# Clouds and Seasonality on Terrestrial Planets with Varying Rotation Rates

Daniel A. Williams Department of Mathematics and Statistics, Faculty of
Environment, Science and Economy, University of Exeter, UK  Xuan Ji(纪璇)
Department of the Geophysical Sciences, University of Chicago, IL, USA  Paul
Corlies Department of Earth, Atmospheric, and Planetary Sciences,
Massachusetts Institute of Technology, Cambridge, MA, USA  Juan M. Lora
Department of Earth and Planetary Sciences, Yale University, New Haven, CT,
USA Daniel A. Williams, Juan M. Lora<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using an idealised climate model incorporating seasonal forcing, we
investigate the impact of rotation rate on the abundance of clouds on an
Earth-like aquaplanet, and the resulting impacts upon albedo and seasonality.
We show that the cloud distribution varies significantly with season,
depending strongly on the rotation rate, and is well explained by the large-
scale circulation and atmospheric state. Planetary albedo displays non-
monotonic behaviour with rotation rate, peaking at around $1/2\Omega_{E}$.
Clouds reduce the surface temperature and total precipitation relative to
simulations without clouds at all rotation rates, and reduce the dependence of
total precipitation on rotation rate, causing non-monotonic behaviour and a
local maximum around $1/8\Omega_{E}$; these effects are related to the impacts
of clouds on the net atmospheric and surface radiative energy budgets. Clouds
also affect the seasonality. The influence of clouds on the extent of the
winter Hadley cell and the intertropical convergence zone is relatively minor
at slow rotation rates ($<1/8\Omega_{E}$), but becomes more pronounced at
intermediate rotation rates, where clouds decrease their maximum latitudes.
The timing of seasonal transitions varies with rotation rate, and the addition
of clouds reduces the seasonal phase lag.

††software: Isca (Vallis et al., 2018), xarray (Hoyer et al., 2022), cartopy
(Elson et al., 2022)

##  1 Introduction

The past 30 years have brought the discovery of roughly 5000 planets beyond
our solar system, most detected by the NASA Kepler (Borucki et al., 2011) and
TESS (Dragomir et al., 2019; Cacciapuoti et al., 2022) missions. Amongst the
catalogue of confirmed exoplanets, the majority are larger and more massive
than Earth (NASA Exoplanet Science Institute, 2020); biases in detection
methods make large, rapidly-orbiting planets easier to detect (Burrows, 2014).
Nevertheless, increasing numbers of smaller terrestrial planets have been
detected (Bryson et al., 2020). The TRAPPIST-1 system (Gillon et al., 2017) is
perhaps the most publicised due to it hosting a large number of planets, but
recently more exoplanets very similar to Earth in size and equilibrium
temperature have been detected (Gilbert et al., 2020). With the development of
improved analytical methods and the launch of the latest generation of
telescopes, it is now possible to characterise the composition of exoplanetary
atmospheres (e.g., Alderson et al., 2022), greatly enhancing the possibility
of using climate models to understand these exotic worlds.

A key aspect of planetary atmospheres is the presence or absence of clouds.
The existence of clouds relies on the presence of a gaseous chemical species
in a planet’s atmosphere that can condense into cloud particles. Clouds on
Earth are a familiar concept, but it is not the only planet where they are
found. Beyond Earth, clouds of varying nature have been observed on other
terrestrial bodies, both within our solar system and beyond. Our nearest
neighbours Venus and Mars both have observable clouds: Venus’s take the form
of a thick global layer of sulphuric acid (Krasnopolsky & Parshev, 1981); on
Mars, tenuous water ice and carbon dioxide ice clouds have been observed
(e.g., Curran et al., 1973; Bell III et al., 1996; Haberle et al., 2017, and
references therein). Saturn’s moon Titan hosts a great variety of clouds, with
a varying seasonal distribution, morphology, and composition (e.g., Griffith,
2006; de Kok et al., 2014; Turtle et al., 2018).

Using visible and near-infrared spectra from telescopes such as the Very Large
Telescope, the Hubble Space Telescope and the James Webb Space Telescope, we
have been able to infer the presence of clouds on exoplanets (Kreidberg et
al., 2014; Sing et al., 2016; Samland et al., 2017; Barstow, 2021). For
instance, Alderson et al. (2022) inferred the existence of clouds in the
atmosphere of WASP-39b through the continuum transit depth observed with JWST.
The sheer variety of exoplanets thus far discovered has also indicated the
presence of exotic condensing species, such as metal oxides and silicates in
the atmospheres of very hot planets (e.g., Helling, 2019).

Clouds have a significant impact on the climate of a planet, both through
their influence on the transport of condensable species in the atmosphere and
on the radiation budget of the climate system (Trenberth & Fasullo, 2012).
Clouds act as a source and sink of moisture, which, via latent heating
associated with phase transitions, modify the temperature and stability of
their atmospheric environment. Their optical properties as effective
scattering media have an impact on both incoming short-wave radiation
(affecting planetary albedo) and long-wave radiation emitted by the planet;
these cloud radiative feedbacks are broadly understood (Webster, 1994;
Pierrehumbert, 2010) and can be reasonably represented in models (Rose &
Scott-Brown, 2021), though clouds also represent one of the larger
uncertainties in state-of-the-art models of Earth’s climate (Ceppi et al.,
2017). Cloud formation is influenced by a number of factors, chiefly the
temperature and specific humidity of an air parcel. These quantities are
themselves influenced by processes such as convection and advection, such that
large-scale atmospheric dynamics are a significant driver of global cloud
distribution.

Despite their importance to the energy budget of the climate system, clouds
are often neglected in planetary climate models due to difficulties in
modelling them accurately. Clouds present a range of complex microphysical
processes that are not fully understood: consider the formation of different
ice crystals (Storelvmo & Tan, 2015), the physics of mixed-phase clouds
(Korolev & Mazin, 2003), or the subsequent formation and growth of raindrops
(Bergeron, 1935). In addition, the processes and dynamics within clouds occur
at small scales, beyond the spatial resolution of general circulation models
(GCMs). These sub-grid processes must be parameterised in order to be
represented in such models, and there is great variation between cloud schemes
of different GCMs. Further, cloud schemes add an additional computational cost
to any model. As a result, clouds are often ignored and excluded, even in
simplified models of the Earth (Thomson & Vallis, 2019a). Nonetheless, given
the clear evidence of clouds on other bodies within our own solar system and
increasing observations and inferences of exoplanetary clouds, as well as
their critical importance to Earth’s climate system, it is pertinent to
attempt to better account for clouds’ impacts in models, especially given
their potential importance to observable quantities such as albedo and
emission or transmission spectra.

Separately, a number of previous studies have investigated how the atmospheric
circulation and climate of a terrestrial planet may be influenced by different
atmospheric and planetary characteristics. Planetary rotation rate is well
understood as a key determiner of atmospheric dynamics, affecting the width
and magnitude of the Hadley circulation (Kaspi & Showman, 2015; Guendelman &
Kaspi, 2018; Singh, 2019; Hill et al., 2022), latitudinal distribution of
extratropical eddies (Eady, 1949; Taylor, 1980), related extratropical jets
(Williams, 1978; Cho & Polvani, 1996; Chemke & Kaspi, 2015a), heat
transport(Liu et al., 2017; Cox et al., 2021), radiative cooling (Zhang,
2023), and general climate and habitability (Yang et al., 2014; Haqq-Misra et
al., 2018; Komacek & Abbot, 2019; Jansen et al., 2019; Guzewich et al., 2020;
Cox et al., 2021; He et al., 2022). The role of rotation rate has often been
studied using idealised GCMs, which strip out factors such as topography in
order to understand the fundamental physics that drive the dynamics and
climate (Schneider, 2006; O’Gorman & Schneider, 2008; Thomson & Vallis,
2019b). More broadly, Kaspi & Showman (2015) used a number of parameter sweeps
to investigate the effects of rotation rate, mean insolation, atmospheric
mass, atmospheric optical density, planetary density and radius on the
resulting atmospheric circulation in an idealised, Earth-like GCM. Their
simulations were based on a clear-sky, equinoctinal aquaplanet with a grey
radiation scheme, constructed using the Flexible Modeling System developed at
GFDL (Held & Suarez, 1994; Anderson, 2004; Frierson et al., 2006). They found
that rotation rate is a key driver of atmospheric dynamics, describing two
principal regimes: fast rotators, characterised by a weaker Hadley circulation
and development of extratropical jets; and slow rotators, with strong Hadley
circulation and a small meridional temperature gradient. O’Gorman & Schneider
(2008) used a similar grey radiation model to study the connection between the
global mean temperature and the hydrological cycle, finding that, as the
climate warms, the global-mean precipitation eventually reaches an asymptotic
value. Whilst these models included moisture and a representation of the
hydrological cycle, all excluded a cloud scheme.

Various past efforts have used GCM simulations with models of higher
complexity to interrogate clouds. For example, Parmentier et al. (2016)
compared cloud presence in exoplanet transit data of hot Jupiters to models
building on the thermal structure from GCMs. They also suggested that cloud
presence and composition may be deduced from asymmetries in the light curves
of transiting cool exoplanets, providing a potential basis for diagnosing
clouds from observational data. Komacek & Abbot (2019) built on the previous
work by Kaspi & Showman (2015), making use of the ExoCAM model (Wolf et al.,
2022), to perform a range of GCM simulations including sea ice, a correlated-k
radiation scheme, and a cloud scheme (Rasch & Kristjánsson, 1998). Their
findings were qualitatively consistent with the results of Kaspi & Showman
(2015), but they observed a larger equator-to-pole temperature gradient. This
may be attributed to their use of a more complex radiative scheme and addition
of a cloud scheme; they also noted that cloud particle size appears to be an
important but uncertain parameter. While they presented an interesting
transition between low and high day-side cloud coverage for synchronously
rotating planets with varying rotation periods, the impact of clouds on
atmospheric dynamics and climate for asynchronous planets was not explicitly
addressed. Yang et al. (2013) and Yang et al. (2014) also investigated clouds
over a range of rotation rates, including tidally-locked systems. They
suggested that the planetary albedo increases to a maximum in the synchronous
case, where a substantial cloud cap forms at the sub-stellar point. Finally,
Guzewich et al. (2020) showed that variations of cloud properties with
rotation rate can alter the observable signatures of otherwise Earth-like
planets in reflected light and thermal emission spectra.

Most of the previous studies generally considered planets without a seasonal
cycle, hence providing a hemispherically symmetric circulation. Whilst
simulating a planet with no obliquity ($\varepsilon=0$) has advantages, we
know from our own solar system that planets rarely conform to this
simplification, as Earth, Mars and Titan all have large effective obliquities
of approximately $\varepsilon=25^{\circ}$. This results in seasonally-varying
radiative forcing that is a key driver of dynamics that are absent in
aseasonal climates (Guendelman & Kaspi, 2018, 2019; Ohno & Zhang, 2019; Singh,
2019; Hill et al., 2022), including the Earth’s monsoon systems (Bordoni &
Schneider, 2008; Geen et al., 2018); in this context, for example, the
migration of the Intertropical Convergence Zone (ITCZ) has been shown to vary
non-monotonically with rotation rate (Faulk et al., 2017; Geen et al., 2018).
Other studies have explored how obliquity influences the climatological
temperature, precipitation and habitability (Ferreira et al., 2014; Kang,
2019; Kodama et al., 2022; Linsenmeier et al., 2015; Lobo & Bordoni, 2020; He
et al., 2022). Furthermore, Lobo & Bordoni (2022) analysed the relationship
between extratropical storminess and longwave radiation, and Hadas et al.
(2023) took cloud albedo into consideration, linking Earth’s large-scale
circulation to planetary albedo, which indicates the importance of using a
more realistic radiative scheme to study the effect of rotation rate.

This work builds on the above studies, especially those by Kaspi & Showman
(2015) and Komacek & Abbot (2019), by interrogating the effects of clouds on
varying planetary climates with idealised GCM simulations including a seasonal
cycle. By covering a large range of rotation rates, we aim to include a swathe
of dynamical regimes with a likely impact on cloud distributions, enabling us
to investigate the response of seasonal cloud behaviour to rotation rate and
the feedback of these clouds on the seasonal climate. We do not consider
tidally locked systems where rotation and orbital rate are synchronous;
tidally locked planets display a distinct dynamical regime (Joshi et al.,
1997; Noda et al., 2017; Haqq-Misra et al., 2018; Pierrehumbert & Hammond,
2019; Wordsworth & Kreidberg, 2022) where dynamics are not longitudinally
invariant (Merlis & Schneider, 2010; Sergeev et al., 2020; Hammond & Lewis,
2021), and the presence and influence of clouds on tidally locked planets have
already been investigated extensively (Yang et al., 2013, 2019; Helling, 2021;
Sergeev et al., 2022).

Hereon, we describe the experimental setup and procedure in Section 2.
Following this we present our results, starting with the effect of rotation
rate on the dynamics and large-scale distribution of clouds, including their
impacts on planetary albedo (Section 3.1). We follow this with results
investigating the consequences of clouds upon climate and seasonality (Section
3.2), and discuss comparisons with previous work and provide a future outlook
in Section 4.

##  2 Methods

###  2.1 Idealised GCM Setup

In order to explore the effect of planetary rotation rate on the dynamics of a
simplified cloudy aquaplanet, we make use of the flexible modelling framework
Isca (Vallis et al., 2018). Isca is not a single model, but a framework based
on the GFDL Flexible Modeling System for constructing a range of idealised
GCMs covering a wide hierarchy of complexity (Thomson & Vallis, 2019a),
allowing the effects of particular processes to be studied in isolation.
Complexity may range from a very simple Newtonian thermal-relaxation model as
described by Held & Suarez (1994) to models that introduce additional
complexity such as moist physics, topography and differing radiation schemes.

The initial base for our setup in Isca can be derived from the moderately
complex Earth-like model described in Thomson & Vallis (2019a). We use a
360-day calendar with all planetary parameters identical to Earth except
rotation rate $\Omega$. We also use a simplified orbit, including the Earth’s
obliquity but neglecting eccentricity such that seasonal forcing is
symmetrical across the northern/southern hemispheres.

Isca uses a spectral dynamical core in spherical coordinates to solve the
primitive equations. We use a spectral resolution of T42 for most experiments,
providing a grid of 64 latitude and 128 longitude cells, each of approximately
$None$$\times$$None$ in size. For higher rotation rate simulations, the
decreasing Rossby deformation scale that results from a higher value of
$\Omega$ demands a higher spectral resolution of T85 (Kaspi & Showman, 2015;
Vallis, 2017). Each model has a vertical grid defined by 50 unevenly spaced
levels up to a pressure height of $0.02\text{\,}\mathrm{hPa}$.

We specify a global mixed-layer slab ocean of $2.5\text{\,}\mathrm{m}$ depth
without prescribed surface Q-fluxes (Jucker & Gerber, 2017). Isca does not
include a dynamical ocean model, so this allows for a closed surface energy
budget with the mixed layer depth setting the thermal inertia of the slab
ocean; the value is low to allow a strong response to seasonal forcing
(Donohoe et al., 2014; Jucker, 2019; Liu et al., 2021). An albedo of $0.2$ is
used to account for the additional albedo that clouds provide (Liu et al.,
2021); the value is used in all experiments for consistency across the
parameter sweep.

To represent moist convection, we use the Simple Betts-Miller (SBM) scheme
described by Frierson (2007). SBM is a quasi-equilibrium scheme (Betts, 1986;
Betts & Miller, 1986) that relaxes to a relative humidity profile of 70 and
temperatures to a moist adiabat over a $7200\text{\,}\mathrm{s}$ timescale
when convective instability occurs. This is coupled with a large-scale
condensation scheme that removes excess vapour beyond saturation in each grid
cell. We use SOCRATES (Edwards & Slingo, 1996; Manners et al., 2016) for the
radiation scheme, which provides higher complexity relative to simpler grey-
radiation schemes: SOCRATES is a comprehensive, multi-band scheme developed by
the UK Met Office which has already been used in applications beyond present-
day Earth (Amundsen et al., 2016; Way et al., 2017). We run SOCRATES with the
included ga7 spectral files, providing 9 long-wave and 6 shortwave bands
(standard configuration for an Earth-like model), and making use of
stratospheric ozone absorption to provide a closer comparison with Earth’s
atmospheric temperature structure.

###  2.2 Cloud Scheme and Rotation Rates

To implement clouds in the simulations, we make use of the SimCloud scheme
developed by Liu et al. (2021). In general, cloud schemes add significant
computational overhead to models since the physics of clouds are complex,
necessitating interface with multiple parts of a GCM. SimCloud aims to provide
a simple diagnostic cloud scheme for examining the impacts of clouds on
climate in idealised models. SimCloud diagnoses clouds by local environmental
variables and only interacts with the radiation scheme. Simplifications
include treating water and ice clouds both as liquid but with different
effective cloud particle radii as derived from observations (Stubenrauch et
al., 2013). Large-scale clouds are diagnosed using the local grid-mean
relative humidity, with a freeze-dry adjustment applied to prevent the
overestimation of polar clouds. Marine low-level clouds such as stratocumulus
are diagnosed using the local vertical temperature profile. We employ the
following options within the scheme for cloud diagnostics (identical to runs
from Liu et al., 2021): a linear large-scale cloud diagnostic formula;
maximum-random assumption for cloud overlap; and the Park-ELF method for low-
level marine clouds. Finally, precipitation in Isca is idealised and decoupled
from the cloud scheme, being driven by either large-scale condensation
(Frierson et al., 2006) or humidity relaxation from the SBM convection scheme
(Frierson, 2007). This allows precipitation formation in models with or
without a cloud scheme present.

We present simulations with the model using 22 different rotation rates. With
Earth’s rotation rate labelled $\Omega_{E}$, we define
$\Omega^{*}=\Omega/\Omega_{E}$, which is the ratio of the planetary rotation
rate compared to Earth’s. $\Omega^{*}$ takes values between $1/128$ and $4$,
extending from very slow to fast rotators, and encompassing the rotation rates
of Earth, Mars and Titan. At each rotation rate, we use configurations of the
model with the cloud scheme enabled and disabled to provide a comparison with
a clear-sky simulation, allowing the impacts from cloud forcing to be more
clearly identified. Each simulation lasts for 15 Earth years, sufficient for
the large-scale dynamics of the troposphere to reach a steady-state; we
discard the initial 10 as a spin-up period, and use the final five years to
provide climatological averages for each experiment.

##  3 Results

###  3.1 Impact of Rotation Rates on Clouds

Figure 1: Global seasonal variation of zonal-mean total cloud water path (CWP)
for a subset of the simulations. Lighter regions represent regions with higher
CWP values.

A number of variables may be used to diagnose the presence of clouds in the
atmosphere. One such quantity is the total cloud water path (CWP), a
vertically integrated quantity defined by

$\textrm{CWP}=\frac{1}{g}\int_{0}^{p_{s}}Cw_{l}\;\textrm{d}p\,,$ (1)

where $g$ is the surface gravity, $C$ the cloud fraction and $w_{l}$ the
liquid mixing ratio at pressure levels $p$. The CWP provides a useful proxy
measure for where the most substantial clouds may be found, particularly those
associated with moist, convective regions. If the value of CWP is high then we
may safely conclude clouds are present, though clouds may also be present in
low-valued regions, as discussed later.

Figure 1 shows the global variation of CWP throughout the seasonal cycle
across a range of rotation rates. It is immediately clear that the average
spatial distribution of CWP varies significantly with both rotation rate and
time of year, with some key identifiable trends. Starting with $\Omega^{*}=1$,
the most Earth-like of the experiments, there is a very clear band of high CWP
that straddles the equatorial region during near-solstice periods
(approximately day 0 and 180) and undergoes a weak latitudinal migration
during the seasonal cycle. This is the ITCZ, where the convergence and ascent
of moist, warm air is highly conducive to the formation of clouds
(Pierrehumbert, 2010).

Considering the more slowly rotating systems, the ITCZ-like cloud band
broadens in latitudinal extent but also develops a strongly seasonally
migratory nature, extending well into what are considered extratropical
latitudes on Earth. At $1/16<\Omega^{*}<1/4$ the ITCZ cloud band transitions
from a near-stationary regime to one wherein its seasonal migration reaches
the polar regions near solstice, consistent with idealised simulations of
Titan (Mitchell et al., 2006). The equinoxes (approximately days 90 and 270)
represent a transitional state where the major cloud band migrates rapidly
across the equatorial region. High values of CWP are predominantly found in
the summer hemisphere, with the winter hemisphere being mostly cloud-free.
This behaviour in the solsticial period becomes ever clearer in the slowest
rotating models, hinting at the development of asymptotic behaviour.

At rotation rates faster than Earth’s, a different regime occurs. The tropical
cloud band appears to become weaker and much more restricted in latitudinal
extent, agreeing with previous results (Yang et al., 2013). At the same time,
new bands of high/low CWP occur at higher latitudes. Relating the position of
these bands to Earth’s climate suggests they are associated with extratropical
storm tracks that develop in an increasingly baroclinic atmosphere (Kaspi &
Schneider, 2013; Shaw et al., 2016); in addition to cloud-free subtropical
regions. These additional bands are most clearly developed near the equinoxes,
which contrasts with the slowly rotating systems where the greatest
distinction is apparent near the solstices.

Figure 2: Variation of cloud area fraction and precipitation rate with
rotation rate. Filled contours show percentage coverage of cloud viewed from
space, with lighter regions being more cloudy; blue lines represent contours
of increasing precipitation. The vertical dotted line represents a rotation
rate of $\Omega^{*}=1/8$.

Whilst Figure 1 gives us a snapshot of a select subset of experiments, it does
not show the transition that occurs over the full range of rotation rates.
Figure 2 shows the zonally averaged behaviour for all rotation rates. In
addition to precipitation, we also show the overall cloud area fraction as
viewed from space; clouds at any height in the atmosphere contribute, with the
maximum-random overlap assumption in the cloud scheme applied.

Figure 2 shows the behaviour of the main ITCZ-like cloud band described above,
traced out by high contours of precipitation; the poleward extent in the
solsticial period at low rotation rates is very clear. Overall, the total
cloud area fraction suggests a much greater abundance of clouds than
previously suggested by the CWP alone. This is because CWP is a vertical
integral of cloud water so suppresses the influence of certain clouds like
shallow marine stratocumulus. This suggests that there are regions aside from
those identified qualitatively in Figure 1 that are cloudy and consequently
contribute to the radiation balance of the climate system. One clear behaviour
nonetheless is the absence of any clouds in the polar regions of the winter
hemisphere at all but the fastest rotation rates. This implies the existence
of a consistent mechanism that suppresses cloud formation at a wide range of
rotation rates. At the highest rotation rates, there is the development of a
cloudy region in the high latitudes around the autumnal equinox. The high
cloud area fraction is co-located with high precipitation rates—it is possible
this is a manifestation of the storm tracks, but it may also be an over-
estimation of polar clouds despite the freeze-dry adjustment in the SimCloud
scheme, as identified in Liu et al. (2021).

Having shown that the spatial distribution of seasonal clouds on a terrestrial
aquaplanet is strongly influenced by the rotation rate, we turn to
understanding some of the mechanisms driving these changes. Figure 3 links the
zonally-averaged distribution in cloud fraction to diagnostic variables
related to the broader atmospheric state. Firstly, the meridional mass
streamfunction, $\psi$, is calculated as

$\psi\left(\vartheta,p\right)=\frac{2\pi a}{g}\cos\vartheta\int
v\left(\vartheta,p\right)\;\textrm{d}p\,,$ (2)

where $\vartheta$ is the latitude, $a$ the planetary radius and $v$ the
meridional wind. This quantity gives us an understanding of the meridional
atmospheric circulation by highlighting where regions of ascending and
descending air occur. On Earth, analysis of $\psi$ indicates the presence of
the Hadley, Ferrel and polar cells, which delineate the principal circulation
regimes of the terrestrial atmosphere. The peak streamfunction magnitude will
vary between models at different rotation rates, as shown by Kaspi & Showman
(2015).

The second diagnostic variable is the equivalent potential temperature
$\theta_{e}$. For any given air parcel, the equivalent potential temperature
is conserved with vertical motion and phase changes of the condensing species.
The precise formula for $\theta_{e}$ is complicated (Paluch, 1979; Emanuel,
1994) so we use an approximated form (Stull, 1988) wherein

$\theta_{e}\approx\left(T+r\frac{L_{v}}{c_{p}}\right)\left(\frac{p_{s}}{p}\right)^{\kappa}\,,$
(3)

where $T$ and $r$ are temperature and vapour mixing ratio respectively.
Constants include enthalpy of vapourisation $L_{v}$, gas heat capacity (at
constant pressure) $c_{p}$, surface pressure $p_{s}$, and $\kappa$, defined by
the ratio of moist to dry gas constants ($R_{v}/R_{d}$). The term $r$ is
substituted for specific humidity $q$ (a model output) since in an Earth-like
climate $q\approx r$ (Vallis, 2017). Equivalent potential temperature is a
useful quantity since it allows us to diagnose the stability of the atmosphere
with respect to convection—clouds are strongly linked to vertical mixing in
moist regions. If $\theta_{e}$ increases with height
($\partial\theta_{e}/\partial z>0$) then the atmosphere is stably stratified
and vertical motions are suppressed. Conversely, if
$\partial\theta_{e}/\partial z<0$, the air is unstable to vertical
perturbations and convection occurs. Moist parcels of air lifted by convection
and thus cooled contribute to cloud formation.

Figure 3: Zonally-averaged relationship between cloud location, atmospheric
circulation and convective stability at a range of rotation rates, averaged
across January. Filled grey contours show grid cell cloud fraction. Mass
streamfunction $\psi$ is denoted by solid contours at $2\text{\,}\mathrm{\char
37\relax}$ intervals up to $10\text{\,}\mathrm{\char 37\relax}$ of the maximum
value for each subplot; $10\text{\,}\mathrm{\char 37\relax}$ is found at inner
edge of main circulation cells. Solid lines denote clockwise circulation;
dashed lines denote anti-clockwise. The maximum values are, respectively,
18.1, 16.9, 10.9, 10.3, 6.5, 5.1, 3.1,
$1.0\text{\times}{10}^{11}\text{\,}\mathrm{kg}\text{\,}{\mathrm{s}}^{-1}$.
Coloured dotted contours show the equivalent potential temperature surfaces in
the lower atmosphere.

Figure 3 shows the January average for eight different rotation rates,
reflecting the climate in northern hemisphere (NH) winter and southern
hemisphere (SH) summer. The variation of the main large-scale meridional
overturning circulation pattern diagnosed by the mass streamfunction, the
Hadley cell, is clear through all the rotation rates: in the slowly rotating
cases the Hadley cell extends into the high mid-latitudes with
ascending/descending branches in the summer/winter hemispheres, a type of
circulation observed on Titan (e.g., Mitchell & Lora, 2016). In these cases,
the bulk of the clouds occur in the mid-latitude and polar regions throughout
the depth of the summer troposphere: at higher levels, extending up to
$200\text{\,}\mathrm{hPa}$, the cloud fraction is highest on the equatorward
flank of the rising branch of the Hadley cell, whereas at lower levels the
cloudiest regions occur at higher latitudes, producing a somewhat tilted
structure. The high cloud fraction found near the ascending branch of the
Hadley cell can be understood through the gradient in $\theta_{e}$, which does
not increase strongly with height and is close to zero in localised regions.
This indicates convection, which saturates and condenses out excess moisture,
thus contributing to cloud formation. Low-level stratocumulus clouds are
additionally under the influence of boundary layer dynamics and instabilities,
which may explain their more cosmopolitan distribution compared to the large-
scale clouds.

In contrast to the summer cloudiness, the winter hemisphere is comparatively
devoid of any large-scale cloud formation in the slowly rotating cases and
there is a clear positive gradient of $\theta_{e}$ with height. The only
exception to an otherwise cloud-free hemisphere is the limited presence of
low-level clouds in the low latitudes. These large-scale distributions can be
explained by considering the large-scale circulation together with the
atmospheric stability. In the winter hemisphere air parcels along the
descending branch of the Hadley cell warm adiabatically. Cloud formation is
effectively inhibited by the downwelling air, which manifests as a consistent
gradient in $\theta_{e}$, which increases with height.

As rotation rate increases, the latitudinal extent of the Hadley cell
decreases, and with it moves the location of peak cloud fraction. At Earth’s
rotation rate $\Omega^{*}=1$, the Hadley cell is much reduced and there is
development of a significant second, thermally indirect cell in the mass
streamfunction, corresponding to the Ferrel cell. Similar trends in the
development of large-scale circulation at near-solstice periods can be seen in
the study by Faulk et al. (2017), supporting the link between large-scale
circulation, convective stability and cloud formation. Whilst low-level clouds
are more widespread at Earth’s rotation rate, the large-scale clouds continue
to align with regions of vertical motion where there is not a clear positive
gradient of $\theta_{e}$ with height. In addition to the deep cloud structure
in the tropics, a secondary region of deep cloud in the winter mid-latitudes
also begins to develop, extending up to $500\text{\,}\mathrm{hPa}$ in height.
This becomes the dominant cloud structure as the rotation rate increases to
higher values and is an indicator of extratropical storm track activity.

At the highest rotation rates the dynamical situation becomes more complex. In
slowly rotating systems energy transport is dominated by the mean-flow,
reflected by the global Hadley circulation. At higher rotation rates energy
transport becomes eddy-dominated (Kaspi & Showman, 2015) and secondary
circulation cells form. As $\Omega$ increases the eddy length scale decreases
(Kaspi & Showman, 2015; Showman et al., 2015; Rhines, 1979), necessitating the
formation of multiple smaller circulation cells and eddy-driven jets (Chemke &
Kaspi, 2015b; Wang et al., 2018), explaining the structure in $\psi$ at the
highest rotation rates. This reflects the highly baroclinic nature of the
winter troposphere in the mid-latitude regions, wherein extratropical cyclones
are prominent. These regions of low pressure promote cloud formation as warm,
moist, poleward-flowing air is lifted along fronts, leading to substantial
large-scale cloud formation. The preferred locations of these systems trace
out the storm tracks, seen as the increased cloud fraction in the mid-
latitudes. As before, regions of high cloud fraction often align with regions
where $\partial\theta_{e}/\partial z$ is not strongly positive. Whilst clouds
become more prevalent in the extratropics, the deep convective cloud band
formerly present in the slower rotation rates dissipates, likely as a result
of a weakening Hadley cell where convective updraughts are less significant.
In contrast to the slowly rotating cases, it is now the summer hemisphere that
is almost devoid of clouds, again with the exception of the low-level clouds
which likely result from boundary layer turbulence.

Figure 4: Planetary albedo at varying rotation rates, split by hemisphere.
Coloured lines are formed by the average of the corresponding season in each
hemisphere: Winter (Northern hemisphere DJF, Southern hemisphere JJA); Spring
(NH MAM, SH SON) and so on. The solid black line represents the resulting
global average over both hemispheres.

Figure 4 shows the overall impact of the presence of clouds, accounting for
changes in spatial distribution with rotation rate, on the planetary albedo.
Albedo is a measure of the incoming versus outgoing shortwave radiation and is
an important quantity for determining the overall radiation budget of a
planet. Both seasonal hemispheric and annual global averages are calculated
for planetary albedo, produced by considering the top-of-atmosphere (TOA)
shortwave radiation

$\alpha=\frac{S_{\uparrow,\textrm{TOA}}}{S_{\downarrow,\textrm{TOA}}}\;,$ (4)

where $S_{\downarrow}$ and $S_{\uparrow}$ are the incoming and outgoing
shortwave radiative fluxes. Following calculation of the TOA albedo $\alpha$
on a grid-cell basis, an area weighting is applied before averaging spatially
and temporally.

Given each simulation uses a surface albedo $\alpha=0.2$, it is immediately
clear that clouds act to increase the global TOA albedo in every season, with
values varying around $\alpha\approx 0.35$. This behaviour is expected since
clouds reflect the incoming shortwave radiation back to space, increasing
$S_{\uparrow}$. At the slowest rotation rates the global albedo appears to
tend asymptotically towards a value of $\alpha\approx 0.33$, whereas at high
rotation rates it decreases again from the maximum value. This perhaps
unexpectedly non-monotonic behaviour can be understood by linking to the
analysis of large-scale cloud formation and atmospheric circulation.

Considering seasonal averages, the variation with rotation becomes more
pronounced. The summer season displays the highest albedo peak of any season
throughout the parameter space, which may be explained by the presence of the
deep cloud band along the ascending Hadley cell branch. Near the peak value,
found near $\Omega^{*}\approx 1/4$, this dominant cloud band is shown by
Figure 3 to be centred around $None$S; during the DJF period the solar zenith
angle will be very high and as such clouds have ample ability to impact the
albedo. Simulations at these rotation rates also display strong low-level
stratocumulus decks, which are known for having an important impact on energy
balance and albedo (Slingo, 1990; Hartmann et al., 1992). At slower rotation
rates the Hadley cell develops a fully pole-to-pole circulation, with peak
cloud formation occurring closer to the summer pole. Conversely, at higher
rotation rates, changes to the Hadley cell circulation result in less
significant cloud formation in the summer hemisphere.

The broad peak in the autumnal albedo can plausibly be explained by
considering the importance of eddies, which is shown by Kaspi & Showman
(2015), in terms of energy flux, to peak around Earth’s rotation rate. Eddies
imply the existence of extratropical cyclones and their associated clouds. At
slower rotation rates the extratropical regions shrink, whilst at higher
rotation rates the eddy length scale decreases, which also acts to shrink the
cloud formation regions. The cloud formation region is further restricted by
the development of convergent subtropical areas that supress cloud formation
(analogous to Hadley/Ferrel cell boundary on Earth).

Overall the winter hemisphere displays a relatively low albedo, which supports
the notion that a high solar zenith angle results in clouds having a smaller
impact on the shortwave radiation budget. Nonetheless, there is a marked
increase at the fastest rotation rates, which may be attributed to cloud
formation associated with storm track activity in the extratropics, as with
the autumnal case.

Finally, spring appears to consistently have low albedo values, which reflects
a relatively cloud-free season during which the solar zenith angle is still
relatively high. Comparing Figures 1 and 2, one may deduce that the limited
cloud present does not have significant CWP, indicating that it is most likely
low-level stratocumulus driven more by the boundary layer, rather than by the
large-scale circulation, explaining the lack of variation with rotation rate.

The cumulative effect of the seasonal albedos produces an unexpected variation
in the annually averaged global albedo, which shows a clear non-monotonic
behaviour. Aside from the key trends at the extrema and the peak value, also
present is a distinctive kink that occurs around $\Omega^{*}=1/8$ where the
albedo has a local minimum—a feature clear in all seasons except summer. This
rotation rate appears to have some significance with respect to atmospheric
dynamics: it represents the transition between a single global Hadley cell
that extends fully from pole to pole to a more restricted form where
additional meridional circulation cells (i.e., Ferrel cells) are present. This
transition at $\Omega^{*}\sim 1/8$ also appears in the results of Kaspi &
Showman (2015) as the rotation rate at which mean meridional heat transport
via the large-scale circulation is overtaken by poleward eddy heat transport.
The transitional range of $\Omega$ between these two dynamical regimes appears
to briefly suppress cloud formation, perhaps as neither mechanism that
normally leads to cloud formation—overturning circulation or baroclinic
instabilty—is dominant.

###  3.2 Impact of Clouds on Climate and Seasonality

Previous studies, ignoring clouds, have shown the impact of rotation rates on
climate and seasonality, including the ITCZ extent, seasonal amplitude, and
latitudinal extent of the maximum temperature (e.g., Faulk et al., 2017; Geen
et al., 2019; Guendelman & Kaspi, 2022). Building on these insights, we
investigate how clouds impact climate and seasonality under varying rotation
rates.

Figure 5: Variation of precipitation and mass streamfunction for different
relative rotation rates: $\Omega^{*}=$ 1/128 (left), 1/8 (middle), and 1
(right) for simulations without clouds (upper row) and simulations with clouds
(middle row). The filled contours depict precipitation (mm d-1). The black
contours show zonal-mean mass streamfunction at the pressure level of its
maximum value with inline labels of values in units of
${10}^{11}\text{\,}\mathrm{kg}\text{\,}{\mathrm{s}}^{-1}$, and all panels
share the common contour interval of
$1\text{\,}\mathrm{kg}\text{\,}{\mathrm{s}}^{-1}$. The third row displays the
inner edge of the winter Hadley cell in the summer hemisphere, and the maximum
zonal mean precipitation (Max ITCZ) in the Northern Hemisphere. The solid
curves show results from the simulations with clouds, while the dotted curves
correspond to the simulations without clouds. The ITCZ is defined as the
latitude of maximum zonal-mean precipitation and it must be associated with
the ascending branch of the Hadley circulation; the edge of the Hadley cell is
defined as the latitude where the streamfunction, taken at the pressure level
of its maximum value, reaches $5\%$ of that maximum value in the summer
hemisphere. The bottom row illustrates the changes in the zonal-mean TOA
energy fluxes (left y-axis) and precipitation (right y-axis) over time at a
latitude of 23∘N. The yellow lines represent the net shortwave radiation at
TOA ($1-\alpha)S\downarrow$, while the pink lines represent outgoing longwave
radiation (OLR). The blue lines depict the zonal-mean precipitation. The
vertical lines represent day of vernal equinox (day 90), northern summer
solstice (day 180), and autumnal equinox (day 270), and day 0 is northern
winter solstice. The presented data are daily averages. The contours of
streamfunction in the first two rows and the curves in the third row are
smoothed by a Gaussian filter with a standard deviation of $\sigma=3$ days.

Figure 5 displays the zonal mean precipitation and mass streamfunction for
both simulations without clouds and simulations with clouds in the top two
rows. Our simulations show that, regardless of the presence of clouds, the
extent of the Hadley cell and ITCZ decreases as rotation rates increase.
However, simulations with clouds exhibit lower maximum precipitation over the
year, indicating that clouds may suppress ITCZ precipitation, while the
streamfunctions are stronger than in the no-clouds case. In the third row, we
present how the inner edge of the winter Hadley cell in the summer hemisphere,
and the ITCZ change seasonally in the NH. In no-cloud simulations, both the
Hadley cell and ITCZ tend to remain at the maximum latitude for longer after
reaching the summer maximum latitude, compared to the simulations with clouds.
In the bottom row, the zonal mean TOA net shortwave radiation
($(1-\alpha)S_{\downarrow}$) and outgoing longwave radiation (OLR) at $None$
latitude are shown separately, along with the change of zonal mean
precipitation.

Our no-cloud simulations exhibit a pattern qualitatively consistent with the
findings of Faulk et al. (2017) regarding slow-rotating planets
($\Omega^{*}\sim 1/8$), with some differences in the latitudes of ITCZ and
Hadley Cell extent. According to Faulk et al. (2017), such planets have global
Hadley cells and are essentially ‘all tropics’ planets, with the ITCZ
remaining at around $\sim\,$$None$. Our no-cloud simulations have the ITCZ
extending closer to the summer pole for slow rotation rates
($\Omega^{*}<1/8$). For the no-clouds case with $\Omega^{*}=1/8$, the maximum
precipitation remains near the summer pole for about 50 days, along with the
ascending branch of the Hadley cell, and the highest precipitation values
exceed $35\text{\,}\mathrm{mm}\text{\,}{\mathrm{d}}^{-1}$.

We locate the ITCZ at the latitude of the maximum zonal-mean precipitation.
However, in the slowest rotating case ($\Omega^{*}=1/128$), a transient
maximum in zonal-mean precipitation is observed at the summer pole (days
180–230). Therefore, we choose the other precipitation maximum at a lower
latitude ($<\,$$None$), which is associated with the ascending branch of the
Hadley cell, as the ITCZ. The transient polar precipitation maximum is also
found by Faulk et al. (2017) in the simulations with $\Omega^{*}<1/6$, while
in our no-clouds simulations, it only appears in the $\Omega^{*}=1/128$ case.
In this simulation, the extent of the ITCZ and Hadley cell is lower than the
case with $\Omega^{*}=1/8$ and the ITCZ is located southward of the ascending
branch of the Hadley cell.

For our simulations with clouds, the highest NH mean precipitation over the
year occurs in early summer for all cases, along with the maximum latitude of
the ITCZ and Hadley cell. In simulations with $\Omega^{*}=1/8$, the
precipitation rapidly decreases after reaching its peak. Interestingly, as the
rotation rate decreases from $\Omega^{*}=1/8$, a second peak of zonal-mean
precipitation occurring in the late summer begins to develop, along with
increases in the latitudes of the ITCZ and Hadley cell edge. At the same time,
the absolute peak value of precipitation increases monotonically as the
rotation rate decreases from $1/8$ to $1/128$. The double-peak precipitation
pattern might relate to the findings on polar precipitation for high-obliquity
cases reported by Lobo & Bordoni (2020), where the peaks were attributed to
the moisture storage term, which is usually negligible in Earth’s climate.

The relationship between TOA radiative energy imbalance and precipitation at a
latitude of $None$ is illustrative. In both cloudy and cloud-free scenarios,
precipitation initiation coincides with the moment when the net energy becomes
positive. At slower rotation rates ($\Omega^{*}=1/128,1/8$), the most intense
precipitation occurs when the ITCZ passes, twice a year. For Earth’s rotation
rate, the ITCZ remains near the $None$ latitude for several months, resulting
in prolonged and intense precipitation. In the no-cloud simulations, the net
shortwave radiation is solely determined by the solar zenith angle, and OLR
remains relatively constant over the course of a year, being minimally
influenced by rotation rate. However, the intensity of precipitation varies
with rotation rate, indicating that strong precipitation is primarily governed
by atmospheric circulation rather than local TOA radiative energy. On the
other hand, in simulations with clouds, precipitation strongly impacts the
local TOA radiation: both shortwave and longwave radiation decrease with
increasing precipitation, reflecting shortwave and longwave cloud radiative
forcing. Overall, the presence of clouds suppresses precipitation, and at all
three rotation rates shown shifts the peak precipitation to earlier in the
season.

Figure 6: Top: variation of global and annual mean precipitation with rotation
rate. Second panel: the shortwave radiative flux absorbed by the atmosphere
(yellow), the net longwave radiative loss from the atmosphere (pink), and the
net atmospheric radiative energy input (black). Third panel: the shortwave
radiative flux absorbed by the surface (yellow), the net longwave radiative
loss of the surface (pink), and the net surface radiative flux. Bottom: the
highest latitude during the year of the maximum zonal mean precipitation (Max
ITCZ) and the inner edge of the winter Hadley cell in the summer hemisphere.
All panels show simulations with clouds (solid lines) and without clouds
(dashed lines). The vertical lines represent rotation rates of $\Omega^{*}$ =
1/8 and 1.

The global average precipitation in both cases, with and without clouds,
changes with the rotation rate. The upper panel of Figure 6 shows the global
annual mean precipitation. The atmospheric net radiative energy is presented
in the middle panel, which aligns with the precipitation variations (O’Gorman
& Schneider, 2008). The simulations without clouds have overall higher global
annual mean precipitation at any rotation rate, and the precipitation
increases with rotation rate monotonically. However, when clouds are present,
a distinctive kink appears in the precipitation curve around $\Omega^{*}=1/8$,
corresponding to the kink in albedo presented in Figure 4. Regarding the
atmospheric energy budget, clouds tend to decrease the absorbed shortwave
radiation flux in the atmosphere, but they also decrease longwave radiative
loss (second panel of Figure 6). Our results suggest that the longwave
radiative forcing from clouds is the dominant effect over the shortwave
radiative forcing at all rotation rates. Examining the surface energy budget
further (the third panel of Figure 6), we observe that clouds decrease the
shortwave radiation flux; they also decrease longwave radiative cooling when
$\Omega^{*}<1/8$ but increase it when $\Omega^{*}>1/8$. In contrast to the
atmospheric radiative budget, the shortwave surface forcing from clouds
dominates over the longwave forcing at all but the highest rotation rates.

The dependence of the winter Hadley cell extent and ITCZ on the rotation rate
is shown in the bottom panel of Figure 6. When $\Omega^{*}<1/8$, there is no
significant difference between non-cloud and cloudy cases, while at around
$\Omega^{*}=1/6$, simulations without clouds exhibit more extended Hadley
cells and therefore ITCZ latitude than those with clouds. As the rotation rate
increases, the discrepancy between the two cases diminishes and eventually
reverses sign around the Earth’s rotation rate. Note that the influence of
clouds on ITCZ latitude is metric-dependent. Here, the extent of ITCZ is
defined by the latitude of the maximum zonal-mean precipitation in the NH, and
the decreases of ITCZ extent caused by clouds appear within
$1/6<\Omega^{*}<1$. If we define it instead by the maximum latitude of the
ITCZ over the year (the peak of the green curve shown in the bottom panels of
Figure 5), the decrease appears within $3/16<\Omega^{*}<3/8$. Regardless of
the metric used, the decrease in ITCZ extent due to clouds only occurs at
intermediate rotation rates.

Figure 7: Daily variation of the Northern Hemisphere (NH) mean surface
temperature with different rotation rates for simulations without (upper) and
with clouds (lower). The vertical lines represent rotation rates of
$\Omega^{*}$ = 1/8 and 1. The horizontal lines represent day of vernal equinox
(day 90), northern summer solstice (day 180), and autumnal equinox (day 270),
and day 0 is northern winter solstice. To highlight the continuous structure
in the upper panel and the distinctive maxima in the lower panel, two open
contour lines are also shown.

Climate seasonality can be considered as the deviation from the annual mean
climate or deviations from hemispheric symmetry for an aquaplanet (Guendelman
& Kaspi, 2022). There are various methods for quantifying these deviations,
and we will focus on temperature first. The daily variation of the NH mean
surface temperature as a function of the rotation rate is shown in Figure 7.
The temperatures in the no-clouds cases are generally higher than those in the
cases with clouds. This is because the presence of clouds raises the overall
albedo.

For simulations without clouds, the NH average temperature reaches the maximum
in early summer and decreases quickly afterwards when $\Omega^{*}<1/8$. The
local temperature peaks at the time when the ITCZ is passing. Locally, there
may be a second peak temperature during the summer occurring when the ITCZ is
migrating southward, but it is not as high as the first one. When
$1/8<\Omega^{*}<1$, the temperature remains constant during the summer for
approximately 100 days. When the rotation rate is that of Earth, the summer
temperature starts to shift relative to the season, and it reaches the maximum
in the middle of the summer, which is consistent with our experience on Earth.

For simulations with clouds, surface temperatures are generally lower,
consistent with the energy balance arguments above. For fast-rotating cases
($\Omega^{*}>1/8$), the overall patterns are similar to those without clouds.
However, a unique characteristic is observed at around $\Omega^{*}=1/8$, where
the temperature exhibits a two-peaked feature with maximum values occurring in
both early and late summer. The maximum temperatures are also slightly higher
than those at somewhat faster rotation rates, which is in contrast to the
behaviour of the simulations without clouds. Finally, for rotation rates below
approximately $\Omega^{*}=1/8$, there is only one peak temperature, occurring
in mid-late summer.

Figure 8: Top: variation of the change in the Northern Hemisphere (NH) mean
surface temperature ($\overline{\langle T\rangle_{\textrm{NH}}}$) with
rotation rate. The shaded areas represent the uncertainty range using the last
five years; the uncertainty is too small to be distinguished for most of
rotation rates. Middle: variation of the day when the NH mean surface
temperature equals the global annual mean surface temperature after summer
($\langle T\rangle_{\textrm{NH}}=\overline{\langle T\rangle_{\textrm{glb}}}$)
with rotation rate, which illustrates the lag of the seasonal transition after
the autumnal equinox (day 270). Bottom: dependence of the amplitude of the
seasonal temperature variation with rotation rate. The pink curve represents
the maximum local temperature variation (max($T-\overline{T}$)), and values
refer to the left $y$-axis. The blue curve represents the difference between
the maximum NH mean temperature and the global annual mean temperature
(max$\left(\langle T\rangle_{\textrm{NH}}-\overline{\langle
T\rangle_{\textrm{glb}}}\right)$ ), and corresponds to the right $y$-axis. All
panels show simulations with clouds (solid lines) and without clouds (dashed
lines). The vertical lines represent rotation rates of $\Omega^{*}$ = 1/8 and
1.

We further analyse three variables that represent the NH mean temperature
($\overline{\langle T\rangle_{\textrm{NH}}}$), the phase lag of seasonal
transition from summer to winter, and seasonal amplitude for both maximum
local variation ($\textrm{max}\left(T-\overline{T}\right)$) and global
variation ($\textrm{max}\left(\langle T\rangle_{\textrm{NH}}-\overline{\langle
T\rangle_{\textrm{glb}}}\right)$), as shown in Figure 8. The top panel of the
figure shows that the NH annual mean temperature changes with rotation rate
for both cases with and without clouds. Overall, the no-clouds cases have a
higher mean temperature at any rotation rate, which is consistent with the
global annual mean precipitation. In the absence of clouds, the temperature
increases monotonically with the rotation rate, but in the presence of clouds,
a kink occurs at around $\Omega^{*}=1/8$, which is associated with the kink of
albedo and precipitation (Figure 4, 6 & 7).

The day of vernal equinox is day 270, but the day when the NH mean surface
temperature equals the global annual mean surface temperature occurs later
than equinox due to the thermal inertia, as shown in the middle panel of
Figure 8. As the rotation rate increases, simulations both with and without
clouds experience increasing seasonal transition lags, meaning that the phase
lag with respect to the radiative forcing becomes greater. The presence of
clouds overall decreases the phase lag and causes Spring temperatures to
arrive earlier than in the no-clouds cases. It is worth noting that the lower
albedo in the no-clouds experiments results in a higher effective emission
temperature, which could, under certain approximations, decrease the radiative
timescale(Mitchell et al., 2014; Ohno & Zhang, 2019; Guendelman & Kaspi, 2022;
Tan, 2022). We might expect a shorter radiative timescale for the cases
without clouds, corresponding to a smaller phase lag and larger variation.
This expectation contradicts our simulation results, prompting a need for a
more detailed calculation with our realistic radiative scheme. Below, we
analyse how other factors influence the seasonal transition.

According to the bottom panel of Figure 8, the amplitude of seasonal
variations of temperature does not vary substantially with rotation rate,
which is consistent with the findings of Guendelman & Kaspi (2022) (cf. their
Fig. 3f). Despite these unclear trends, differences between the clouds and no-
clouds cases becomes significant when the rotation rate is relatively high
($\Omega^{*}>1$) and low ($\Omega^{*}<1/8$).

Guendelman & Kaspi (2022) pointed out a key factor controlling seasonal
transitions is atmospheric heat transport efficiency. They concluded that the
amplitude of temperature variation decreases as heat transport increases.
Their results show that, as rotation rate increases, heat transport becomes
less efficient, leading to an increase in the equator-to-pole temperature
difference. This observation aligns with previous work on aseasonal planets,
including simulations without clouds (e.g., Showman et al., 2015; Cox et al.,
2021) and simulations with clouds (e.g., Komacek & Abbot, 2019; Liu et al.,
2017). In our simulations without clouds, the annual mean equator-to-pole
temperature difference increases with rotation rate. However, in cases with
clouds, the trend is non-monotonic. The temperature difference is higher than
the no-cloud case when $\Omega^{*}=1/128$, generally decreases with rotation
rate until $\Omega^{*}=1$, and then increases for higher rotation rates. We
suggest that accounting for both clouds and seasonality results in a different
trend than that from considering heat transport changes with rotation rate
alone.

Considering seasonal variation, we calculated the average equator-to-pole
temperature difference for the summer hemisphere in the three months preceding
the seasonal transition. In no-cloud simulations, the temperature difference
remains relatively constant for $\Omega^{*}<1/2$ and increases with rotation
rate for $\Omega^{*}>1/2$. As depicted in the middle panel of Figure 8, a
higher rotation rate, which corresponds to a lower heat transport efficiency,
is associated with a larger phase lag. In our simulations, the temperature
difference in no-cloud simulations is significantly higher than in simulations
with clouds when $\Omega^{*}\gtrsim 1/8$. With a higher temperature
difference, heat transport efficiency is lower, and consequently, a larger
phase lag for no-clouds cases compared to cases with clouds is expected.

However, it is worth noting that for simulations with clouds, the temperature
difference remains consistently low ($<$20K) for all rotation rates and
decreases with rotation rates for $\Omega^{*}<1$, but increases with rotation
rates beyond that. Our analysis suggests that heat transport efficiency alone
cannot account for the seasonal transition. The radiative timescale also plays
a significant role in controlling the seasonal transition (Guendelman & Kaspi,
2022; Tan, 2022; Ohno & Zhang, 2019). Moreover, the radiative feedback may
vary with rotation rate in our more realistic radiative and cloud schemes.
Thus, further theoretical research that incorporates the evolving radiative
feedback with rotation rate is imperative to comprehensively understand the
seasonal transition.

Figure 9: Left: seasonal variation of the temperature, which is defined by the
difference between the Northern Hemisphere (NH) averaged surface temperature
and the global annual mean surface temperature. Middle: the maximum
streamfunction in NH. Right: The latitude of the maximum zonal mean
precipitation (ITCZ). The filled shaded contours represent the results from
no-cloud simulations and the values are shown in the colourbars, while the
open contours represent the results from cloud simulations and the values are
shown in inline labels. For comparison, the contours are plotted at the same
levels for no-clouds and clouds simulations. To highlight the difference in
seasonal transition, the zero contour lines are bold black (clouds) and gray
(no-clouds) for the left and right panels. The contours of streamfunction and
ITCZ latitude are smoothed by a Gaussian filter with a standard deviation of
$\sigma=3$ days. The horizontal lines represent day of vernal equinox (day
90), northern summer solstice (day 180), and autumnal equinox (day 270), and
day 0 is northern winter solstice. The vertical lines represent rotation rates
of $\Omega^{*}$ = 1/8 and 1.

To provide more insight into how the rotation rate affects the seasonal
transition, we present additional information in Figure 9. Specifically, from
left to right, the panels depict the seasonal variation of temperature in the
NH, the maximum streamfunction in the NH, and the latitude of the ITCZ for
both no-clouds (filled shaded contours) and clouds (open contours) cases, as a
function of rotation rate and day of the year. In the left panel, the contours
are noticeably tilted upward as the rotation rate increases, consistent with
the increases in phase lag shown in the middle panel of Figure 8. The general
trend is that clouds amplify the seasonal amplitude at slower rotation rates,
while at $\Omega^{*}>1/8$, they suppress it. The tilt of the contours with
rotation rate is slightly less in the cases with clouds. A distinctive feature
occurs in summer around $\Omega^{*}=1/8$ for no-clouds cases, indicating that
the NH average temperature reaches its maximum in early summer and cools
rapidly afterwards. This is also shown in the top panel of Figure 7. This
feature is specific to no-clouds simulations, and is not related to the kink
of albedo and precipitation observed in cloudy cases (Figures 4; 6–8). The
contour plots of the zonal mean temperature at latitudes $40^{\circ}$N and
$60^{\circ}$N also confirm this feature, with the maximum being reached
earlier at lower latitudes, coinciding with the time of ITCZ northward
migration.

The middle panel of Figure 9 shows the maximum mass streamfunction in the NH.
For both cases with and without clouds, the maximum values throughout the year
do not vary much with rotation when $\Omega^{*}<1/8$, and decrease with
rotation rate when $\Omega^{*}>1/8$. The presence of clouds overall increases
the maximum streamfunction value, but does not substantially alter its
variation with rotation.

A similar trend is found for the latitude of the ITCZ (right panel of Figure
9). The seasonal transition occurs earlier in cases with clouds than without
clouds, as discussed above (right bottom panel of Figure 5 and middle panel of
Figure 8). The two-peak feature of the ITCZ latitude, shown for the cloudy
case with $\Omega^{*}=1/128$ in Figure 5, can be seen at around day 190 and
day 210 for all slowly-rotating cases, and disappears when the rotation rate
reaches $\Omega^{*}\sim 1/16$. At intermediate rotation rates, clouds appear
to cause solstitial excursions of the ITCZ to occur somewhat earlier in the
season, though the modest trend of the equatorial crossing with rotation rate
is similar in both cases with and without clouds.

Our findings about the impact of clouds on climate, seasonality, and its
changes with rotation rate discussed above all agree qualitatively with the
results of previous studies (Faulk et al., 2017; Guendelman & Kaspi, 2022;
Jansen et al., 2019). While the influence of clouds on seasonality can be
attributed to their impact on the energy budget to leading order, it is
important to note that our analysis is qualitative. Therefore, further
investigation is warranted to gain a more detailed and quantitative
understanding of the underlying mechanisms, and to validate our hypotheses.

##  4 Summary and Discussion

Using a suite of simulations with an idealised terrestrial aquaplanet GCM, we
demonstrate the relationship between rotation rate, cloud distribution and
planetary albedo under the influence of seasonal forcing. We show that
rotation rate influences the cloud distribution via the large-scale
circulation; consequentially, cloud feedbacks impact the seasonality of the
circulation itself. Our key findings are summarised as follows:

  1. 1.

The seasonal cycle leads to a dramatic shift in cloud distribution with
rotation rate. At slow rotation rates, cloud area fraction displays a dipolar
behaviour near the solstices, as the summer hemisphere is very cloudy whilst
the winter hemisphere is relatively cloud-free. At high rotation rates, the
clearest distinction between seasons occurs near the equinox rather than the
solstice, with the autumnal hemisphere significantly cloudier than the spring
hemisphere. The latitudinal distribution of clouds at high rotation rate is
more complex with the development of separate equatorial and mid-latitude
regions of cloudiness.

  2. 2.

Cloud distribution is well explained by the large-scale circulation, with the
cloudiest regions associated with those undergoing ascent within the
atmosphere and having a small (or negative) gradient in equivalent potential
temperature. Broadly speaking, low-level clouds occur with a more global
distribution, more strongly influenced by the boundary layer.

  3. 3.

The seasonal cloud distribution greatly impacts the planetary albedo as a
function of rotation rate, with substantial inter-seasonal variability.
Planetary albedo displays a non-monotonic behaviour that peaks at around
$\Omega^{*}\approx 1/2$, decreasing towards slower rotations as the ITCZ and
peak cloud distribution move poleward, and likewise towards faster rotation
rates as eddy length scale decreases, reducing the scale of extratropical
cyclonic systems that contribute to cloud formation. The local minimum in
albedo at $\Omega^{*}\approx 1/8$ is likely linked to the transition point
between a circulation that is mean-flow dominated at slow rotation rates to
eddy dominated at high rotation rates.

  4. 4.

The introduction of clouds leads to a decrease in annual global mean
precipitation compared to cloud-free simulations at all rotation rates,
related to their impact on the energy budget. Without clouds, precipitation
exhibits a monotonic increase with increasing rotation rates. Once clouds are
introduced, a peak in precipitation develops around $\Omega^{*}\approx 1/8$.
Precipitation remains relatively constant for rotation rates below
$\Omega^{*}<1/8$ and increases with rotation rates above $\Omega^{*}>1/4$.

  5. 5.

At slow rotation rates ($\Omega^{*}<1/8$) the inner edge of the winter Hadley
cell and the ITCZ are found near $None$ latitude, with little distinction
between cloudy and cloud-free cases. At higher rotation rates, a significant
difference between the two cases develops, which changes the sign at the
fastest rotation rates.

  6. 6.

Increasing rotation rate increases the magnitude of the seasonal phase lag in
both cloudy and cloud-free simulations. The addition of clouds reduces the
seasonal phase lag.

Our results cover a large range of rotation rates, from slow rotators such as
Titan to fast rotators such as Earth. This covers a similar parameter space to
the simulation grid performed by Kaspi & Showman (2015), but we do not include
simulations equivalent to their fastest ($\Omega^{*}>4$). This is justified on
practical and scientific bases: Chemke & Kaspi (2015b) demonstrated that for
fast rotating Earth-like systems, the eddy-driven jets migrate latitudinally
even in the absence of seasonal forcing, an effect that does not occur at
slower rotation rates. This would be incompatible with any analysis of
seasonal averages, particularly with the link demonstrated between large-scale
circulation and cloud distribution—we note that our fastest rotating models
$\Omega^{*}\gtrsim 3$ may be affected by this. Running models of higher
rotation rates also necessitates the use of higher spectral resolutions and
lower model timesteps, imposing a significant penalty on model run-time. At
the other end, Yang et al. (2014) found that slowly rotating planets exhibit
strong convergence and convection in the substellar region, leading to the
formation of extensive optically thick clouds and a significant increase in
planetary albedo. Their results show an increase in planetary albedo with
decreasing rotation rates when $\Omega^{*}<1/8$. However, in our simulations,
we did not investigate the tidally-locked atmospheric dynamical regime, even
at extremely low rotation rates such as $\Omega^{*}=1/128$. Our simulations
did not exhibit a day-night contrast but rather revealed zonal banded patterns
in the daily mean (24-hour mean) climatology.

Regarding how seasonality varies with rotation rate, Faulk et al. (2017) used
a more idealised moist GCM, also based on the GFDL spectral dynamical core, to
investigate the dependence of ITCZ migration on rotation rate. Their model did
not include the effects of clouds and used an idealised gray radiative
transfer scheme. As discussed in Section 3.2, our results generally agree with
theirs, except for the maximum latitude of the ITCZ and Hadley cell extent for
cases with $\Omega^{*}<1/8$. Faulk et al. (2017) showed that the ITCZ remains
approximately at $\sim\,$$None$ while the Hadley cell extent remains at
$None$–$None$, while, in our simulations, both the ITCZ and Hadley cell
migrate to $None$ in cases with and without clouds (Figure 5 & 6). We
speculate that the difference in the radiation scheme, which leads to
insolation and longwave differences, may help determine the location of the
ITCZ, which suggests important model dependence. Further, our use of a low
mixed-layer depth for the slab ocean allowed the ITCZ to migrate farther
poleward in our simulations. Separately, Guendelman & Kaspi (2022) and Tan
(2022) investigated how the phase lag of seasonal transition varies with
rotation rate and concluded that, with a slower rotation rate, the increase in
meridional heat transport would decrease the phase lag, which is also seen in
our simulations (Figure 8 & 9).

We compare our results of simulations with clouds to the results of Komacek &
Abbot (2019). Komacek & Abbot used a GCM of higher complexity (ExoCAM; Wolf et
al., 2022) that includes a correlated-$k$ radiation scheme, allows sea ice to
form, and uses sub-grid parameterisations for clouds from Rasch & Kristjánsson
(1998). However, their simulations assumed zero obliquity and no seasonal
effects. Therefore, the absolute values of streamfunction and jet speed in
their results are different from ours. To account for this, we normalise the
values to Earth’s rotation rate, and we find similar trends: the maximum
streamfunction decreases with rotation rate, and the maximum jet speed peaks
at around $\Omega^{*}=1/8$. Our results regarding the latitude of the edge of
the Hadley cell and the jet also overlap with those of Komacek & Abbot (2019).
However, our results on how cloud fraction changes with rotation rate do not
agree. In ExoCAM simulations, cloud fraction overall decreases with rotation
rate, while our Isca simulations show an increasing trend. There is no clear
albedo kink occurring around $\Omega^{*}=1/8$ in Komacek & Abbot (2019),
possibly due to their sparse data points on rotation rate but also likely as a
consequence of the aseasonality and model configuration differences. The
intrinsic difference in cloud parameterisation between the two models, in
particular, warrants further research.

It is likely that there may be differences between our and others’ results
simply through the use of a particular cloud scheme; Komacek & Abbot (2019)
noted that their results may differ from GCMs using alternative cloud
parameterisations. Each cloud scheme is often tuned to reasonably represent
the distribution of clouds on present-day Earth. We acknowledge that our
simulations present a novel use of the SimCloud scheme, differing primarily
from the work by Liu et al. (2021) in using an aquaplanet setup and altering
the rotation rate. To our understanding, however, none of our changes should
invalidate the simple cloud parameterisation itself. Indeed, our study aims at
a fundamental understanding of the distribution of clouds in terms of key
planetary parameters enabled by a more idealised representation without the
complications introduced by effects, such as land-sea heat capacity gradients
and orographic uplift, that influence cloud formation.

As well as helping to elucidate the impact of clouds in terrestrial planetary
climate systems, our results may be of use to the observation and
characterisation of exoplanets. Proposed missions such as the Habitable Worlds
Observatory (HWO) aim to study Earth-like exoplanets in reflected light via
direct imaging methods. If an Earth-like planet displays an inhomogenous
distribution of reflected light (albedo) across the planetary disk, it may be
indicative of a cloud-bearing atmosphere (Cowan et al., 2009; Lustig-Yaeger et
al., 2018). With sufficiently long monitoring, it may be possible to infer the
rotation rate of the planet by studying changes in the albedo distribution,
whose distribution and seasonality we have shown to be related to rotation
rate. Regardless, there is ample scope for relating this theoretical work to
future observational campaigns.

We give thanks to the organisers of the Rossbypalooza 2022 Summer School at
the University of Chicago, at which the investigation that led to this
manuscript started. DAW was supported by an STFC studentship (ST/V506667/1);
this work was partially supported by NASA grant No. 80NSSC21K1718, which is
part of the Habitable Worlds program. For the purpose of open access, the
author has applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising. The research data supporting this
publication are openly available at doi://10.5281/zenodo.8075444.

We thank T. Komacek for providing the output data from ExoCAM simulations for
comparison. We also extend thanks to a number of people who offered
constructive comments and ideas during the undertaking of this work, in
alphabetical order: D. Abbot, M. Battalio, B. Fan, E. Kite, N. Lewis, D.
Sergeev, W. Seviour, S. Thomson, G. Vallis, and A. Warren. This work was
completed in part with resources provided by the University of Chicago
Research Computing Center. We finally would like to thank two reviewers for
their help in clarifying and improving the paper.

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