Figure 4(f)), contributes to the majority of KM in the upper
troposphere on the dayside (Figure 12(c)), while zonal-mean zonal winds in the
tidally locked coordinates are very weak (Figure 4(c)). In addition, the
backflow from the nightside to the substellar point contributes to the
secondary maximum in KM near the surface of the dayside, but with a relatively
smaller wind speed due to surface friction. The stationary planetary waves
contribute a portion of KE, which is centered on the nightside and extends to
the dayside (Figure 12(d)).

Note that the superrotation is transformed to an eddy component in the tidally
locked coordinates (Figure 3). Thus, the kinetic energy of the superrotation
contributes a portion of KE rather than KM. In general, KE here is a measure
of any wind that deviates from the global overturning circulation.

The conversion from PM to PE is inefficient throughout the atmosphere due to
the WTGs and weak baroclinic activity (Figure 12(e)). PE is converted to KE on
the nightside and around the substellar point, but KE is converted back to PE
on the dayside, which is mainly related to the cross-isobaric motion of the
equatorial superrotation (Figure 12(f)). The combine action of the conversion
between PE and KE makes the global-mean conversion from PE to KE be
inefficient. However, the ultimate reason is that the atmosphere is nearly
barotropic. The conversion between KE and KM is dominant by KM converting to
KE, which occurs near the terminator (Figure 12(g)). It is caused by the
interactions between the global overturning circulation and the eddy
components, and the details are discussed in section 3.5. PM is converted to
KM near the surface and in the upper troposphere around the substellar point,
by the motion along the pressure gradient in the global overturning
circulation (Figures 4(f) and 12(h)). KM is converted back to PM in a limited
region, and the net conversion is from PM to KM.

The structure of the conversion from PE to KE is common on a slowly rotating
tidally locked planets. This is due to the fact that the crests of stationary
Rossby waves usually lie near the eastern terminator and the troughs usually
lie near the western terminator (e.g., Carone et al., 2015; Hammond &
Pierrehumbert, 2018; Wang & Yang, 2021). That is, the equatorial superrotation
acts against the pressure gradient force as it crosses the dayside, converting
KE to PE, and vice versa.

Comparing to Earth, the big difference is that PM on the slowly rotating
tidally locked planet is much smaller, which is mainly due to the WTGs caused
by the smaller planetary rotation rate, i.e., 1/60 of that on Earth. Moreover,
the strong and wide temperature inversion away from the substellar region
makes the atmosphere be more stable than Earth and more difficult to generate
motions, which also results in small PM. Another difference from Earth is that
the energy conversion involved in baroclinic activity is inefficient, while
the conversion from PM to KM is efficient, as a consequence of that a small
planetary rotation rate makes the atmosphere be more barotropic and makes the
thermal forcing tend to generate a strong circulation rather than temperature
gradients (e.g., Edson et al., 2011; Noda et al., 2017; Komacek et al., 2019).

###  3.5 Comparison of LEC between standard and tidally locked coordinates

In this section, we compare the LEC between standard and tidally locked
coordinates applied to rapidly and slowly rotating tidally locked planets.
Briefly, the LEC in the standard coordinates is beneficial to describe PM
related to equator–pole temperature contrasts, KM related to zonal-mean zonal
winds (e.g., the equatorial superrotation), the conversion between PM and KM,
and wave–mean flow interactions. While the LEC in the tidally locked
coordinates is beneficial to describe PM related to day–night temperature
contrasts, KM related to the global overturning circulation, and the
conversion between them. To gain more insight, we recalculate the LEC on the
rapidly rotating tidally locked planet in the tidally locked coordinates and
the LEC on the slowly rotating tidally locked planet in the standard
coordinates, and compare them with the results in Sections 3.3 and 3.4.

####  3.5.1 Comparison on rapidly rotating, tidally locked terrestrial planet

Figure 13: Same as Figure 7 but on the rapidly rotating tidally locked
terrestrial planet. Panels are shown in the tidally locked coordinates. Figure
14: Climatic elements on the rapidly rotating tidally locked planet in the
standard coordinates (left column) and in the tidally locked coordinates
(right column). Upper row: surface temperature (contours) and the near-surface
winds (vectors). Middle row: air temperature (contour) and winds (vector) at
500 hPa. Lower row: deviations of geopotential height from the global means
(contour) and winds (vector) at 200 hPa. Black dot represents the substellar
point, and black triangle represents the North Pole in the standard
coordinates.

The global-mean vertical integrals of the LEC on the rapidly rotating tidally
locked planet in the tidally locked coordinates are shown in Figure 8(b), and
their structures are shown in Figure 13.

On a rapidly rotating tidally locked planet, the air temperature and
geopotential height are zonally homogeneous in the free atmosphere, and the
sharp contrasts are between the equatorial and polar regions (Figures 14(c)
and (e)). The winds in the free atmosphere also behave as the zonal jets.
These equator–pole contrasts become eddy components in the tidally locked
coordinates, and so do the zonal jets (Figures 14(d) and (f)). Thus, PE and KE
in the tidally locked coordinates are much larger than those in the standard
coordinates, respectively, which may lead a misconception that eddies dominate
this planet rather than large-scale circulation (Figure 8). Temperature and
winds exhibit a day–night asymmetry only very close to the surface (Figure
14(a)), so that PM and KM in the tidally locked coordinates are smaller than
those in the standard coordinates and centered only close to the surface
(Figures 13(a) and (c)).

Since both zonal-mean zonal winds and waves belong to eddy components in the
tidally locked coordinates, the conversion between KE and KM here no longer
describes the wave–mean flow interactions. The structure of this conversion
rate is complicated and atypical, and the corresponding dynamical process is
unclear on this planet (Figure 13(g)). The zonal-mean meridional wind speed in
the tidally locked coordinates is generally larger than that in the standard
coordinates, because meridional winds in the standard coordinates are reversed
between the day and night hemispheres. For example, the former is over 10 m
s-1 on this planet, while the latter is 4 m s-1. Thus, the conversion from PM
to KM in the tidally locked coordinates is more efficient than that in the
standard coordinates (Figure 8).

The integrals and cross-sections of energy and conversion rates in the tidally
locked coordinates can be understood from the distributions of air
temperature, geopotential height, and horizontal winds. However, they are more
complicated and lead to less insight into the atmospheric circulation on this
planet than those in the standard coordinates, so we omit them here.

Note that the total potential energy, i.e., the sum of PM and PE, is about
$44.8\times 10^{5}$ J m-2 and is the same in both two coordinates (Figure 8).
Likewise, the total kinetic energy is about $78.4\times 10^{5}$ J m-2 and is
also the same in the two coordinates. They do not depend on the choice of the
coordinates.

####  3.5.2 Comparison on slowly rotating, tidally locked terrestrial planet

Figure 15: Same as Figure 7 but on the slowly rotating tidally locked
terrestrial planet. Panels are shown in the standard coordinates. Figure 16:
Decomposition of temperatures and winds on the slowly rotating tidally locked
terrestrial planet in the standard coordinates. (a) Meridional contrast of
zonal-mean surface temperature; (b) eddy surface temperature; (c) zonal-mean
zonal and meridional winds at 300 hPa; (d) eddy winds at 300 hPa.

The global-mean vertical integrals of the LEC on the slowly rotating tidally
locked planet in the standard coordinates are shown in Figure 11(b), and their
structures are shown in Figure 15.

On a slowly rotating tidally locked planet, the day–night temperature contrast
is usually larger than the meridional contrast. For example, the day–night
surface temperature contrast in our experiment is larger than 80 K, while the
meridional contrast is smaller than 30 K (Figures 16(a) and (b)). Likewise,
the wind speeds of the global overturning circulation are larger than those of
the zonal-mean zonal winds, for example, about 20 m s-1 for the former and
about 5 m s-1 for the latter at 300 hPa in our experiment (Figures 16(c) and
(d)). Thus, in contrast to the LEC in the tidally locked coordinates, PM and
KM are smaller in the standard coordinates, but PE and KE are larger than
those in the tidally locked coordinates (Figure 11), as the day–night
temperature contrast and the overturning circulation belong to eddy components
here (Figures 15(b) and (d)).

The main conversion path in the standard coordinates becomes PE to KE, which
may lead to a misconception that the conversion is dominated by the baroclinic
activity (Figure 15(f)). Indeed, the main conversion occurs in the large-scale
dynamical process, where the day–night temperature contrast induced by the
uneven stellar radiation generates large-scale upwellings and downwellings.
However, the meaning of the conversion between KE and KM in the standard
coordinates is clear. Its structure is consistent with wave–mean flow
interactions, where momentum is transported to the equator by waves to
maintain the equatorial superrotation (Figure 15(g)). By contrast, the
conversion between KE and KM in the tidally locked coordinates is complex. On
this planet, the tidally locked zonal-mean zonal winds are nearly zero, but
the tidally locked zonal-mean meridional winds are strong (Figures 4(c) and
(f)), so that the conversion between KE and KM in the tidally locked
coordinates is more related to the shear instability of the meridional winds,
i.e., the global overturning circulation (see Equation (B7) in Appendix B). In
our calculations, the bulk of the conversion from KM to KE is contributed by
the horizontal shear of the global overturning circulation, which causes the
accumulation of the large-scale momentum and the conversion to the vortex
momentum (see the second term in the right-hand side in Equation (B7), figure
not shown).

Note that the total potential energy is about $2.8\times 10^{5}$ J m-2, the
total kinetic energy is about $9.9\times 10^{5}$ J m-2, and they are the same
in the two coordinates, respectively (Figure 11).

##  4 Summary and Discussions

In this study, we employ the Lorenz energy cycle (LEC) to understand the
atmospheric circulation on tidally locked terrestrial planets. We use ExoCAM
to simulate atmospheric circulation on both rapidly and slowly rotating
tidally locked terrestrial planets, calculate their LECs, and compare them
with that on Earth. The main conclusions are as follows:

  1. (1)

On the rapidly rotating tidally locked planet, the mean potential energy PM
and eddy potential energy PE are comparable to those on Earth, because both
them have similar steep meridional temperature gradients. The mean kinetic
energy KM is much larger than that on Earth, mainly related to much larger
wind speeds. The two paths of energy conversion, P${}_{\rm
M}\rightarrow$P${}_{\rm E}\rightarrow$KE and P${}_{\rm M}\rightarrow$K${}_{\rm
M}\rightarrow$KE, are both effective. The former path is mainly associated
with baroclinic instabilities, and the latter path is associated with large-
scale thermal-driven circulation and barotropic instabilities. These suggest
that the atmosphere of rapidly rotating tidally locked planets is in a mixed
dynamical regime of single-cell circulation and baroclinic eddies.

  2. (2)

On the slowly rotating tidally locked planet, PM and PE are small. This is
because the slow rotation rate makes the planet be in a weak temperature
gradient regime. Meanwhile, the temperature inversion that lies over the
entire nightside and part of the dayside makes the atmosphere be very stable,
also contributes to the small potential energy. KM and KE are comparable to
those on Earth. However, in the tidally locked coordinates, KM is the measure
of the global overturning circulation, and KE is the measure of waves and
zonal-mean zonal jets including the equatorial superrotation. The main path of
energy conversion is P${}_{\rm M}\rightarrow$K${}_{\rm M}\rightarrow$KE,
associate with cross-isobaric motions in the global overturning circulation
and interactions between the global overturning circulation and eddy
components.

In this study, the main factor discussed to affect the LEC is the planetary
rotation rate. Although only two experiments have been performed, the rough
tendency is that air temperature gradients and wind speed decrease as the
rotation rate becomes smaller, as can be found in previous studies (e.g.,
Carone et al., 2015, 2016; Noda et al., 2017). As a result, the total
available energy stored in the atmosphere is less on planets with slower
rotation. Meanwhile, a smaller planetary rotation rate makes the atmosphere
less baroclinic (Komacek et al., 2019). Thus, the conversion paths associated
with baroclinic instabilities are less efficient, and the conversion between
PM and KM becomes dominant.

The planetary rotation rate also determines the ratio of the Rossby
deformation radius to the planetary radius, and hence a regime transition in
the atmosphere (e.g., Carone et al., 2015; Haqq-Misra et al., 2018). On a slow
rotator, the atmospheric state has a large day–night asymmetry, and a LEC in
the tidally locked coordinates is more appropriate. Since the planet becomes a
rapid rotator, the atmospheric state tends to be zonally homogeneous, so that
a LEC in the standard coordinates is more appropriate. For an Earth-like
tidally locked planet, the rotation period that separates the rapid and slow
rotators is usually 5–10 Earth days (Edson et al., 2011; Yang et al., 2014;
Noda et al., 2017). However, both forms of LEC have shortcomings, as the LEC
in the standard coordinates may not be robust very close to the surface, and
the LEC in the tidally locked coordinates is disabled in describing wave–mean
flow interactions. There may be a new form of LEC that can be applied to both
rapid and slow rotators, which needs to be investigated in the future.

Other factors may also affect the LEC, but are not included in this study. For
example, both day–night temperature contrast and wind speeds monotonically
decrease as the background air pressure is increased (Kite et al., 2011;
Leconte et al., 2013; Wordsworth, 2015; Zhang & Yang, 2020). They would make
the available potential and kinetic energy per unit mass of air decrease, but
the change of the total atmospheric energy is unclear, as the total mass of
the atmosphere is increased. Atmospheric compositions should also have impacts
on the LEC. Ding & Wordsworth (2019) showed that excluding greenhouse gases
from the atmosphere would decrease the day–night temperature contrast and wind
speeds; Wang & Yang (2022) showed that the atmospheric circulation in pure N2
atmosphere without any greenhouse gas is very weak. They suggest that the
potential and kinetic energy would be smaller in an atmosphere without any
greenhouse gas.

In this study, we do not calculate the generation rates of potential energy
and the dissipation rates of kinetic energy, which are inevitable to close the
LEC. However, they could be estimated from the remnants of four conversion
rates by assuming an equilibrium state (e.g., Peixóto & Oort, 1992; Li et al.,
2007; Pan et al., 2017). By doing so, our calculations obtain the total
generation rates of potential energy (= total dissipation rates of kinetic
energy) with values of 2.61, 4.24, and 2.39 W m-2 on the three planets,
respectively. Moreover, the generation and dissipation rates can be used to
obtain a more realistic efficiency of the atmospheric heat engine, which is
regarded as the ratio of the total dissipation rate of kinetic energy to the
mean net incoming solar radiation. On Earth, the total dissipation rate is
2.61 W m-2 and the mean net incoming solar radiation is 238 W m-2. They lead
to $\eta\approx 1.1\%$, which is much smaller than the ideal limit based on
the Carnot’s heat engine, $\eta\approx 10\%$ (Peixóto & Oort, 1992). Likewise,
the LEC will also provide a smaller efficiency than the ideal limit on tidally
locked planets, for example, $2.3\%$ (4.24 W m-2/187 W m-2) for the rapidly
rotating planet and $1.4\%$ (2.39 W m-2/175 W m-2) for the slowly rotating
planet in our experiments. Koll & Abbot (2016) estimated the surface wind
speed on tidally locked planets by using the ideal efficiency of atmospheric
heat engine, and the efficiency we obtained may help to optimize their
estimations.

The formulas of the LEC in this study is only applied to atmospheres on
terrestrial planets. Based on the shallow atmosphere approximation, we do not
consider the kinetic energy of vertical motion or the conversion rates
contributed from corresponding metric terms (e.g., $-u_{TL}\omega_{TL}/a$ in
Equation (9)). However, this approximation may be invalid when the vertical
motion is comparable to the horizontal motion. That means that the primitive
equations and the LEC in this study may be inapplicable for gas planets. They
must be corrected by considering all the terms involved in the vertical
motion. The application of the LEC to gas giants is useful to estimate the
efficiency of the planetary heat engine, which is different from that of a
terrestrial planet. In solar system, gas giants (e.g., Jupiter and Saturn)
receive less solar radiation than terrestrial planets (e.g., Venus and Earth)
but hold stronger winds, suggesting a more efficient planetary heat engine,
which may be due to the absence of solid surface or atmospheric compositions
(Showman et al., 2009; Ingersoll, 2013). This estimate may be beneficial for
predicting wind speeds including superrotation and for quantifying the energy
sources of jet streams on gas giants. Moreover, the LEC may be a useful way to
evaluate various models for gas giants by comparing the results computed from
these models.

##  Appendix A transformation relation

In order to transform the physical quantities in the standard coordinates to
their counterparts in the tidally locked coordinates, we derive a
transformation relation between the two coordinates. This relation has been
derived by Koll & Abbot (2015), but with a left-hand coordinate system (i.e.,
$\mathbf{\hat{e}_{i}}\times\mathbf{\hat{e}_{j}}=-\mathbf{\hat{e}_{k}}$, see
their Figure 1(b)). Here we make sure that the two coordinates are right-hand
systems (i.e.,
$\mathbf{\hat{e}_{i}}\times\mathbf{\hat{e}_{j}}=\mathbf{\hat{e}_{k}}$). We put
the substellar point at latitude/longitude
$\left(\phi,\lambda\right)=\left(0^{\circ},180^{\circ}\right)$ in the standard
coordinates and at tidally locked latitude $\phi_{TL}=90^{\circ}$ in the
tidally locked coordinates, and also put the South and North Poles in the
standard coordinates at
$\left(\phi_{TL},\lambda_{TL}\right)=\left(0^{\circ},0^{\circ}\right)$ and
$\left(\phi_{TL},\lambda_{TL}\right)=\left(0^{\circ},180^{\circ}\right)$,
respectively (Figure 2). We transform the two coordinates into the Cartesian
coordinates, where x-axis links the spherical core and the substellar point,
and z-axis links the spherical core and the North Pole in the standard
coordinates, so that

$\displaystyle x$ $\displaystyle=$ $\displaystyle-r\cos\lambda\cos\phi,$
$\displaystyle y$ $\displaystyle=$ $\displaystyle-r\sin\lambda\cos\phi,$
$\displaystyle z$ $\displaystyle=$ $\displaystyle r\sin\phi,$ (A1)

and

$\displaystyle x$ $\displaystyle=$ $\displaystyle r\sin\phi_{TL},$
$\displaystyle y$ $\displaystyle=$ $\displaystyle
r\sin\lambda_{TL}\cos\phi_{TL},$ $\displaystyle z$ $\displaystyle=$
$\displaystyle-r\cos\lambda_{TL}\cos\phi_{TL},$ (A2)

where $r$ is the radial distance from the center of the planet. Combining
Equations (A) and (A) yields the transformation relations between the two
coordinates:

$\displaystyle\lambda_{TL}$ $\displaystyle=$
$\displaystyle\tan^{-1}\left(\frac{\sin\lambda}{\tan\phi}\right),$
$\displaystyle\phi_{TL}$ $\displaystyle=$
$\displaystyle\sin^{-1}\left(-\cos\lambda\cos\phi\right),$
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\tan^{-1}\left(\frac{\sin\lambda_{TL}}{\tan\phi_{TL}}\right),$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\sin^{-1}\left(-\cos\lambda_{TL}\cos\phi_{TL}\right),$ (A3)

where $\lambda$ and $\lambda_{TL}$ belong to $[0,2\pi]$, and $\phi$ and
$\phi_{TL}$ belong to $[-\pi/2,\pi/2]$. Because $\cos\phi\geq 0$ and
$\cos\phi_{TL}\geq 0$, Equations (A) and (A) also yield the sign relations:

$\displaystyle\frac{\cos\lambda}{\left|\cos\lambda\right|}$ $\displaystyle=$
$\displaystyle-\frac{\sin\phi_{TL}}{\left|\sin\phi_{TL}\right|}=-\frac{\tan\phi_{TL}}{\left|\tan\phi_{TL}\right|},$
$\displaystyle\frac{\sin\lambda}{\left|\sin\lambda\right|}$ $\displaystyle=$
$\displaystyle-\frac{\sin\lambda_{TL}}{\left|\sin\lambda_{TL}\right|},$ (A4)

which are used to calculate the transformation of trigonometric functions.

The transformation relations for the scalars including the temperature and the
geopotential height follow the transformation of the coordinates. That is, the
value of a scalar in the tidally locked coordinates is equal to the value at
the corresponding longitude and latitude in the standard coordinates, i.e.,

$A(\lambda_{TL},\phi_{TL})=A(\lambda(\lambda_{TL},\phi_{TL}),\phi(\lambda_{TL},\phi_{TL})),$
(A5)

where the transformation of the coordinates is from Equation (A).

The winds in the standard coordinates are
$(u,v,w)\equiv(r\cos\phi(D\lambda/Dt),r(D\phi/Dt),Dr/Dt)$. Likewise, the winds
in the tidally locked coordinates are

$\displaystyle u_{TL}$ $\displaystyle\equiv$ $\displaystyle
r\cos\phi_{TL}\frac{D\lambda_{TL}}{Dt}$ $\displaystyle=$ $\displaystyle
r\cos\phi_{TL}\left(\frac{\partial\lambda_{TL}}{\partial\lambda}\frac{D\lambda}{Dt}+\frac{\partial\lambda_{TL}}{\partial\phi}\frac{D\phi}{Dt}\right)$
$\displaystyle=$
$\displaystyle\cos\phi_{TL}\left(\frac{\partial\lambda_{TL}}{\partial\lambda}\frac{u}{\cos\phi}+\frac{\partial\lambda_{TL}}{\partial\phi}v\right),$
$\displaystyle v_{TL}$ $\displaystyle\equiv$ $\displaystyle
r\frac{D\phi_{TL}}{Dt}$ $\displaystyle=$ $\displaystyle
r\left(\frac{\partial\phi_{TL}}{\partial\lambda}\frac{D\lambda}{Dt}+\frac{\partial\phi_{TL}}{\partial\phi}\frac{D\phi}{Dt}\right)$
$\displaystyle=$
$\displaystyle\frac{\partial\phi_{TL}}{\partial\lambda}\frac{u}{\cos\phi}+\frac{\partial\phi_{TL}}{\partial\phi}v,$
$\displaystyle w_{TL}$ $\displaystyle=$ $\displaystyle\frac{Dr}{Dt}=w.$ (A6)

Substituting Equation (A) into Equation (A), we obtain

$\left[\begin{array}[]{lll}u_{TL}\\\ v_{TL}\\\
w_{TL}\end{array}\right]=\left[\begin{array}[]{lll}\frac{\cos\lambda\sin\phi}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}&\frac{-\sin\lambda}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}&0\\\
\frac{\sin\lambda}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}&\frac{\cos\lambda\sin\phi}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}&0\\\
0&0&1\end{array}\right]\left[\begin{array}[]{lll}u\\\ v\\\
w\end{array}\right].$ (A7)

The general transformation procedure is to calculate $u_{TL}$ and $v_{TL}$ at
an arbitrary point in the standard coordinates using Equation (A7), and then
to determine the position of the point in the tidally locked coordinates using
Equation (A). Substituting Equations (A) and (A) into the transformation
matrix of the winds yields

$\displaystyle\frac{\cos\lambda\sin\phi}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}$
$\displaystyle=$
$\displaystyle-\frac{\left|\cos\lambda\right|\sin\phi}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}\frac{\tan\phi_{TL}}{\left|\tan\phi_{TL}\right|}$
$\displaystyle=$
$\displaystyle-\frac{\sin\phi}{\sqrt{\tan^{2}\lambda+\sin^{2}\phi}}\frac{\tan\phi_{TL}}{\left|\tan\phi_{TL}\right|}$
$\displaystyle=$
$\displaystyle\frac{\cos\lambda_{TL}\cos\phi_{TL}\tan\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\cos^{2}\lambda_{TL}\sin^{2}\phi_{TL}}}$
$\displaystyle=$
$\displaystyle\frac{\cos\lambda_{TL}\tan\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}},$
$\displaystyle\frac{\sin\lambda}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\lambda_{TL}}{\left|\sin\lambda_{TL}\right|}\frac{\left|\sin\lambda\right|}{\sqrt{1-\cos^{2}\lambda\cos^{2}\phi}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\lambda_{TL}}{\left|\sin\lambda_{TL}\right|}\frac{1}{\sqrt{1+\cot^{2}\lambda\sin^{2}\phi}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\lambda_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\cos^{2}\lambda_{TL}\sin^{2}\phi_{TL}}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\lambda_{TL}\sec\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}},$

so that Equation (A7) can also be expressed in terms of the tidally locked
coordinates,

$\left[\begin{array}[]{lll}u_{TL}\\\ v_{TL}\\\
w_{TL}\end{array}\right]=\left[\begin{array}[]{lll}\frac{\cos\lambda_{TL}\tan\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}}&\frac{\sin\lambda_{TL}\sec\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}}&0\\\
-\frac{\sin\lambda_{TL}\sec\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}}&\frac{\cos\lambda_{TL}\tan\phi_{TL}}{\sqrt{\sin^{2}\lambda_{TL}+\tan^{2}\phi_{TL}}}&0\\\
0&0&1\end{array}\right]\left[\begin{array}[]{lll}u\\\ v\\\
w\end{array}\right],$ (A8)

which can be used to directly calculate the transformation of some special
wind fields in the standard coordinates, such as a uniform zonal-mean zonal
winds. It is easy to prove that

$\displaystyle u_{TL}^{2}+v_{TL}^{2}+w_{TL}^{2}=u^{2}+v^{2}+w^{2},$

so that the transformation between the standard and the tidally locked
coordinates satisfies the conservation of energy.

##  Appendix B formulas of LEC

The formulas of the LEC in the standard coordinates have been derived by
Peixóto & Oort (1974, 1992). We follow their method and obtain the LEC in the
tidally locked coordinates. Since the primitive equations in the standard and
the tidally locked coordinates are the same, the formulas of the LEC in the
two coordinates are also the same, albeit with different details. For example,
$u$ is the winds along the longitude in the standard coordinates but along the
tidally locked longitude in the tidally locked coordinates. The zonal mean is
the average along the longitude in the standard coordinates, but is the
average along the tidally locked longitude in the tidally locked coordinates,
so do the deviations from zonal means. The formulas and the descriptions of
the LEC are shown below.

  1. $\bullet$

Mean available potential energy

${\rm
P_{M}}=\frac{c_{p}}{2}\int\gamma\left[\bar{T}\right]^{{}^{\prime\prime}2}dm,$
(B1)

which mainly depends on the departures of zonal-mean isotherms from their
global means. The zonal-mean isotherms are always tilted by large-scale uneven
heating or cooling, while their global means are horizontal (Figure 1(b)).

  2. $\bullet$

Eddy available potential energy

${\rm
P_{E}}=\frac{c_{p}}{2}\int\gamma\left[\overline{T^{{}^{\prime}2}}+\bar{T}^{*2}\right]dm,$
(B2)

which mainly depends on the temporal variability of temperatures and the
departures from temporal- and zonal-mean temperatures. PE is also generated by
uneven heating or cooling, but usually in smaller scales.

  3. $\bullet$

Mean kinetic energy

${\rm
K_{M}}=\frac{1}{2}\int\left(\left[\bar{u}\right]^{2}+\left[\bar{v}\right]^{2}\right)dm,$
(B3)

which depends on the strengths of jet streams and overturning circulation.

  4. $\bullet$

Eddy kinetic energy

${\rm
K_{E}}=\frac{1}{2}\int\left[\overline{u^{{}^{\prime}2}}+\overline{v^{{}^{\prime}2}}+\bar{u}^{*2}+\bar{v}^{*2}\right]dm,$
(B4)

which depends on the wind speeds of transient and stationary eddies.

  5. $\bullet$

Conversion rate from PM to PE

$\displaystyle{\rm C\left(P_{M},P_{E}\right)}=-\int
c_{p}\gamma\left[\overline{v^{{}^{\prime}}T^{{}^{\prime}}}+\bar{v}^{*}\bar{T}^{*}\right]\frac{\partial\left[\bar{T}\right]}{a\partial\phi}dm-\int
c_{p}p^{-\kappa}\left[\overline{\omega^{{}^{\prime}}T^{{}^{\prime}}}+\bar{\omega}^{*}\bar{T}^{*}\right]\frac{\partial}{\partial
p}\left(\gamma p^{\kappa}\left[\bar{T}\right]^{{}^{\prime\prime}}\right)dm,$
(B5)

which mainly depends on the heat transport by baroclinic eddies. In this
process, the isotherms in the longitude-latitude cross-section are warped,
which reduces the zonal-mean meridional temperature gradients but leads to
additional variance in the longitude direction, equivalent to converting PM to
PE (Figure 1(c)).

  6. $\bullet$

Conversion rate from PE to KE

$\displaystyle{\rm C\left(P_{E},K_{E}\right)}=-\int
g\left[\overline{\frac{u^{{}^{\prime}}\partial
Z^{{}^{\prime}}}{a\cos\phi\partial\lambda}}+\overline{\frac{v^{{}^{\prime}}\partial
Z^{{}^{\prime}}}{a\partial\phi}}+\frac{\bar{u}^{*}\partial\bar{Z}^{*}}{a\cos\phi\partial\lambda}+\frac{\bar{v}^{*}\partial\bar{Z}^{*}}{a\partial\phi}\right]dm,$
(B6)

which depends on the cross-isobaric motions in eddies. The pressure gradient
force would do work on the air parcel which moves along the pressure
gradients, and convert PE to KE. Otherwise, the air parcel moving against the
pressure gradient force would convert KE to PE (Figure 1(d)).

  7. $\bullet$

Conversion rate from KE to KM

$\displaystyle{\rm C\left(K_{E},K_{M}\right)}$ $\displaystyle=$
$\displaystyle\int\left[\overline{v^{{}^{\prime}}u^{{}^{\prime}}}+\bar{v}^{*}\bar{u}^{*}\right]\cos\phi\frac{\partial\left[\bar{u}\right]/\cos\phi}{a\partial\phi}dm+\int\left[\overline{v^{{}^{\prime}2}}+\bar{v}^{*2}\right]\frac{\partial\left[\bar{v}\right]}{a\partial\phi}dm$
(B7)
$\displaystyle+\int\left[\overline{\omega^{{}^{\prime}}u^{{}^{\prime}}}+\bar{\omega}^{*}\bar{u}^{*}\right]\frac{\partial\left[\bar{u}\right]}{\partial
p}dm+\int\left[\overline{\omega^{{}^{\prime}}v^{{}^{\prime}}}+\bar{\omega}^{*}\bar{v}^{*}\right]\frac{\partial\left[\bar{v}\right]}{\partial
p}dm$
$\displaystyle-\int\left[\overline{u^{{}^{\prime}2}}+\bar{u}^{*2}\right]\left[\bar{v}\right]\frac{\tan\phi}{a}dm,$

which normally depends on the wave–mean flow interactions. Under a positive
$\beta$-plane, eddies with group velocity directed away from the source region
would transport momentum back to accelerate the zonal winds, which leads to a
conversion from KE to KM; the shear instability of the zonal-mean winds would
generate eddies, which leads to a conversion from KM to KE (Figure 1(e)).

  8. $\bullet$

Conversion rate from PM to KM

$\displaystyle{\rm C\left(P_{M},K_{M}\right)}=-\int
g\left[\bar{v}\right]\frac{\partial\left[\bar{Z}\right]}{a\partial\phi}dm,$
(B8)

which depends on the same processes as $\rm C(P_{E},K_{E})$ but in the large-
scale circulation.

We do not calculate the generation rates of potential energy ($\rm G(P_{M})$
and $\rm G(P_{E})$) and the dissipation rates of kinetic energy ($\rm
D(K_{M})$ and $\rm D(K_{E})$). This is because it is hard to list all diabatic
heating, cooling, and damping processes from the reanalysis data and the GCMs’
outputs.

The formulas of $\rm C(P_{M},K_{M})$ and $\rm C(P_{E},K_{E})$ are so-called
‘v$\cdot$grad z’ form. There is another form called ‘$\omega\cdot\alpha$’
($\alpha$ is specific volume), i.e.,
$\left[\bar{\omega}\right]\left[\bar{\alpha}\right]$ and
$\left[\overline{\omega^{{}^{\prime}}\alpha^{{}^{\prime}}}+\bar{\omega}^{*}\bar{\alpha}^{*}\right]$,
respectively. The ‘$\omega\cdot\alpha$’ form is associated with different
physical aspect of the conversion, that is, vertical motions of warm and cold
air. The two forms may display different spatial distributions, but lead to
the same global-mean vertical integrals (Peixóto & Oort, 1992; Marques et al.,
2009; Kim & Kim, 2013). Thus, different forms have tiny impacts on our
conclusions. We use the ‘v$\cdot$grad z’ form in this study, because wind
velocities and geopotential height can be directly read from the reanalysis
data and GCMs’ outputs.

We are grateful to Eric Wolf for the release of the model ExoCAM and to D.B.
Koll for the release of the changing-coordinates codes and helpful
discussions. J.Y. acknowledges support from the National Natural Science
Foundation of China (NSFC) under grants 42161144011 and 42075046. The model
ExoCAM is released at https://github.com/storyofthewolf/ExoCAM. The
transformation codes for tidally locked coordinates are released at
https://github.com/ddbkoll/tidallylocked-coordinates. The calculation codes
for the LEC are released at https://doi.org/10.5281/zenodo.7472396. The
simulation data in this study are archived at
https://doi.org/10.5281/zenodo.7476074.

## References

  * Carone et al. (2015) Carone, L., Keppens, R., & Decin, L. 2015, Monthly Notices of the Royal Astronomical Society, 453, 2412, doi: 10.1093/mnras/stv1752
  * Carone et al. (2016) —. 2016, Monthly Notices of the Royal Astronomical Society, 461, 1981, doi: 10.1093/mnras/stw1265
  * Debras et al. (2020) Debras, F., Mayne, N., Baraffe, I., et al. 2020, A&A, 633, A2, doi: 10.1051/0004-6361/201936110
  * Del Genio et al. (1993) Del Genio, A. D., Zhou, W., & Eichler, T. P. 1993, Icarus, 101, 1, doi: https://doi.org/10.1006/icar.1993.1001
  * Ding & Pierrehumbert (2020) Ding, F., & Pierrehumbert, R. T. 2020, ApJ, 901, L33, doi: 10.3847/2041-8213/abb941
  * Ding & Wordsworth (2019) Ding, F., & Wordsworth, R. D. 2019, ApJ, 878, 117, doi: 10.3847/1538-4357/ab204f
  * Ding & Wordsworth (2020) —. 2020, ApJ, 891, L18, doi: 10.3847/2041-8213/ab77d1
  * Ding & Wordsworth (2021) —. 2021, The Planetary Science Journal, 2, 201, doi: 10.3847/psj/ac2236
  * Duan & Wu (2005) Duan, A., & Wu, G. 2005, Science in China Series D: Earth Sciences, 48, 1293, doi: 10.1360/04yd0042
  * Edson et al. (2011) Edson, A., Lee, S., Bannon, P., Kasting, J. F., & Pollard, D. 2011, Icarus, 212, 1, doi: https://doi.org/10.1016/j.icarus.2010.11.023
  * Gill (1980) Gill, A. E. 1980, Q. J. R. Meteorol. Soc., 106, 447, doi: 10.1002/qj.49710644905
  * Hammond & Lewis (2021) Hammond, M., & Lewis, N. T. 2021, Proceedings of the National Academy of Sciences, 118, doi: 10.1073/pnas.2022705118
  * Hammond & Pierrehumbert (2018) Hammond, M., & Pierrehumbert, R. T. 2018, ApJ, 869, 65, doi: 10.3847/1538-4357/aaec03
  * Hammond et al. (2020) Hammond, M., Tsai, S.-M., & Pierrehumbert, R. T. 2020, ApJ, 901, 78, doi: 10.3847/1538-4357/abb08b
  * Haqq-Misra et al. (2018) Haqq-Misra, J., Wolf, E. T., Joshi, M., Zhang, X., & Kopparapu, R. K. 2018, ApJ, 852, 67, doi: 10.3847/1538-4357/aa9f1f
  * Harnik & Chang (2003) Harnik, N., & Chang, E. K. M. 2003, Journal of Climate, 16, 480 , doi: 10.1175/1520-0442(2003)016<0480:STVASI>2.0.CO;2
  * Hernández-Deckers & von Storch (2010) Hernández-Deckers, D., & von Storch, J.-S. 2010, Journal of Climate, 23, 3874, doi: 10.1175/2010JCLI3176.1
  * Holton & Hakim (2013) Holton, J. R., & Hakim, G. J. 2013, An introduction to Dynamic Meteorology: 5th (London: Academic Press.), doi: https://doi.org/10.1016/C2009-0-63394-8
  * Ingersoll (2013) Ingersoll, A. P. 2013, Planetary Climates (New Jersey: Princeton University Press)
  * Joshi et al. (1997) Joshi, M., Haberle, R., & Reynolds, R. 1997, Icarus, 129, 450 , doi: https://doi.org/10.1006/icar.1997.5793
  * Joshi et al. (2020) Joshi, M. M., Elvidge, A. D., Wordsworth, R., & Sergeev, D. 2020, The Astrophysical Journal Letters, 892, L33, doi: 10.3847/2041-8213/ab7fb3
  * Kanno & Iwasaki (2022) Kanno, Y., & Iwasaki, T. 2022, Journal of Geophysical Research: Atmospheres, 127, e2021JD036380, doi: https://doi.org/10.1029/2021JD036380
  * Kim & Choi (2017) Kim, W., & Choi, Y.-S. 2017, Climate Dynamics, 49, 3605, doi: 10.1007/s00382-017-3533-0
  * Kim & Kim (2013) Kim, Y.-H., & Kim, M.-K. 2013, Climate Dynamics, 40, 1499, doi: 10.1007/s00382-012-1358-4
  * Kite et al. (2011) Kite, E. S., Gaidos, E., & Manga, M. 2011, ApJ, 743, 41, doi: 10.1088/0004-637x/743/1/41
  * Koll & Abbot (2015) Koll, D. D. B., & Abbot, D. S. 2015, ApJ, 802, 21, doi: 10.1088/0004-637x/802/1/21
  * Koll & Abbot (2016) —. 2016, ApJ, 825, 99, doi: 10.3847/0004-637x/825/2/99
  * Komacek et al. (2019) Komacek, T. D., Jansen, M. F., Wolf, E. T., & Abbot, D. S. 2019, ApJ, 883, 46, doi: 10.3847/1538-4357/ab3980
  * Kundu et al. (2016) Kundu, P. K., Cohen, I. M., & Dowling, D. R. 2016, in Fluid Mechanics (Sixth Edition), sixth edition edn., ed. P. K. Kundu, I. M. Cohen, & D. R. Dowling (Boston: Academic Press), 49–76, doi: https://doi.org/10.1016/B978-0-12-405935-1.00002-2
  * Leconte et al. (2013) Leconte, J., Forget, F., Charnay, B., et al. 2013, A&A, 554, A69, doi: 10.1051/0004-6361/201321042
  * Lee & Richardson (2010) Lee, C., & Richardson, M. I. 2010, Journal of Geophysical Research: Planets, 115, doi: https://doi.org/10.1029/2009JE003490
  * Li et al. (2007) Li, L., Ingersoll, A. P., Jiang, X., Feldman, D., & Yung, Y. L. 2007, Geophysical Research Letters, 34, doi: https://doi.org/10.1029/2007GL029985
  * Lorenz (1955) Lorenz, E. N. 1955, Tellus, 7, 157, doi: 10.3402/tellusa.v7i2.8796
  * Marques et al. (2010) Marques, C., Rocha, A., & Corte-Real, J. 2010, Dynamics of Atmospheres and Oceans, 50, 375, doi: https://doi.org/10.1016/j.dynatmoce.2010.03.003
  * Marques et al. (2011) Marques, C. A. F., Rocha, A., & Corte-Real, J. 2011, Climate Dynamics, 36, 1767, doi: 10.1007/s00382-010-0828-9
  * Marques et al. (2009) Marques, C. A. F., Rocha, A., Corte-Real, J., et al. 2009, International Journal of Climatology, 29, 159, doi: https://doi.org/10.1002/joc.1704
  * Matsuno (1966) Matsuno, T. 1966, J. Meteorol. Soc. Japan, 44, 25, doi: 10.2151/jmsj1965.44.1_25
  * Mendonça (2019) Mendonça, J. M. 2019, Monthly Notices of the Royal Astronomical Society, 491, 1456, doi: 10.1093/mnras/stz3050
  * Merlis & Schneider (2010) Merlis, T. M., & Schneider, T. 2010, Journal of Advances in Modeling Earth Systems, 2, doi: https://doi.org/10.3894/JAMES.2010.2.13
  * Michaelides (2021) Michaelides, S. 2021, Climate, 9, doi: 10.3390/cli9120180
  * Noda et al. (2017) Noda, S., Ishiwatari, M., Nakajima, K., et al. 2017, Icarus, 282, 1, doi: https://doi.org/10.1016/j.icarus.2016.09.004
  * Olbers et al. (2012) Olbers, D., Willebrand, J., & Eden, C. 2012, Ocean Dynamics (Berlin, Heidelberg: Springer), doi: https://doi.org/10.1007/978-3-642-23450-7
  * Pan et al. (2017) Pan, Y., Li, L., Jiang, X., et al. 2017, Nature Communications, 8, 14367, doi: 10.1038/ncomms14367
  * Peixóto & Oort (1974) Peixóto, J. P., & Oort, A. H. 1974, Journal of Geophysical Research (1896-1977), 79, 2149, doi: https://doi.org/10.1029/JC079i015p02149
  * Peixóto & Oort (1992) —. 1992, Physics of Climate (New York: American Institute of Physics)
  * Perez-Becker & Showman (2013) Perez-Becker, D., & Showman, A. P. 2013, ApJ, 776, 134, doi: 10.1088/0004-637x/776/2/134
  * Pierrehumbert (2010) Pierrehumbert, R. T. 2010, ApJ, 726, L8, doi: 10.1088/2041-8205/726/1/l8
  * Pierrehumbert & Hammond (2019) Pierrehumbert, R. T., & Hammond, M. 2019, Annual Review of Fluid Mechanics, 51, 275, doi: 10.1146/annurev-fluid-010518-040516
  * Sergeev et al. (2022) Sergeev, D. E., Fauchez, T. J., Turbet, M., et al. 2022, The Planetary Science Journal, 3, 212, doi: 10.3847/PSJ/ac6cf2
  * Showman et al. (2009) Showman, A. P., Cho, J. Y.-K., & Menou, K. 2009, Atmospheric Circulation of Exoplanets, doi: 10.48550/arXiv.0911.3170
  * Showman & Polvani (2010) Showman, A. P., & Polvani, L. M. 2010, Geophys. Res. Lett, 37, doi: 10.1029/2010GL044343
  * Showman & Polvani (2011) —. 2011, ApJ, 738, 71, doi: 10.1088/0004-637x/738/1/71
  * Song & Yang (2021) Song, X., & Yang, J. 2021, Frontiers in Astronomy and Space Sciences, 8, doi: 10.3389/fspas.2021.708023
  * Trenberth (1991) Trenberth, K. E. 1991, Journal of Atmospheric Sciences, 48, 2159 , doi: 10.1175/1520-0469(1991)048<2159:STITSH>2.0.CO;2
  * Tsai et al. (2014) Tsai, S.-M., Dobbs-Dixon, I., & Gu, P.-G. 2014, ApJ, 793, 141, doi: 10.1088/0004-637x/793/2/141
  * Turbet et al. (2022) Turbet, M., Fauchez, T. J., Sergeev, D. E., et al. 2022, The Planetary Science Journal, 3, 211, doi: 10.3847/PSJ/ac6cf0
  * Ulbrich & Speth (1991) Ulbrich, U., & Speth, P. 1991, Meteorology and Atmospheric Physics, 45, 125, doi: 10.1007/BF01029650
  * Vallis (2019) Vallis, G. K. 2019, Essentials of atmospheric and oceanic dynamics (Cambridge, U.K.: Cambridge University Press), doi: 10.1017/9781107588431
  * von Storch et al. (2012) von Storch, J.-S., Eden, C., Fast, I., et al. 2012, Journal of Physical Oceanography, 42, 2185, doi: 10.1175/JPO-D-12-079.1
  * Wang & Yang (2021) Wang, S., & Yang, J. 2021, ApJ, 907, 28, doi: 10.3847/1538-4357/abcf2a
  * Wang & Yang (2022) —. 2022, The Planetary Science Journal, 3, 171, doi: 10.3847/psj/ac6d65
  * Wolf & Toon (2014) Wolf, E., & Toon, O. 2014, Astrobiology, 14, 241, doi: 10.1089/ast.2013.1112
  * Wolf et al. (2022) Wolf, E. T., Kopparapu, R., Haqq-Misra, J., & Fauchez, T. J. 2022, The Planetary Science Journal, 3, 7, doi: 10.3847/psj/ac3f3d
  * Wolf et al. (2017) Wolf, E. T., Shields, A. L., Kopparapu, R. K., Haqq-Misra, J., & Toon, O. B. 2017, ApJ, 837, 107, doi: 10.3847/1538-4357/aa5ffc
  * Wolf & Toon (2015) Wolf, E. T., & Toon, O. B. 2015, Journal of Geophysical Research: Atmospheres, 120, 5775, doi: https://doi.org/10.1002/2015JD023302
  * Wordsworth (2015) Wordsworth, R. 2015, ApJ, 806, 180, doi: 10.1088/0004-637x/806/2/180
  * Yamamoto & Takahashi (2006) Yamamoto, M., & Takahashi, M. 2006, Journal of the Atmospheric Sciences, 63, 3296, doi: 10.1175/JAS3859.1
  * Yang et al. (2014) Yang, J., Boué, G., Fabrycky, D. C., & Abbot, D. S. 2014, ApJ, 787, L2, doi: 10.1088/2041-8205/787/1/L2
  * Yang et al. (2013) Yang, J., Cowan, N. B., & Abbot, D. S. 2013, ApJ, 771, L45, doi: 10.1088/2041-8205/771/2/l45
  * Zhang & Yang (2020) Zhang, Y., & Yang, J. 2020, ApJ, 901, L36, doi: 10.3847/2041-8213/abb87f