hyperrefSuppressing link with empty target

11institutetext: Université Paris-Saclay, UVSQ, CNRS, CEA, Maison de la
Simulation, 91191, Gif-sur-Yvette, France,
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK

# Flows, Circulations, and Energy Transport in the Outer and Deep Atmospheres
of Synchronous and Non-synchronous Hot Jupiters

F. Sainsbury-Martinez 1122<EMAIL_ADDRESS>Pascal Tremblin ,
11

(Received ### ## 2023; accepted ### ##, 2023)

###### Abstract

Aims. Recent studies have shown that vertical enthalpy transport can explain
the inflated radii of highly irradiated gaseous exoplanets. Simultaneously,
they have also shown that rotation can influence this transport, leading to
highly irradiated, rapidly rotating, objects that are uninflated. Here we
explore the flows which underpin this transport, including the impact of
synchronous/non-synchronous rotation.

Methods. We use DYNAMICO to run a series of long-timescale, HD209458b-like,
atmospheric models at various surface rotation rates. For models that are
tidally-locked, we consider rotation rates between $1/16^{\mathrm{th}}$ and
$40$ times the rotation rate of HD209458b, whilst for non-synchronous models
we consider the range $1/8^{\mathrm{th}}$ to $4$ times HD209458b.

Results. We find that our synchronous models fall into one of three
$\Omega$-dependent regimes: at low $\Omega$, we find that the outer atmosphere
dynamics are driven by a divergent day-night wind, which drives weak vertical
transport and can lead to the formation of a night-side hot-spot. At
intermediate $\Omega$, we find classical hot Jupiter dynamics, whilst at high
$\Omega$ we find a strong Coriolis effect that suppresses off-equator
dynamics, including the jet-driving standing waves, thus also reducing
vertical transport. As for non-synchronicity, when small, we find that it has
little effect on the dynamics. However as it grows, we find that temporal
variations prevents the formation of the persistent structures that drive
large scale dynamics and transport.

Conclusions. We find that rotation can significantly impact the atmospheric
dynamics of irradiated exoplanets, including vertical enthalpy advection,
which may help to explain the scatter in the hot Jupiter radius-irradiation
relation. We have also identified a seemingly robust atmospheric feature at
slow rotation: a night-side hot-spot. As this may have important implications
for both the phase curve and atmospheric chemistry, we suggest that this study
be followed up with next-generation GCMs that robustly models radiation and
chemistry.

###### Key Words.:

Planets and satellites: interiors - Planets and satellites: atmospheres -
Planets and satellites: fundamental parameters - Planets: HD209458b -
Hydrodynamics

##  1 Introduction

Thanks to the continuing efforts of the observational and instrumental
astrophysics community, the parameter regime in which hot Jupiters can be
found has expanded greatly (see, for example, Shporer et al. 2017; Hjorth et
al. 2019; Huang et al. 2020; Schlecker et al. 2020; Dong et al. 2021; Stangret
et al. 2022; Ulmer-Moll et al. 2022). In particular, there has been a
significant increase in the range of orbital radii in which hot Jupiters can
be observed. Simultaneously, numerical studies of hot Jupiter, and hot Brown
Dwarf, atmospheres has revealed that rotation can play a significant role on
both the horizontal and vertical dynamics (Showman et al. 2014; Sainsbury-
Martinez et al. 2021). For example, Sainsbury-Martinez et al. (2021), found
that, in addition to the link between radius inflation and surface
irradiation, a link might also exist between the level of radius inflation
observed and the surface rotation rate of the exoplanet in question.
Specifically, they suggested that the lack of observed radius inflation for
the ultra-short-orbit hot Brown Dwarf SDSS1411b could be explained by the
influence that rotation has on the atmospheric dynamics.

Examples of how the Coriolis effect can influence atmospheric dynamics can be
found in our own solar system. Generally, planetary atmospheric dynamics
depend upon the influence of both surface irradiation and rotation, with the
later effect leading to the formation of three distinct atmospheric regimes.
At slow rotation rates, e.g. Venus and Titan, atmospheric dynamics are
dominated by planetary scale Hadley cells (Hadley 1735; Schneider 2006;
Showman et al. 2013) which mix temperature horizontally, resulting in weak
latitudinal temperature gradients. This is known as Cyclostrophic circulation,
and leads to significant super-rotation of the winds, as well as significant,
circumpolar, vortexes, all of which is described well by Thermal Wind Balance
(Flasar et al. 2010).
Increasing the rotation rate, we come to the dynamical regime occupied by both
Earth and Mars: At low latitudes, the Coriolis effect remains relatively weak,
and Hadley cells develop (the tropics), However as we move to higher latitudes
(i.e. the extra-tropics - $\theta>30^{\circ}$ for Earth), the regime changes
and rotational effects start to play a much more significant role. This
results in the wind changing from being Cyclostrophic to Geostrophic, which in
turn is paired with a change in thermal/angular-momentum transport from Hadley
circulations to baroclinic instabilities (which drive the extra-tropical
cyclones/anti-cyclones that dominate much of Europe’s and the US’s weather -
Hoskins & Valdes 1990; Chang et al. 2002; Laurila et al. 2021). As a
consequence of this change in dynamics, meridional heat transport has
weakened, resulting in latitudinal temperature gradients that are comparable
to vertical gradients.
The strength of the meridional heat transport continues to decrease as we move
into the rapid rotation regime, as seen on Jupiter, Saturn, and the other
gaseous solar-system giants. Here the Coriolis effect plays a significant role
at all latitudes, leading to a Geostrophic atmosphere dominated by the
infamous banded jet structure (Ingersoll & Vasavada 1998) (with baroclinic
instabilities driving mixing/angular-momentum-transport between bands).

A similar change in dynamics is expected to occur in exoplanetary atmospheres,
albeit over a significantly wider parameter space thanks to the much wider
variety of planetary type/orbital radii combinations found outside of our
solar system. One such combination found only outside of our solar-system is
the hot Jupiter, a Jupiter-like, gaseous planet which is in a very short orbit
and hence is very highly irradiated. Whilst a number of recent studies have
explore how atmospheric dynamics changes with orbital radii (Showman et al.
2014; Sainsbury-Martinez et al. 2023a), these come with the proviso that the
models link effects caused by changes in both surface irradiation and
planetary rotation, making it difficult to explore the effects that rotation
alone has on the dynamics.

As such, and in order to isolate how rotation affects the atmospheric dynamics
(with a particular focus on vertical heat transport, which is believed to be
the primary driver of radius inflation - see Sainsbury-Martinez et al. 2019,
2021; Sainsbury-Martinez et al. 2023b), here we present a series of deep,
HD209458b-like atmospheric models in which everything but the rotation rate is
held constant. Specifically the rotation rate will be varied between
$\Omega_{\mathrm{rot}}=0.0625\Omega_{0}$ and
$\Omega_{\mathrm{rot}}=40\Omega_{0}$, where $\Omega_{0}=2.1\times
10^{-5}\mathrm{s^{-1}}$ is the canonical angular rotation rate of HD209458b,
and the atmospheric models will be run until the deep atmosphere (with
$P_{max}=200$\mathrm{bar}$$) has equilibrated. These values where chosen such
that our models would span the rotation regimes found in our own solar system,
and we will investigate cases in which the rotation is both synchronous and
non-synchronous.

The structure of this work is as follows: In section 2 we introduce the model
we use to explore the effects of rotation on atmospheric dynamics and
observables: This includes both a brief description of our 3D global
circulation model (GCM) DYNAMICO, as well as a more indepth discussion of the
parametrisations we use. Then, in section 3, we discuss how rotation
influences the atmospheric dynamics of synchronously rotating hot Jupiters,
starting with zonal-mean dynamics before moving onto the horizontal and
vertical winds and heat transport, which we explore via both the helmholtz
wind decomposition as well as the enthalpy flux. Together these allow us to
investigate how rotation influences deep atmospheric heating, and hence
observed radius inflation. We then, in section 4, perform a similar analysis
for a set of hot Jupiter atmospheric models in which the orbital period and
planetary rotation period are allowed to diverge (i.e. non-synchronous
rotation), investigating how sensitive the standing-wave driven dynamics, and
hence the vertical heat transport, are to longitudinal shifts in the surface
irradiation profile. We finish, in section 5, with concluding remarks,
discussing the implications of our results and suggestions for future studies,
with next-generation GCMs, which will investigate the robustness of our
results. In particular, we are very interested in confirming if the formation
of a night-side hot-spot at slow rotation rates is just a feature of our
atmospheric models, or if it is a robust feature that may have important
implications for both the chemistry and observations of more slowly rotating
hot Jupiters (i.e. hot Jupiters in longer orbits around massive/bright stars).

##  2 Model Setup

In order to investigate how rotation impacts atmospheric dynamics in both the
observable outer atmosphere, and the more quiescent (but still dynamically
important) deep atmosphere, we adopt the simplified atmospheric setup
Sainsbury-Martinez et al. (2019) used to model the deep atmosphere of
HD209458b. This is necessitated by both the high computational cost of
evolving the quiescent deep atmosphere to steady state (even when we
initialise the model with an adiabat close to the expected equilibrium -
Rauscher & Menou (2010); Mayne et al. (2014); Sainsbury-Martinez et al.
(2021)) as well as the large range of rotation rates considered: from
$\Omega_{\mathrm{rot}}=0.0625\Omega_{0}$ and
$\Omega_{\mathrm{rot}}=40\Omega_{0}$, where
$\Omega_{0}=\Omega_{\textrm{HD209458b}}=2.1\times 10^{-5}\mathrm{s^{-1}}$.
Here, we give a very brief overview of the highly computationally efficient
GCM DYNAMICO (subsection 2.1), which uses a Newtonian Cooling approach to
model the radiative forcing (subsection 2.2), which we use to run a large
array of atmospheric models to steady state at all pressures, and from which a
smaller subset of models which represent the different dynamical regimes
observed are selected (subsection 2.3).

###  2.1 DYNAMICO

DYNAMICO is one of a number of highly computationally efficient global
circulation models which has been developed to run on the next-generation of
Exascale supercomputers, enabling atmospheric models of resolutions and
regimes that where previously out of reach. To achieve this, DYNAMICO solves
the primitive equations of meteorology (Dubos & Voitus 2014; Vallis 2006) on a
spherical icosahedral grid (Dubos et al. 2015). Here we configure this
horizontal-grid such that the angular resolution of each grid-cell is
approximately $2.5^{\circ}$. Vertically, the levels of the pressure-grid are
linearly spaced in $\log\left(P\right)$ space with a maximum pressure of
$200$\mathrm{bar}$$ and a minimum pressure of $7\times 10^{-3}$\mathrm{bar}$$.
This pressure range was selected such that we model both the highly irradiated
outer atmosphere where irradiation always dominates over transport as well as
a sufficiently large region of the deep atmosphere such that the effects of
differing vertical heat transport rates are clearly visible. Note that the
vertical boundaries of our model are closed and stress-free such that the only
means of energy transport into and out-of the system are the numerical-
dissipation (implemented as a hyperdiffusion filter with a fixed dissipation
timescale) required to stabilise the system against grid-scale noise, and the
Newtonian cooling relaxation scheme which we use to model the stellar
irradiation (subsection 2.2).

Following the calibration procedures of Sainsbury-Martinez et al. (2019) and
Sainsbury-Martinez et al. (2021), we select a numerical dissipation timescale
of $\tau_{dissip}=2500\,\mathrm{s}$ in order to ensure the stability of the
equatorially confined jets that develop in our most rapidly rotating models
(e.g. 1(c)). These narrow jets with their high latitudinal velocity gradients
also result in us setting a 90 second model time-step, which balances the need
to correctly reproduce model physics, whilst also minimising the computational
cost modelling equilibrium deep atmospheres.

###  2.2 Radiative forcing via Newtonian cooling

As discussed above, for computational efficiency reasons, we do not directly
model either the incident stellar irradiation or resulting night-side thermal
emission. Instead, as discussed in Sainsbury-Martinez et al. (2019), we take a
Newtonian cooling approach in which the outer atmosphere is relaxed onto a
spatially varying equilibrium temperature profile $T_{eq}$ over a time-scale
based upon the local radiative time-scale $\tau_{rad}$. In DYNAMICO, this is
achieved by adding a source term to the temperature evolution equation:

$\frac{\partial T\left(P,\theta,\phi\right)}{\partial
t}=-\frac{T\left(P,\theta,\phi\right)-T_{eq}\left(P,\theta,\phi\right)}{\tau_{rad}\left(P\right)}\,.$
(1)

For numerical reasons, we express the equilibrium temperature profile
($T_{eq}\left(P,\theta,\phi\right)$) onto which the atmosphere is relaxed, in
terms of the night-side temperature profile $T_{n}\left(P\right)$ (Iro et al.
2005) and a pressure dependent day-night temperature difference
($\Delta{T}(P)$) which has been calibrated such as to reproduce observations
of the hot Jupiter HD209458b:

$\displaystyle T_{eq}\left(P,\theta,\phi\right)$
$\displaystyle=T_{n}\left(P\right)$
$\displaystyle+max|0,\Delta{T}\left(P\right)\cdot cos\left(\theta\right)\cdot
cos\left(\phi-180\right)|$
$\displaystyle+\Delta{T}(P)\cos\left(\theta\right)\max\left[0,\cos(\phi-\pi)\right],$
(2)

where the night-side temperature profile is parametrised as series of linear
in $\log(P)$ space interpolations between the following T-P points

$T_{night}=\left(800,1100,1800\right)\,@\,P=\left(10^{-6},1,(\geq)10\right)\textrm{
bar},$ (3)

and the day-night temperature difference takes the form

$\displaystyle\Delta T(P)$
$\displaystyle=\left\\{\begin{array}[]{ll}600$\mathrm{K}$&\textrm{if
}P<0.01$\mathrm{bar}$\\\ 600$\mathrm{K}$\log(P/(0.01))&\textrm{if
}0.01$\mathrm{bar}$<P<10$\mathrm{bar}$\\\ 0&\textrm{if
}P>10$\mathrm{bar}$\end{array}\right.$ (7)

The radiative time-scale over which this relaxation occurs is also based upon
the work of Iro et al. (2005), with the linear in $\log(P)$ space
interpolation taking the form:

$\log(\tau_{rad})=\left(2.5,5.0,7.5,\log(\tau_{rad}^{d})\right)\,@\,P=\left(10^{-6},1,10,200\right)\textrm{
bar}.$ (8)

Note that, for the majority of models discussed here, we set the deep
radiative time-scale to infinity $\tau_{rad}^{d}=\infty$, thus disabling
radiative forcing in the deep atmosphere where it is expected to be
significantly slower than advection/transport (Sainsbury-Martinez et al.
2019).

###  2.3 HD209458b-like Atmospheric Models

In order to explore how rotation impacts the outer and deep atmospheric
dynamics and observables of highly irradiated, gaseous, exoplanets, we have
explored synchronously rotating atmospheres at 28 different rotation rates
(between $\Omega_{\mathrm{rot}}=0.0625\Omega_{0}$ and
$\Omega_{\mathrm{rot}}=40\Omega_{0}$ \- discussed in section 3) and non-
synchronously rotating atmospheres at 13 different planetary rotation rates
(between $\Omega_{\mathrm{rot}}=0.25\Omega_{0}$ and
$\Omega_{\mathrm{rot}}=4\Omega_{0}$, with the orbital period held constant at
that of HD209458b - discussed in section 4).
The parameters for these models are given in Table 1 and are the same for both
synchronous and non-synchronous models, with one major difference. In the
synchronous models, the Newtonian Cooling profile is fixed in both latitude
and longitude, representing the irradiation of a tidally-locked exoplanet. On
the other hand, in our non-synchronous models, the longitude of the Newtonian
Cooling (forcing) profile is updated every time-step based upon the difference
between the planetary and orbital rotation rates:

$\displaystyle\Delta\Omega$
$\displaystyle=\Omega_{\mathrm{rot}}-\Omega_{\mathrm{orb}}$ (9)
$\displaystyle\phi_{net}$
$\displaystyle=\left(t*\Delta\Omega\right)\bmod(360).$ (10)

Quantity (units) | Description | Value
---|---|---
dt ($\mathrm{s}$) | Time-step | 90
$N^{\circ}$ | Angular Resolution | $\leavevmode\nobreak\ 2.7^{\circ}$
$P_{top}$ ($\mathrm{bar}$) | Pressure at Top | $7\times 10^{-3}$
$P_{bottom}$ ($\mathrm{bar}$) | Pressure at Bottom | 200
$g$ ($\mathrm{m}\text{\,}{\mathrm{s}}^{-2}$) | Gravity | 8.0
$R_{HJ}$ ($\mathrm{m}$) | HJ Radius | $10^{8}$
$\Omega_{0}$ (${\mathrm{s}}^{-1}$) | Base HJ Angular Rotation | $2.1\times 10^{-5}$
$c_{p}$ ($\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$) | Specific Heat | 13226.5
$\mathcal{R}$ ($\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$) | Ideal Gas Constant | 3779.0
Table 1: Parameters for Simulations

(a) $0.125\Omega_{0}$

(b) $1\Omega_{0}$

(c) $15\Omega_{0}$

Figure 1: Zonally and temporally averaged zonal wind profiles for three
HD209458b-like models with rotation rates of; $0.125\Omega_{0}$ \- top,
$\Omega_{0}$ \- middle, and $15\Omega_{0}$ \- bottom.

(a) $R_{0}$

(b) $u$

Figure 2: Latitudinal Rossby number, $R_{0}$ \- top, and zonal-mean zonal-
wind, $u$ \- bottom, profiles in the outer atmospheres ($P\simeq
0.016\textrm{bar}$) of five HD209458b-like models at different rotation rates
spanning the range $0.125\Omega_{0}\rightarrow 15\Omega_{0}$.

(a) $0.125\Omega_{0}$

(b) $1\Omega_{0}$

(c) $15\Omega_{0}$

Figure 3: Longitudinally averaged profiles of the zonal wind at five different
latitudes, spanning the region from the equator ($50^{\circ}$) to the northern
pole ($0^{\circ}$), for three HD209458b-like models at different rotation
rates: $0.125\Omega_{0}$ \- left, $\Omega_{0}$ \- centre, and $15\Omega_{0}$
\- right. These three rotation rates have been chosen to emphasise the change
in zonal flow dynamics with rotation regime. Note: Positive wind speeds
indicate eastwards flows.

##  3 Synchronous Rotation

We start by exploring how rotation affects the atmospheric dynamics of
synchronously rotating exoplanets, focusing our analysis on three rotation
rates ($0.125\Omega_{0}$, $\Omega_{0}$, $15\Omega_{0}$) which are typical of
the three dynamical regions observed. We return to our full sample in
subsection 3.5 where we investigate the influence of rotation on vertical heat
transport.

###  3.1 Zonal-Mean Zonal-Winds

We begin our study by exploring how the zonal-mean zonal-winds change with
rotation rate. Specifically, as previously alluded to, we find that our models
all fit into one of three dynamical regimes (or the transition between
regimes), as shown in Figure 1.

At slower rotation rates ($\Omega_{\mathrm{rot}}<0.5\Omega_{0}$ \- 1(a)), we
find that a relatively slow, very shallow, and latitudinally extended ‘jet’
appears to dominate the zonal-mean zonal-wind (this can also be seen in
vertical slices of the zonal-mean zonal-wind: 3(a)). This is paired with a
rather quiescent mid to deep atmosphere
($\left|U\right|_{max}<150$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ for
$P>0.6$\mathrm{bar}$$), which implies that deep mixing may be suppressed in
this regime.
As $\Omega_{\mathrm{rot}}$ increases (between
$0.5\Omega_{0}<\Omega_{\mathrm{rot}}\lesssim 5\Omega_{0}$ for our models), we
find that the zonal-mean zonal wind (1(b)) resembles that found in the
classical hot Jupiter regime: a strong zonal jet that extends all the way from
the irradiated outer atmosphere to the advective deep atmosphere
($P>10$\mathrm{bar}$$), flanked by weaker high latitude counter-flows (see
3(b)) which are typically associated with the zonal jet driving standing wave
pattern (see subsection 3.4 and Showman & Polvani 2011). Note that this jet is
slightly narrower than the ‘jet’ that forms in the slow rotation regime. As
shown in 2(b), the latitudinal extent of the jet is highly sensitive to the
rotation rate, likely due to the influence that rotation plays, via the
Coriolis effect, on off-equator dynamics (see 2(a)).
This can be seen in our $\Omega_{\mathrm{rot}}=15\Omega_{0}$ model where we
find (1(c)) an even narrower zonal jet which extends all the way from the
outer atmosphere to the very bottom of the simulation domain (3(c)). Once
again we also find that this jet is braced by weaker counter flows, however
due to the Coriolis effect, these counter flows have become more latitudinally
confined, revealing even weaker easterly flows near the poles. This pattern of
alternating easterly and westerly flows is reminiscent of the wind structure
found for; gas giants in our solar system, the weakly irradiated and rapidly
rotating hot Jupiter models of Showman et al. (2014), and the rapidly rotating
brown Dwarf SDSS1411b (Sainsbury-Martinez et al. 2021).
This narrowing of the jet and suppression of off-equator flows continues as we
move to even higher rotation rates. However here another effect also becomes
apparent: a reduction in the total zonal transport by the jet, which can be
seen in 7(a) as a reduction in the mean zonal wind speed with increasing $Ro$
(an effect we discuss in more detail in subsection 3.4). As we discuss in
subsection 3.4, this is likely due to the Coriolis suppression of the jet-
driving off-equator standing Rossby and Kelvin waves.

(a) $0.125\Omega_{0}$

(b) $1\Omega_{0}$

(c) $15\Omega_{0}$

Figure 4: Zonally and temporally averaged meridional circulation
streamfunctions for three HD209458b-like models with rotation rates of;
$0.125\Omega_{0}$ \- top, $\Omega_{0}$ \- middle, and $15\Omega_{0}$ \-
bottom. Note that the meridional circulation profile is plotted on a log scale
with clockwise circulations shown in red and anti-clockwise circulations shown
in blue. At the base rotation rate, for example, these circulations combine at
the equator to drive an equatorial downflow at almost all pressures.

###  3.2 Meridional Circulation Streamfunction

Moving onto the meridional circulation profile, specifically the
streamfunction of the mass-flux on the meridional (i.e. latitudinal-pressure)
plane, we again find that our models fall into three distinct regimes (Figure
4). This is in agreement with the results of Sainsbury-Martinez et al. (2019),
who suggests that the meridional circulation, and more generally the vertical
dynamics, are driven by the horizontal winds.
Starting in the slowly rotating regime, we find (4(a)) a relatively weak
meridional circulation profile that generally consists of a single circulation
cell in each hemisphere. The only exception to this is the very deepest
regions of the simulation domain which has not yet equilibrated due to the low
local wind speeds (1(a)) and hence long dynamical time-scales. In the outer
atmosphere, where the influence of rotation on the global overturning
circulation is strongest, these circulations are driven by the net-eastward
zonal wind, and as a result we find clockwise/anti-clockwise circulation cells
in the southern/northern hemisphere respectively, cells that combine at the
equator to drive a weak, net, downflow. However, as we move deeper
($P>0.5$\mathrm{bar}$$) and the net zonal-wind fades to near zero(see 3(a)),
the sense of these circulations is reversed, leading to an equatorial
upwelling that is typical of a global overturning circulation that is barely
influence by rotation (Hammond & Lewis 2021).
Moving towards the classical hot Jupiter rotation regime, we find a meridional
circulation profile (4(b)) similar to that found by Sainsbury-Martinez et al.
2019: a pair of meridional circulation cells in each hemisphere which combine
together to drive an equatorial downflow from the outer atmosphere to the
bottom of the simulation domain (albeit with a slight equatorial upflow at
very low pressures where $\tau_{rad}\ll\tau_{adv}$ and hence no jet has
formed: 3(b)), a mass conserving upflow at mid-latitudes, and finally weak
polar downflows. Further, this circulation is almost an order of magnitude
stronger than that found in the other rotation regimes, which may help to
explain this regimes strong vertical enthalpy advection (see subsection 3.5).
Finally, in the rapidly rotating regime, we find a meridional circulation
profile with multiple circulation cells per hemisphere (4(c)), much like what
was found for SDSS1411b by Sainsbury-Martinez et al. (2021). These circulation
cells combine at the equator to drive a relatively weak downflow, and, much
like the zonal wind, the circulation cells become significantly weaker as we
move towards higher latitudes. Furthermore, both this effect, as well as the
number of circulation cells in each hemisphere, increases with $\Omega$, an
effect which is likely linked to banded zonal-wind structure that also
develops (3(c)).

(a) $0.125\Omega_{0}$ \- 0.0026 bar

(b) $1\Omega_{0}$ \- 0.0026 bar

(c) $15\Omega_{0}$ \- 0.0026 bar

(d) $0.125\Omega_{0}$ \- 0.016 bar

(e) $1\Omega_{0}$ \- 0.016 bar

(f) $15\Omega_{0}$ \- 0.016 bar

(g) $0.125\Omega_{0}$ \- 0.2 bar

(h) $1\Omega_{0}$ \- 0.2 bar

(i) $15\Omega_{0}$ \- 0.2 bar

Figure 5:  Temporally averaged zonal wind (arrows) and temperature profile
(map) at three different pressures (0.0026 bar - top - to 0.2 bar - bottom)
for three HD209458b-like models at different rotation rates ranging from
$0.125\Omega_{0}$ \- left - to $15\Omega_{0}$ \- right.

###  3.3 Horizontal-Wind and Temperature Profiles

In order to better understand the dynamics discussed above, we next
investigate the horizontal wind and temperature profiles at select pressure
levels for three HD209458b-like models which fall into our three regimes of
interest, as shown in Figure 5. Note that pressure levels have been chosen so
as to emphasise the difference between the dynamical regimes (i.e. the
differences associated with varying both rotation rate and relative
radiative/advective dynamical strength).

We start our analysis with the slowly rotating regime (left column of Figure
5). Here, in the outer atmosphere, 5(a), we find that the horizontal wind
primarily flows from the day-side to the night-side with almost no asymmetry -
i.e. an almost isotropic divergence of the wind on the hot day-side away from
the substellar point and convergence of wind on the cold night-side. However
almost isotropic does not mean completely: we find that the easterly flow from
the day to night-side is faster than its westerly equivalent, and that the
convergence point of the winds on the night-side is west of the anti-stellar
point. Both of these effects can be linked to the, weak, influence of rotation
on the atmosphere, either in the form of standing waves attempting to drive an
equatorial jet (and hence introducing a preferred direction to flows), or the
preferential enhancement/suppression of flows with/against the direction of
rotation.
Moving deeper into the atmosphere( 5(d)) we start to find evidence that
horizontal advection is shaping the thermal structure as the influence of
radiation wanes. Of particular note are the equatorial winds, which are now
easterly at all longitudes, although the influence of divergent day-night
winds remains with the equatorial winds being significantly slower just west
of the substellar point (see subsection 3.4 for more details). We also catch
our first glimpse of a rather unusual feature that is exclusive to the slowly
rotating regime: a slight hot-spot on the night-side, where the divergent
winds converge (as we discuss in Figure 9, this is not a coincidence).
This night-side hot-spot grows in strength as we move towards mid-pressures
($0.2$\mathrm{bar}$$ \- 5(g)), resulting in a rather unusual wind and
temperature structure: the hottest region of the atmosphere has moved from the
day-side to just west of the anti-stellar point, where the divergent day-night
winds converge. Further, this night-side hot-spot, and the resulting
temperature gradients, drive a significant equatorial wind from the night-side
to the day-side, which appears in the zonal-mean zonal-wind profile as part of
the jet, despite it not forming in the same way as the classical hot Jupiter
zonal-mean equatorial jet.
If robust, the formation of this night-side hot spot, which occurs for all of
our HD209458b-like models that fall into the slowly rotating regime, may have
important implications for our understanding of the dynamics of such objects.
For example, as discussed in subsection 3.5 and shown in 10(a), the night-side
hot-spot is associated with a thermal inversion, the presence of which might
have strong implications for both the observed phase curve (depending upon the
wavelength/molecule observed, and hence the pressure probed) as well as the
chemistry, since the inversion can act as a cold-trap, leading to
condensation/rain-out and hence changing the, observable, composition of the
atmosphere (which in turn may effect the thermal structure due to changes in
absorption/emission). Note that, as discussed above, this feature is common to
all our slowly rotating regime models, and, in fact, its presence is almost a
defining feature of the slowly rotating regime. This is because, as $\Omega$
increases, rotational influences shape the day-night flow, shifting the hot-
spot even further westwards, away from the anti-stellar point and towards the
terminator, until it eventually combines with the easterly advected day-side
hot-spot, leading to the formation of the thermal butterfly that is emblematic
of the classical hot-Jupiter regime at a transitional rotation rate of around
$\Omega=0.5\rightarrow 0.6\Omega_{0}$.

Moving into the classical hot Jupiter regime, which we plot on the middle
column of Figure 5, we find dynamics that are in good agreement with previous
studies of hot Jupiters, such as Sainsbury-Martinez et al. (2019).
For example, in the outer atmosphere (5(b)), we find slight advection from the
day-side towards the eastern terminator, forming ‘wings’ that: a) trace the
off-equator divergent/convergent flows, which are increasingly rotationally
influenced, and hence have become longitudinally asymmetric, with eastern
flows compressed and, higher latitude, western flows stretched longitudinally;
and b) will eventually form, deeper in the atmosphere, a key part of the
thermal butterfly structure that is synonymous with the classical regime. We
also find that, unlike in the slowly rotating regime, super-rotating flows are
present at all longitudes on the equator, albeit with the proviso that winds
west of the sub-stellar point are slightly suppressed/weakened by the
divergent component of the wind (see subsection 3.4).
Evidence for the formation of foreshadowed thermal butterfly increases as we
move deeper, 5(e), with the influence that rotation has on the dynamics also
becoming increasingly clear. For example, the high latitude flows have become
increasingly asymmetric, shifting the ‘wings’ associated with convergent flows
closer towards the terminator and the day-side hot-spot. Together wind the
growing strength of the equatorial jet at all longitudes (although it does
remain slightly weaker just west of the sub-stellar point) which drives the
eastward advection (by approximately $20^{\circ}$) of the near equator day-
side hot-spot, we are starting to see the basis of the structure that will
form the thermal butterfly.
A structure that becomes apparent as we move into the mid atmosphere
($0.2$\mathrm{bar}$$ \- 5(h)), where we find that the day-side hot-spot is now
located $\sim 50^{\circ}$ east of the sub-stellar point. Interestingly, we
also find a secondary peak in the temperature profile even further east,
across the terminator and on the night-side - this is a remnant of the ‘wings’
seen at lower pressures and can also be considered as a sign that the off-
equator heat transport was sufficiently rotationally influenced as to prevent
the formation of the night-side hot-spot found in the slowly rotating regime.
Further, whilst the off-equator flows are mostly hidden by the equatorial jet,
the visible remnants suggest that, at high latitudes, the easterly circulation
that previously had been confined to the day-side has been almost completely
suppressed, leaving only a westerly day-night flow, which appears in the
zonal-mean zonal-wind as a counterflow (1(b)).
Note that, as we increase the rotation rate in the classical regime, the
influence of rotation grows, with off-equator flows playing an increasing role
in shaping the circulation, leading to westward and poleward advection at mid-
latitudes and more distinct ‘wings’ to the thermal butterfly.

This rotational influence becomes increasingly apparent as we move into the
rapidly rotating regime, which we plot on the right-hand column of Figure 5.
Here, in the outer atmosphere (5(c)) we find that the off-equator wind
dynamics are very strongly suppressed (by the Coriolis effect), particularly
on the cold night-side, resulting in only a weak day-side circulation from
west of the sub-stellar point to the east via mid-latitudes. This leaves the
strong super-rotating equatorial jet as the dominant flow, but even this is
influenced by rotation, with the jet being visibly narrower than its classical
regime counterpart. Consequently, this highly rotationally-confined wind
structure drives thermal advection that is limited to equatorial regions, with
weak eastwards advection at the equator (weak due to the reduced jet speed
that latitudinally confined standing waves can drive), weak westwards
advection at low latitudes, and basically no heat transport at high latitudes
(i.e. above $\pm\sim 30^{\circ}$).
Moving deeper, 5(f) and 5(i), we find a very similar structure but with one
significant difference: As the radiative forcing weakens, the eastwards
equatorial and westwards low-latitude advection strengthens, leading to
stronger zonal mixing, such as the band of equatorial day-night heating found
at 0.2 bar (5(i)). Note that as we move even deeper into the atmosphere, this
zonal mixing grows, leading to an equatorial warm band between 5 and
$20$\mathrm{bar}$$.
Note that, at even faster rotation rates, we find that both the equatorial
jet, as well as the low-latitude counterflows become increasingly
latitudinally confined, which has implications for not only the thermal
structure, but also the strength of the zonal winds (as we discuss in
subsection 3.4), and hence the vertical transport of potential-
temperature/enthalpy (as we discuss in subsection 3.5).

(a) $0.125\Omega_{0}$ \- Divergent

(b) $1\Omega_{0}$ \- Divergent

(c) $15\Omega_{0}$ \- Divergent

(d) $0.125\Omega_{0}$ \- Rotational

(e) $1\Omega_{0}$ \- Rotational

(f) $15\Omega_{0}$ \- Rotational

(g) $0.125\Omega_{0}$ \- Eddy

(h) $1\Omega_{0}$ \- Eddy

(i) $15\Omega_{0}$ \- Eddy

Figure 6: Helmholtz decomposition of the radially averaged (over the outer and
mid atmosphere) horizontal wind of three, synchronous, HD209458b-like
atmospheric models with rotation rates ranging from 0.125$\Omega_{0}$ to
15$\Omega_{0}$. The top row plots the divergent component ($H_{div}$) of the
Helmholtz decomposition, the middle row plots the rotational component
($H_{rot}$) of the decomposition, and the bottom row shows the eddy component
($H_{eddy}=H_{rot}-\left<H_{rot}\right>$) of the decomposition. Note the
change (decrease) in mean velocity as the rotation rate increases.

###  3.4 Helmholtz Wind Decomposition

Our above analysis has revealed that the horizontal winds are highly sensitive
to both rotation rate and pressure (which acts as a proxy for the relative
influence of radiative forcing to advective transport). In order to
disentangle these effects, we turn to the Helmholtz wind decomposition, which
has long been a staple of Earth atmospheric studies (Dutton 1986), and which
has recently been applied to both hot Jupiters (Hammond & Lewis 2021) and
tidally-locked rocky exoplanets (Sergeev et al. 2022).
Briefly, a Helmholtz decomposition can be used to split the horizontal wind at
a select pressure level (or over an averaged pressure range), $\bm{u}=(u,v)$,
into divergent (i.e. ‘vorticity free’) and rotational (i.e. ‘divergence free’)
components (Dutton 1986):

$\displaystyle\bm{u}$ $\displaystyle=\bm{u}_{d}+\bm{u}_{r}$ (11)
$\displaystyle=-\bm{\nabla}\chi+\bm{k}\times\bm{\nabla}\psi,$ (12)

where $\bm{k}$ denotes the unit vector in the (eastward) zonal direction,
$\chi$ is the velocity potential function, $\psi$ is the velocity stream-
function, and both $\chi$ and $\psi$ can be linked to the divergence $\delta$
/ vorticity $w$ directly:

$\displaystyle\nabla^{2}\chi$ $\displaystyle=\delta$ (13)
$\displaystyle\nabla^{2}\psi$ $\displaystyle=w.$ (14)

Additionally, in order to isolate the equatorial super-rotating jet from any
other rotational wind dynamics it may be masking, we further split the
rotational component, $\bm{u}_{r}$ into a zonal-mean component $\bm{u}_{z}$
and an eddy-wind component $\bm{u}_{e}$:

$\displaystyle\bm{u}_{z}$ $\displaystyle=\left<\bm{u}_{r}\right>$ (15)
$\displaystyle\bm{u}_{e}$ $\displaystyle=\bm{u}_{r}-\bm{u}_{z},$ (16)

where $\left<\right>$ indicates the zonal-mean.
As for exactly what these components represent for a tidally-locked
atmosphere: $\bm{u}_{d}$ represents flows that diverge from the hot-spot on
the day-side and converge somewhere on the cold night-side, forming a closed
cycle when combined with an upwelling below the day-side hot-spot and the
resulting downwelling on the nightside; $\bm{u}_{r}$ represents dynamics
driven by angular momentum transport via stationary Rossby and Kelvin waves -
in typical hot Jupiters these standing waves transport angular momentum from
mid-latitudes to the equator, resulting in slight westward flows at mid-
latitudes and a super-rotating flow (jet) at the equator (Showman & Polvani
2011); and finally, as mentioned above, $\bm{u}_{e}$ and $\bm{u}_{z}$
represent a further decomposition of $\bm{u}_{r}$, allowing us to explore the
transport by standing waves in cases where the presence of a super-rotating
equatorial jet would completely dominate the dynamics.

In Figure 6, we plot $\bm{u}_{d}$ (top), $\bm{u}_{r}$ (middle), and
$\bm{u}_{e}$ (bottom), for the three different rotation rates discussed in
subsection 3.1, in order to investigate how the balance between different
components of the radially-averaged (over the mid/outer atmosphere where $P<1$
bar) horizontal winds changes with rotation rate.

We start by analysing the winds in the slowly rotating regime, which is
plotted in the left column of Figure 6. Here, our analysis reveals that, in
the outer and mid atmosphere, the primary component which make up the total
horizontal wind is the divergent component
($\left|\bm{u}_{d}\right|=554\mathrm{\,ms^{-1}}$), which is, on average, a
factor of two faster than the rotational component
($\left|\bm{u}_{r}\right|=291\mathrm{\,ms^{-1}}$). This makes sense, and is in
good agreement with the wind structure discussed in subsection 3.1. To start,
off-equator, we find flows that trace the dominant, divergent, day-side to
night-side flow (compare 6(a) and 5(a)), and which are driven by the strong
radiative forcing and day-night temperature gradient. Whereas at the equator,
we find that the zonal wind associated with rotational dynamics opposes the
divergent wind west of the sub-stellar point whilst also reinforcing the flow
to the east, leading to to observed asymmetry in equatorial wind speed.
Decomposing the rotational wind further reveals the driving force behind this
equatorial jet: as shown in 6(g), the eddy component of the wind consists of
an $m=1$ circulation structure (i.e. two circulation cells per hemisphere), a
structure which is associated with Kelvin and Rossby standing waves and which
is commonly attributed as being the driving mechanism behind super-rotating
jets on hot Jupiters (Showman & Polvani 2011). Briefly, this mechanism works
as follows: longitudinal variations in the radiative forcing (i.e. the strong
day/night heating/temperature contrast) trigger the formation of standing,
planetary-scale, Rossby and Kelvin waves, with the Kelvin waves straddling the
equator, and the Rossby waves lying at mid latitudes (with their exact
location depending upon the rotation rate). This standing wave pattern then
essentially acts to pump eastward angular momentum from mid-latitudes to the
equator, inducing equatorial superrotation and mid-latitude counterflows. Note
that this angular momentum pumping is also associated with an eastwards tilt
of the standing wave pattern, a tilt which is strongly associated with
Coriolis forces, and hence is only found at more rapid rotation rates
(6(h)-l).

As we increase the rotation rate and move into the classical hot Jupiter
regime, shown in the middle column of Figure 6, we find that the balance has
shifted in favour of the rotational component
($\left|\bm{u}_{r}\right|=692\mathrm{\,ms^{-1}}$) over the divergent component
($\left|\bm{u}_{d}\right|=241\mathrm{\,ms^{-1}}$), with the eddy component
(6(h) \- $\left|\bm{u}_{e}\right|=283\mathrm{\,ms^{-1}}$) starting to exhibit
signs of the titled standing wave pattern associated with a strong
rotationally influenced zonal jet, as seen in 6(e) and subsection 3.1. This
dominance of rotation driven dynamics is also reflected in the thermal/wind
structures shown in Figure 5. For example, comparing 6(e) with 5(b)/f/g
reveals the thermal butterfly formation mechanism: a strong jet drives
equatorial advection whilst the $m=1$ circulations drives the formation of the
off-equator temperature wings east of the sub-stellar point as well as the
slightly cooler point found on the equator just behind the tip of the
butterfly. Both of these features are associated with the rotational wind,
with the divergent wind (6(b)) only acting to slightly slowdown/speedup the
jet west/east of the sub-stellar point.

Moving into the rapidly rotating regime, as shown on right-hand column of
Figure 6, we start to see strong evidence of the Coriolis driven reshaping of
the winds. For example, we find that the rotational wind
($\left|\bm{u}_{r}\right|=455\mathrm{\,ms^{-1}}$) completely dominates the
divergent wind ($\left|\bm{u}_{d}\right|=32\mathrm{\,ms^{-1}}$), leading to a
divergent wind whose contributions are insignificant at all but the lowest of
pressures where radiative forcing dominates. However this change has not come
about because of an enhancement of the rotational wind (which is slower here
than either slower rotation regime), rather we find that Coriolis effects have
started to seriously suppress off-equator flows, essentially destroying the
divergent wind and leaving a highly latitudinally compressed and extremely
titled eddy wind (6(i) \- $\left|\bm{u}_{e}\right|=83\mathrm{\,ms^{-1}}$) that
struggles to generate a broad zonal jet, but does still manage to drive at
least some off-equator advection east of the sub-stellar point (e.g. 5(c)).
Evidence for this compression can be seen in 1(c), where we find that the off-
equator counter flows associated with the standing wave pattern have moved to
low latitudes. We also find that, off-equator, both components of the wind are
notably weaker on the night-side than the day-side, an effect which we
attribute to the tilt of the eddy winds and the standing wave pattern leading
to a preferred direction for circulations and essentially zero night-side
flow.
It is worth mentioning that the compression/suppression of the off-equator
flows only grows in strength as we increase the rotation rate further. This
has devastating consequences for overall zonal advection/transport: as the
standing-wave structure gets further compressed/suppressed by Coriolis forces,
its ability to pump off-axis, eastwards, angular momentum towards the equator
is reduced, resulting in a slight slowdown in equatorial jet speeds, as shown
in 7(a), as well as a narrowing of the jet, as shown in 2(b). Together this
leads to a significant reduction is zonal transport except at the equator, the
effects of which can be seen in Figure 5. As we discuss below, this can have
significant consequences for the vertical advection of enthalpy in rapidly
rotating atmospheres, and may help to explain the lack of inflation observed
in such objects.

(a) $R_{0}$ vs. $\left<U\right>_{equator}$

(b) $R_{0}$ vs. $\left<W\right>_{pole}$

Figure 7: A pair of plots showing how the mean zonal wind at the equator - top
- and mean vertical wind near the northern pole - bottom - vary with global
mean Rossby number for 29 HD209458b-like models spanning the rotation range
$0.125\Omega_{0}\rightarrow 15\Omega_{0}$.  Figure 8: Global mean global
vertical enthalpy flux for 29 HD209458b-like models spanning the rotation
range $0.125\Omega_{0}\rightarrow 40\Omega_{0}$. Note that negative fluxes
indicate net downward enthalpy (potential temperature) transport.

(a) $0.125\Omega_{0}$ \- Zonal - 0.0026 bar

(b) $0.125\Omega_{0}$ \- Zonal - 0.1 bar

(c) $0.125\Omega_{0}$ \- Meridional - 0.016 bar

(d) $0.125\Omega_{0}$ \- Vertical - 0.016 bar

(e) $1\Omega_{0}$ \- Zonal - 0.0026 bar

(f) $1\Omega_{0}$ \- Zonal - 0.1 bar

(g) $1\Omega_{0}$ \- Meridional - 0.016 bar

(h) $1\Omega_{0}$ \- Vertical - 0.016 bar

(i) $15\Omega_{0}$ \- Zonal - 0.0026 bar

(j) $15\Omega_{0}$ \- Zonal - 0.1 bar

(k) $15\Omega_{0}$ \- Meridional - 0.016 bar

(l) $15\Omega_{0}$ \- Vertical - 0.016 bar

Figure 9: Maps showing the Zonal, Meridional, and Vertical Enthalpy transport
at select pressure levels for three exemplary HD209458b-like models with
rotation rates of; $0.125\Omega_{0}$ \- top, $\Omega_{0}$ \- middle, and
$15\Omega_{0}$ \- bottom. Here, positive (red) fluxes represent
eastward/polar/outwards flows for the zonal/meridional/vertical enthalpy flux
maps respectively. Note: We include two maps of the zonal enthalpy transport
at different pressure levels in order to emphasise how the zonal advection
changes with height.

(a) $0.125\Omega_{0}$

(b) $1\Omega_{0}$

(c) $15\Omega_{0}$

Figure 10: Latitudinally averaged (over the equatorial regions) T-P profiles
for three HD209458b-like models at different rotation rates; $0.125\Omega_{0}$
\- top, $1\Omega_{0}$ \- middle, and $15\Omega_{0}$ bottom. Each plot includes
profiles from 6 different longitudes, ranging from the sub-stellar point
(whose equilibrium profile is shown in red) eastwards to the anti-stellar
point (whose equilibrium profile is shown in blue) on the night-side. Note
that we have excluded the deepest regions of the atmosphere
($P>50\textrm{bar}$) from these plots so as to emphasise the changes that
occur with longitude at lower pressures, whilst still showing how the deep
atmospheric heating changes (compared to the initial adiabat, shown as a dark
grey dash) with rotation rate.

(a) $0.125\Omega_{0}$ \- Longitudinal Variations

(b) $1\Omega_{0}$ \- Longitudinal Variations

(c) $15\Omega_{0}$ \- Longitudinal Variations

(d) $0.125\Omega_{0}$ \- Global Mean

(e) $1\Omega_{0}$ \- Global Mean

(f) $15\Omega_{0}$ \- Global Mean

Figure 11: Longitudinal variations (top) and global means (bottom) of the
vertical enthalpy flux $F_{\mathcal{H}}$ profiles for three exemplary
HD209458b-like models; $0.125\Omega_{0}$ \- top, $\Omega_{0}$ \- middle, and
$4\Omega_{0}$ \- bottom. In the top row, we plot vertical enthalpy flux
profiles at 5 different longitudes, ranging from the sub-stellar point to just
west of the anti-stellar point, as well as the mean enthalpy flux profile.
However, since the mean flux is significantly smaller than the local fluxes,
we replot the mean profiles in the bottom row such as to better demonstrate
the vertical variations in enthalpy transport.

###  3.5 Enthalpy Transport

We expect differences in the horizontal, and hence vertical, wind dynamics to
also manifest as significant differences in horizontal/vertical energy
transport. In order to explore this, we consider the enthalpy flux:

$F_{H}\left(r,\theta,\phi\right)=\rho
c_{p}T\bm{U}\left(r,\theta,\phi\right)\,,$ (17)

where $\rho$ is the density, $T$ is the temperature, and $c_{p}$ is the
specific heat. We plot the zonal, meridional, and vertical enthalpy flux for
select pressure levels, chosen to emphasise differences in outer and deep
atmosphere transport, for our three dynamical regimes in Figure 9.

Starting in the slowly rotating regime, shown in the top row of Figure 9, we
find that the zonal and meridional transport in the outer atmosphere (9(a)/c)
is dominated by the divergent day-night wind, with eastward/westward enthalpy
transport east/west of the sub-stellar point, and polewards/equatorwards
meridional transport of the day/night side. Taken together, these flows
suggest that enthalpy is transported from the irradiated day-side to slightly
west of the anti-stellar point on the night-side, i.e. exactly where the
night-side hot-spot eventually forms (5(d)).
The vertical advection of enthalpy in the outer atmosphere (9(d)) reinforces
this link between horizontal transport and the night-side hot-spot, with a
significant down-flow developing exactly where the night-side hot-spot forms,
tracing the convergence of the horizontal winds. We also find a significant
upwelling on the day-side, aligned with the hot-spot, leading to a global
overturning-like circulation that is typical of atmospheres whose dynamics are
shaped by divergent flows.
The impact that the night-side hot-spot has on enthalpy transport becomes even
more apparent at higher pressures. For example, in the mid-atmosphere, the
zonal enthalpy transport (9(b)) is now no longer controlled just by the
divergent day-night flows (although divergence driven dynamics remain dominant
at high latitudes). Instead, we find significant, equatorial, enthalpy
transport eastwards from the night-side towards the day-side/sub-stellar-
point, driven by the ‘reversed’ day/night temperature difference. Note that
this night to day transport essentially acts to close the overturning-like
circulation described above, which has important implications for the deep
atmosphere. Specifically, it means that the deep atmosphere is relatively
quiescent and not as strongly linked to the outer atmosphere, leading to
almost no deep heating and hence the suppression of advectively driven radius
inflation (10(a)). Evidence for this can be seen in both 11(a) and 11(d),
which show the vertical enthalpy transport against pressure for various
longitudes/the global mean respectively: here we find a significant vertical
enthalpy flux in the mid-atmosphere, around the same pressure as the night-
side hot-spot, as well as a net (global mean) downward enthalpy flux in the
deep atmosphere. Note that, despite the similarities in enthalpy flux
magnitudes in local regions (compare 11(a) and 11(b)), we find that overall
(i.e. when taking the global mean/calculating the net flux), the downward
enthalpy advection is almost an order of magnitude lower here than in the
classical hot Jupiter regime (compare 11(d) and 11(e)). This can be linked to
the different vertical wind structures that the different horizontal wind
profiles drive, see for example the much narrower downflow in 9(d) as opposed
to 9(h).
Finally, we note that the results of this enthalpy transport are clearly
visible in the T-P profiles of the mid/outer atmosphere, as shown in 10(a).
For example, on the night-side between $\sim 0.05\rightarrow\sim
2$\mathrm{bar}$$ we find clear evidence of the night-side hot-spot, including
the significant cold-trap associated with the thermal inversion. We also find
a reduced day-night temperature difference, suggesting that divergent day-
night energy transport is highly efficient in the outer atmosphere, and that
the day-night contrast of hot Jupiters might be sensitive to rotation rate.

Moving onto the classical hot Jupiter regime, which we plot on the middle row
of Figure 9, we find that the primary driver of enthalpy transport is now the
rotational component of the wind. For example, at low pressures, we find that
both the zonal and meridional enthalpy transport (9(e)/g) traces the eddy
component of the wind (6(h)). As a result we find a significant asymmetry
between westwards and eastwards day-night enthalpy transport: westwards of the
sub-stellar point we find broad poleward transport which crosses the entire
night-side, whereas eastwards of the sub-stellar point we find a much more
longitudinally constrained profile linked with the off-equator wings that from
in the outer and mid atmosphere (e.g. 5(e)). In turn, these off-equator wings
are associated with a strong downward vertical enthalpy transport (9(h)) that
is somewhat balanced by a broad upwelling near the sub-stellar point. Together
forming a global circulation reminiscent of the global ‘overturning’
circulation.
However unlike in the slowly rotating regime, we find that this global
‘overturning’ circulation extends deep into the atmosphere, where radiative
forcing is weak, thus connecting the outer and deep atmospheres, and allowing
for deep heating (10(b)). As for the reason why this circulation penetrates
deeper into the atmosphere, this is because of the strong, deep zonal jet
which drives strong mixing near the equator, matching what we find in the
meridional circulation streamfunction (4(b)). Evidence for this deeper zonal
jet can be seen in 9(f), which reveals strong eastwards advection near the
equator at all longitudes, paired with weak off-equator transport linked to
the equatorward pumping of eastwards angular momentum from mid-latitudes.
Overall, these strong zonal/equatorial dynamics drive significant vertical
enthalpy transport, as shown in 11(b)/e, where we find the highest net/global-
mean downwards vertical enthalpy of our three regimes. A strong enthalpy flux
which drives significant deep heating (see 10(b) and the dashed line in the
deep atmosphere which shows the initial adiabat), suggesting that the planet
is inflated relative to both more slowly rotating hot Jupiters and 1D
atmospheric models (Sainsbury-Martinez et al. 2019).

Finally we come to the rapidly rotating regime, which we plot on the middle
row of Figure 9. Here we find that advective heat transport has become
increasingly confined to low latitudes, with the equatorial jet acting as the
main source of horizontal transport at almost all pressures. For example, in
the outer atmosphere, 9(i)/k, we find horizontal enthalpy transport which is
either driven by the jet (zonally) or by the very highly tilted and
latitudinally constrained $m=1$ standing wave pattern (meridional). The jet
transports enthalpy eastwards from the sub-substellar point, across the night-
side, leading to the band of equatorial heating/mixing seen in 5(i), whilst
the tilted standing wave primarily drives transport on the day-side
(poleward/westward advection west of the sub-stellar point and
equatorward/eastward advection to the east) and leads to the warped/compressed
thermal butterfly also seen in 5(i).
The influence of rotation on the enthalpy transport is also evident in the
vertical enthalpy flow which is highly confined to low-latitude/equatorial
regions. For example, as shown in 9(l) both the days-side upwelling and night-
side downflow are confined to be within $\sim 20^{\circ}$ of the equator,
which explains the lack of high-latitude downflows found at low Rossby numbers
(7(b)). Further, due to the day-side confinement of the off-equator
circulations, we find that the night-side downflows tend to be closer to the
terminators/day-side than in the more slowly rotating regimes, although this
changes as we move deeper into the atmosphere and the downflows become more
tightly associated with a zonally-uniform zonal jet, as seen in 9(j).
Overall, the Coriolis driven confinement of the horizontal and vertical
enthalpy fluxes has a suppressing effect on the driving of deep atmospheric
heating. That is to say that we find that, as shown in 11(c) downward enthalpy
advection is significantly weaker than both of the other regimes considered
here, suggesting that deep heating, and hence radius inflation, should be
strongly suppressed. And the global-mean deep T-P profile confirms that this
is indeed the case. However. if we confine our investigation of the deep
atmosphere to equatorial regions, as we do in 10(c), we do find slight hints
of deep heating. This occurs because the local vertical enthalpy advection is
orders of magnitude stronger at the equator than elsewhere, and the
latitudinal confinement for the off-equator circulations mean that latitudinal
mixing is weak. With time, this mixing might lead to global heating of the
deep atmosphere, albeit to a much lesser extent than in the classical regime,
however a quick test with even weak radiative forcing back towards the initial
adiabat (designed to reproduce the slight radiative cooling of the hot deep
atmosphere) is more than enough to prevent this heating. As such, we feel
confident in arguing that models in the rapid rotation regime should show
little to no radius inflation, in agreement with the SDSS1411b results of
Sainsbury-Martinez et al. (2021).

##  4 Non-Synchronous Rotation

Many of the dynamics found in the synchronously rotating regime (section 3)
rely on the presence of an $m=1$ standing wave (Figure 6), which develops
thanks to the strong, persistent, day-night forcing associated tidally-locked
irradiation (Showman & Polvani 2011). However for warm Jupiters with longer
orbital radii (i.e. $a=0.2$\mathrm{ua}$$ for a Sun-like star, like HD209458A,
even closer for an M-Dwarf), the synchronisation time-scale starts to approach
the system/planetary age.
Thus there is a strong possibility that next-generation exoplanet hunting
missions will start to detect warm/hot Jupiters whose rotation period is not
perfectly synchronised with their orbital period. This might have profound
implications for their atmospheric dynamics. As such, we next explore how both
small and large differences between the orbital and rotation period (i.e. non-
synchronicity) affect the dynamics and resulting heat transport of such
objects. Here our models fall into two regimes: the first is models in which
the difference between the orbital ($\Omega_{\mathrm{orb}}$) and rotation
($\Omega_{\mathrm{rot}}$) angular rotation rates is a fraction of the overall
rotation rate (e.g.
$\Omega_{\mathrm{rot}}=0.9\Omega_{\mathrm{orb}}=0.9\Omega_{0}$ \- more
generally when the difference is less than $\sim 15\%$ of the rotation rate -
here that is the rotation rate of HD209458b), and the effect of non-
synchronicity is relatively minor. And the second regime is when the
difference is significant (e.g.
$\Omega_{\mathrm{rot}}=4\Omega_{\mathrm{orb}}=4\Omega_{0}$), on the order of
or greater than either the orbital or angular rotation rate, leading to
significant disruption of the driving mechanisms discussed in section 3.

(a) $\Omega_{\mathrm{rot}}=0.9\Omega_{0}$ \- Non-Synchronous

(b) $\Omega_{\mathrm{rot}}=4.0\Omega_{0}$ \- Non-Synchronous

(c) $\Omega=0.9\Omega_{0}$ \- Synchronous

(d) $\Omega=4.0\Omega_{0}$ \- Synchronous

Figure 12:  Zonally and temporally averaged zonal wind profiles for four
HD209458b-like models, two of which are non-synchronously rotating (top) and
two of which are synchronously rotating (bottom), with rotation rates chosen
to fall into our weak-non-synchronicity (left -
$\Omega_{\mathrm{rot}}=0.9\Omega_{\mathrm{orb}}=0.9\Omega_{0}$ in the non-
synchronous model and
$\Omega_{\mathrm{rot}}=\Omega_{\mathrm{orb}}=0.9\Omega_{0}$ in its synchronous
equivalent) and strong-non-synchronicity (right -
$\Omega_{\mathrm{rot}}=4\Omega_{\mathrm{orb}}=4.0\Omega_{0}$ in the non-
synchronous model and
$\Omega_{\mathrm{rot}}=\Omega_{\mathrm{orb}}=4.0\Omega_{0}$ in its synchronous
equivalent) regimes.  Figure 13:  Longitudinally averaged zonal wind profiles
in the outer atmospheres (P $\simeq 0.016\textrm{bar}$) of four non-
synchronous HD209458b-like models, with rotation rates
($\Omega_{\mathrm{rot}}$) chosen to fall into our weak-non-synchronicity
($\Omega_{\mathrm{rot}}=0.9\Omega_{0}$ \- green, and
$\Omega_{\mathrm{rot}}=2.0\Omega_{0}$ \- teal) and strong-non-synchronicity
($\Omega_{\mathrm{rot}}=0.5\Omega_{0}$ \- yellow, and
$\Omega_{\mathrm{rot}}=4.0\Omega_{0}$ \- orange) regimes.

###  4.1 Zonal-Mean Zonal-Winds

We start by exploring how the zonal-mean zonal-wind changes as we increase the
difference between the rotational and orbital angular rotation rates (i.e. as
we increase the non-synchronicity).
To that end we plot, on the top row of Figure 12 the zonal-mean zonal-wind for
models with both a small difference between the orbital and rotational angular
rotation rates ($\Omega_{\mathrm{rot}}=0.9\Omega_{\mathrm{orb}}=0.9\Omega_{0}$
\- left - weakly non-synchronous) and a large difference
($\Omega_{\mathrm{rot}}=4\Omega_{\mathrm{orb}}=4.0\Omega_{0}$ \- right -
highly non-synchronous). Further, to aid in our comparisons, we also plot
models with the same rotation rates, but with their orbital periods adjusted
such that the rotation is once again synchronous
($\Omega_{\mathrm{rot}}=\Omega_{\mathrm{orb}}=0.9\Omega_{0}$ on the left and
$\Omega_{\mathrm{rot}}=\Omega_{\mathrm{orb}}=0.9\Omega_{0}$ on the right).
We start in the weakly non-synchronous regime, where we find a zonal-mean
zonal-wind (12(a)) that is very similar to its synchronously-rotating
counterpart (12(c)), with differences on the order of that found when changing
the averaging period of the temporal mean. This suggests that, when the
difference between the rotational and orbital periods is small, the day-night
temperature forcing that develops due to the strong stellar irradiation is
sufficient to drive a long-lived standing Rossby and Kevin wave pattern
leading to the development of a super-rotating jet equivalent to that found in
the synchronous regime.
However as we increase the difference between the orbital and rotational
angular rotation rates, we find that the differences between non-synchronous
(12(b)) and synchronous (12(d)) models with the same planetary rotation rate
starts to grow. In this highly non-synchronous regime, we find that: a) both
eastwards (equatorial jet) and westwards (mid-latitude) winds are slower than
in the synchronous regime and b) the weak eastwards equatorial jet that does
develop is notably shallower than its synchronous counterpart, which in turn
implies that vertical enthalpy advection/heating will be weakened/suppressed
(see subsection 4.2 and subsection 4.4). These changes suggest that the time-
dependent shift in the outer atmospheric irradiation (caused by the difference
between the rotational and orbital period) hampers the formation of the
standing wave pattern which drives both on and off-equator dynamics. We
investigate the exact nature of these changes in subsection 4.3.

We can better understand how a difference between the rotational and orbital
periods can affect the shape/speed of the zonal jet by looking at the
latitudinal variations in the zonal-mean outer atmosphere wind, as shown in
Figure 13. Comparing this with 2(b), we can see that, when the level of non-
synchronicity is small (e.g.
$\Omega_{\mathrm{rot}}=0.9\Omega_{\mathrm{orb}}$), the jet closely resembles
that found in the synchronous regime. However as the difference between the
rotational and orbital periods grows, things start to change, with both the
peak jet speed and jet width decreasing. Eventually we reach the point
($\Omega_{\mathrm{rot}}=4.0\Omega_{\mathrm{orb}}$) that off-equator
counterflows have almost dropped to zero, which reinforces our conclusion that
it is a lack of off-equator standing-waves which leads to the lack of a strong
super-rotating jet in highly non-synchronous atmospheres.
Note that the influence of rotation can also be seen in Figure 13, with models
in which we double/half the rotation rate with respect to the orbital period
showing similar peak jet speeds and off-equator flows, but different jet
widths: the jet appears wider in the more slowly rotating model due to the
increased influence of divergent winds on the dynamics, as was seen in the
synchronous regime (subsection 3.4). A similar story holds true for the cases
discussed here. For example, our model with
$\Omega_{rot}=1.1\Omega_{\mathrm{orb}}$ is very similar to the case with
$\Omega_{\mathrm{rot}}=0.9\Omega_{\mathrm{orb}}$, just with a slightly
narrower jet. This is because, in both cases, the level of non-synchronicity
is low so the main driver of differences is the relative influence of
rotation.

(a) $\Omega_{\mathrm{rot}}=0.9\Omega_{0}$ \- Non-Synchronous

(b) $\Omega_{\mathrm{rot}}=4.0\Omega_{0}$ \- Non-Synchronous

(c) $\Omega=0.9\Omega_{0}$ \- Synchronous

(d) $\Omega=4.0\Omega_{0}$ \- Synchronous

Figure 14:  Zonally and temporally averaged meridional circulation
streamfunctions for four HD209458b-like models: two non-synchronous models
(top), one in the weak-non-synchronicity regime (left) and one on the strong-
non-synchronicity regime (right), paired with two synchronous models at the
same rotation rates (bottom).

###  4.2 Meridional Circulation

We next explore how the changes induced by a difference between the rotational
and orbital angular rotation rates impacts the meridional mass stream-
function, as shown in Figure 14.

As was the case for the zonal-mean zonal-wind, when the difference between the
rotational and orbital periods is small (i.e. in the weakly non-synchronous
regime), we find a circulation profile that is broadly equivalent to that
found in for a synchronous model (Figure 4), just a little weaker. In turn,
this suggests that vertical enthalpy advection, and hence deep heating, should
remain significant. We explore if this is the case below (subsection 4.4).
However, as the difference between the rotational and orbital periods grows,
the above no longer holds true: Whilst we do find general agreement, visually,
with our equivalently rotating synchronous model, including the development of
an additional circulation cells in each hemisphere as the influence of
rotation increases, we also find that the strength of these circulations is
significantly reduced, reaching a peak difference of almost 2 orders of
magnitude. Furthermore, when we look at snapshots instead of the temporal
average, we find that the circulation is highly time dependent, particularly
in the deep atmosphere. Taken together, these two factors suggest that the
vertical advection of enthalpy is likely to be significantly suppressed as the
difference between the rotational and orbital periods grows.

(a) 0.016 bar

(b) 0.2 bar

Figure 15:  Temporally averaged zonal wind (arrows) and temperature profile
(map) at two different pressures (0.0026 bar - left, and 0.2 bar - right) for
a non-synchronous HD209458b-like model in which the difference between the
rotational and orbital periods is large
($\Omega_{\mathrm{rot}}=4\Omega_{\mathrm{orb}}=4\Omega_{0}$). Here we have
forgone showing the weakly non-synchronous regime as we have already
established its similarity to its synchronous equivalent. Note that the x-axis
of these plots is relative-longitude, indicating that we have taken the
temporal mean of the wind/temperature profile in the frame of reference of the
irradiation.

###  4.3 Horizontal-Wind and Temperature Profiles

In order to better understand how differences between the rotational and
orbital periods impact the dynamics, we next explore how the horizontal
wind/temperatures profiles in the outer atmosphere (where the influence of the
time-dependent radiative forcing profile is likely to be largest) change with
increasing non-synchronicity.

Briefly, when the difference between the orbital and rotational periods is
small, we find a wind/temperature structure that is almost identical to that
found in the classical hot Jupiter regime. This includes a strong equatorial
jet which is driven by off-equator $m=1$ standing wave circulations, as well
as the classical thermal butterfly. Note that we find slightly more asymmetry
in the flows here than in the synchronous case, asymmetry which we attribute
to the time-dependent radiative forcing profile at least slightly disrupting
the $m=1$ standing wave structure. Other evidence for the time-dependent
radiative forcing can be found in the temperature profile of the non-
synchronous mean night-side, which a hint hotter here than in an equivalent
synchronous model, likely as a result of remnant thermal energy (that is still
dissipating) from when that physical region of the atmosphere was irradiated.

On the other hand, when the difference between the rotational and orbital
periods is large, other than in the fully radiatively controlled regions of
the outer atmosphere, significant changes to the wind and temperature
structure become evident, changes which we can be seen in Figure 15. For
example, at $0.016$\mathrm{bar}$$, we find strong thermal ‘wings’ that extend
from mid-latitudes on the ‘mean’ (where the mean is taken in the frame of
reference of the irradiation) day-side to near the equator on the ‘mean’
night-side, ‘wings’ which trace the relatively weak off-equator standing wave
circulation the develops here. These ‘wings’ only increase in strength as we
move deeper into the atmosphere ($0.2$\mathrm{bar}$$ \- 15(b)), where we find
that the non-synchronous mean day-side temperature profile has becoming
increasingly smeared eastwards of the (moving) sub-stellar point. Whilst this
description make the temperature profile sound a little like the classical
thermal butterfly, differences in advective transport mean that this is not
the case.
The formation of the classical thermal butterfly is driven by the eastwards
advection of heat at the equator and the westwards advection of heat at higher
latitudes. However, as seen in subsection 4.1 and 15(b), the horizontal winds
are much weaker in a highly non-synchronous atmosphere than its synchronous
equivalent. For example, the eddy wind in our
$\Omega_{\mathrm{rot}}=4\Omega_{\mathrm{orb}}=4\Omega_{0}$ model is a third of
that found in the synchronous
$\Omega_{\mathrm{rot}}=\Omega_{\mathrm{orb}}=4\Omega_{0}$ model
($\left|u_{e}\right|=63$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ vs
$\left|u_{e}\right|=179$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$), whilst the
zonal component is less than a quarter of its synchronous counterpart
($\left|u_{z}\right|=146$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ vs
$\left|u_{z}\right|=645$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$). Instead,
analysis of snapshots of the temperature/wind profiles reveal that the ‘mean’
dynamics found in Figure 15 are the result of the sub-stellar point sweeping
across the planets ‘surface’, from east to west, at a rate that depends upon
the difference between the rotational and orbital angular rotation rates. This
leads to the smeared thermal structure seen in 15(b), which occurs due to
residual night-side temperature from when that region of the atmosphere was
irradiated (i.e. the same effect which lead to slight night-side heating in
the weakly non-synchronous regime). Essentially, we find that increasing the
difference between the orbital and rotational periods decreases the day-night
temperature difference by acting as a form of ‘advection’ from the day-side to
the night-side.
Note that the dynamics are slightly different when the rotational period is
less than the orbital period. This occurs for two reasons, the first is that
the direction of the hot-spot shift is reversed westwards rather than
eastwards) and the second in the decreasing influence of rotation at low
$\Omega_{\mathrm{rot}}$ resulting in divergent rather than rotational
dynamics. Note that the aforementioned smearing of the temperature profile by
the shifting sub-stellar point, as well as the above factors, means that we
also no longer find any evidence for the formation of the night-side hot-spot
that was a defining feature of our slow-rotation-regime models (subsection
3.3).
Overall we find that, regardless of its sign, when the difference between the
rotational and orbital angular rotation rates is large the underlying dynamics
which lead to the features found in the synchronous regime are unable to
develop.

Figure 16: Global mean vertical enthalpy flux for 13 non-synchronous
HD209458b-like models spanning the rotation range
$\Omega_{\mathrm{rot}}=0.5\Omega_{Orbit}\rightarrow 4\Omega_{Orbit}$, where
$\Omega_{Orbit}=\Omega_{0}$ is held constant. Note that negative fluxes
indicate net downward enthalpy transport.

###  4.4 Vertical Enthalpy Transport

We finish our analysis with a brief investigation into how the above changes
to the flows and circulations due to a difference between the rotational and
orbital periods impacts the global mean vertical enthalpy flux, as shown in
Figure 16.
When the difference between the rotational and orbital periods is small, we
find global mean vertical enthalpy fluxes that are at most 25$\%$ weaker than
that found in an equivalent synchronous model. However as the difference
between the rotational and orbital periods grows, and we enter the highly non-
synchronous regime in which the horizontal dynamics have significantly slowed
relative to a synchronous equivalent, we find that the vertical enthalpy
transport has become increasingly suppressed, eventually dropping by over two
orders of magnitude when
$\Omega_{\mathrm{rot}}=4\Omega_{\mathrm{orb}}=4\Omega_{0}$.
This suggests that only weakly non-synchronous atmospheres should exhibit any
significant deep heating/radius inflation. And indeed an analysis of the deep
Temperature-Pressure profiles reveals this to be the case, with our weakly
non-synchronous models showing signs of warming from their initial adiabat
whilst the deep atmospheres of our highly non-synchronous models are highly
sensitive to even the weakest of deep radiative forcing, suggesting that the
link between the deep and outer atmosphere is nebulous at best in these
models. Note that, as reflected in the weaker vertical enthalpy fluxes, we do
find that the deep adiabats of our weakly non-synchronous atmospheres is
slightly cooler than a synchronous equivalent.

##  5 Discussion and Conclusion

In this work, we have investigated how the planetary rotation rate affects the
atmospheric dynamics, including the vertical energy transport, of an
HD209458b-like hot Jupiter (with $\Omega_{0}=2.1\times
10^{5}\,\mathrm{s^{-1}}$ based upon the model of Sainsbury-Martinez et al.
2019) for both synchronous ($\Omega_{\mathrm{rot}}=0.125\Omega_{0}\rightarrow
40\Omega_{0}$ \- 28 models) and non-synchronous
($\Omega_{\mathrm{rot}}=0.5\Omega_{0}\rightarrow 4\Omega_{0}$ \- 13 models)
rotation.
This analysis revealed that, for the rotation rate ranges considered, our
models fall into one of a number of distinct regimes.

For synchronous rotation we find three distinct dynamical regimes: slow
rotation when $\Omega_{\mathrm{rot}}<0.5\Omega_{0}$, the classical hot Jupiter
regime when $\sim 0.5\Omega_{0}<\Omega_{\mathrm{rot}}<\sim 5\Omega_{0}$ (an
analysis of which can be found in the main text), and the rapidly rotating
regime when $\Omega_{\mathrm{rot}}\gg 5\Omega_{0}$.

We find that the slowly rotating regime is characterised by a slower zonal-
mean zonal-jet (1(a)) which does not extend deep into the atmosphere
($P>0.5$\mathrm{bar}$$ is quiescent). The underlying cause of this slow wind
can be understood via the helmholtz wind decomposition, which reveals that the
strongest component of the horizontal wind, in the outer atmosphere, is the
divergent component associated with a global overturning circulation as heat
rises on the day-side, diverges from the sub-stellar point, before converging
on the night-side, just to the west of the sub-stellar point. Note that the
exact location at which the winds converge is dependent upon the rotation
rate, with the convergence point shifting westwards towards the eastern
terminator as the influence of rotation grows. This divergent wind acts as the
main driver of horizontal enthalpy transport, leading to the formation of a
rather unexpected feature: a night-side hot-spot (5(g) and 10(a)). This night-
side hot-spot appears to be a robust feature of the slowly rotating regime and
plays a role in shaping the mid-atmosphere dynamics. For example, around
$0.2$\mathrm{bar}$$, we find that the night-side hot spot is hotter than any
point on the day-side, an effect which reshapes the winds, driving a strong
night-day flow.
As a result we find that the global overturning circulation, associated with
the divergent day-side wind and a broad sub-stellar point upflow, is closed in
the mid-atmosphere, helping to explain the quiescent deep atmosphere. This
effect can be seen in the meridional circulation profile, where we find an
equatorial downflow that is confined to low pressures, and horizontal mixing
at the bottom of the jet, where the night-side hot-spot forms. Finally this
lack of a strong equatorial downflow that connects the outer atmosphere with
the deep, $P>10\,$\mathrm{bar}$$, atmosphere means that the net deep heating
is also weak (11(a) and 11(d)), leading to a deep atmosphere that is not
significantly heated (10(a)), thus suggesting that slowly rotating warm/hot
Jupiters should exhibit little to no radius inflation.
It is important to note that this wind structure and the associated night-side
hot spot may also have implications beyond suppressing radius inflation. For
example, linked with the night-side hot-spot is a thermal inversion and cold-
trap (10(a)), an effect which may have important implications for observations
of these planets and in particular their atmospheric chemistry: these
observations are starting to become possible thanks to next-generation
telescopes, such as JWST, that allow us to characterise hot/warm gaseous
exoplanets on longer orbits around hotter stars.

On the other hand, we find that the rapidly rotating regime is characterised
by a narrow, but deep, equatorial jet braced by easterly counterflows (1(c)).
This is a result of the increasing influence that the Coriolis effect plays on
the atmospheric dynamics, suppressing high-latitude off-equator flows.
Specifically we find that, as the rotation rate rises, the standing $m=1$
Rossby and Kelvin waves which drive the archetypal super-rotating jet become
increasingly tilted and confined to low latitudes, leading not only to the
narrower jet seen here as the circulation cells become latitudinally
compressed, but also a weakening of the overall equatorial winds as the
angular momentum pumping (from mid-latitudes to the equator) mechanism becomes
less efficient. Note that this compression only grows as we increase the
rotation rate, leading to noticeably slow jets at very high
$\Omega_{\mathrm{rot}}$ (7(a)). This latitudinal compression of the zonal-
winds and off-equator flows also impacts the advection of enthalpy away from
the sub-stellar point. For example, we find that the thermal butterfly
(5(c)/h/l) has become somewhat confined to equatorial regions, with eastwards
advection at the equator, westwards/polewards advection at low latitudes and
almost no advection at higher latitudes where the Coriolis forces are
strongest (and winds are weakest).
Vertically, the latitudinal compression and resulting banding of the zonal-
mean zonal-wind leads to a meridional circulation profile (4(c)) which
contains multiple circulation cells in each hemisphere, although these
circulations cells do still combine at the equator to drive a weakened
downflow at all pressures, connecting the outer and deep atmospheres. Note
that, much like the zonal wind, the meridional circulation is significantly
weakened at high latitudes thanks to the Coriolis effect, hence most vertical
mixing is also confined to low-latitudes (i.e. the equatorial region). This
has important implications for the vertical enthalpy transport and hence
heating of the deep atmosphere, both of which we find to be globally weak
(9(l), 11(f), and 10(c)). Note however that this is just the global value. If
we instead consider the equatorial mean (where vertical mixing is strongest),
we find a net downward enthalpy flux comparable to that found in the classical
hot Jupiter regime, and which leads to slight warming of the equatorial
regions deep T-P profile (and only the equatorial regions due to the
quiescence of the off-equator deep atmospheric winds). As such, we might
expect such objects to be slightly puffed up near the equator, although such
an effect is likely to remain minor and may be entangled with other effects
that rotation and irradiation has on the atmospheres shape.
This may help to explain the radius inflation, or lack-thereof, observed for
ultra-hot Jupiters: planets with an orbital period of less than around 20
hours tend to exhibit little to no radius inflation e.g. Wasp-18b - Arcangeli
et al. 2019 \- or Wasp-43b - Hellier et al. 2011), whereas planets with
orbital periods of $>22$ hours start to show signs of inflation (e.g.
Wasp-103b - Gillon et al. 2014 \- or KELT-16b - Oberst et al. 2017), similar
to the transition seen with rotation rate here, although so far our models
have not identified such a tight transition. This is also in agreement with
the results of Sainsbury-Martinez et al. (2021), who found similar dynamics
and a similar lack of radius inflation for SDSS1411b.

Thanks to JWST, as well as other next-generation ground and space based
telescopes, there is likely to be in influx of long orbital period hot/warm
Jupiters in the near future. Not only would these objects likely fall into the
slowly rotating regime, it is also increasingly likely that at least some will
not be perfectly synchronously rotating. This can occur because the
synchronisation time-scale increases with orbital radii, approaching the
system age for a hot/warm Jupiter orbiting a sun-like star at
$0.2$\mathrm{ua}$$. There are also a number of scenarios which might lead to a
non-synchronous but highly irradiated hot Jupiter being observed. For example,
we know that hot Jupiters cannot have formed so close to their host star and
hence must have undergone planetary migration (Lin et al. 1996; Heller 2019).
If we where to observe a hot Jupiter that had recently, or is still
undergoing, migration, then it might not yet have had time to synchronise.
As a result, and since many of the dynamics observed in hot Jupiter
atmospheres are directly or indirectly linked to the strong day/night thermal
forcing (from the day/night divergent flows in the slowly rotating regime to
the thermally driven, $m=1$ Kelvin and Rossby standing waves which are
responsible for the equatorial jet when rotational effects dominate), we next
investigated how varying levels of non-synchronicity, as characterised by a
difference between the varying planetary rotation rate and a fixed orbital
period affects atmospheric dynamics. Here we found two additional regimes,
depending upon the difference between the planetary rotational and orbital
periods: when the difference is small, on the order of a few percent ($<15\%$)
of the smaller period, we find dynamics that are qualitatively similar to the
synchronous regime, just a little weaker. On the other hand when the
difference is large, we find significant differences in the dynamics,
especially in the driving of the super-rotating jet.

Briefly, when the difference between the rotational and orbital periods is
small, we find dynamics, including vertical enthalpy transport, that are
structurally similar to those found in the synchronous regime, but just a
little bit weaker. The biggest difference comes about when we look at the
horizontal temperature profile in the outer atmosphere, where we find that the
‘sweeping’ of the stellar irradiation essentially acts as a form of day-night
transport, with residual heat leading to a warmer night-side and hence a
reduced day-night temperature difference.
The above holds true when we also vary $\Omega_{\mathrm{orb}}$. As such, we
expect small difference between the rotational and orbital periods to have
little effect on observable features, such as the phase curve (hot-spot
location) or radius. Furthermore, since the synchronisation timescale rapidly
decreases with decreasing orbital radii it is highly likely that the majority
of warm/hot Jupiters detected in the near future will be either synchronously
rotating or only weakly non-synchronous. Whilst this is good in one sense,
i.e. we can understand the dynamics of such objects quite easily, it also
means that, with current tools, there is a distinct possibility that some
weakly non-synchronous objects (which are potentially young, depending upon
the local synchronisation time-scale), might be erroneously labelled as
tidally-locked.

But what about a scenario in which a planet has recently migrated onto a
shorter orbit and not yet synchronised its rotation with its host star,
leading to a large difference between the rotational and orbital angular
rotation rates? Here our models suggest that the dynamics should be distinct
from those found in the synchronous regime. For example, we find a zonal-mean
zonal-jet (12(b)) which is both slower and shallower than its synchronous
counterpart. This has a knock-on effect on the vertical dynamics, weakening
both the meridional circulation and vertical enthalpy flux, the latter of
which rapidly decreases as the difference between the rotational and orbital
periods grows (Figure 16). The underlying cause of these changes can be best
understood by exploring the horizontal wind/temperature profiles in the outer
atmosphere (Figure 15). One immediate observation is that the thermal
butterfly that typically forms in the mid-atmosphere of non-slowly-rotating
hot Jupiters is not present. Instead, we find that the temperature profile has
become longitudinally smeared, with the strength of the smearing depending
upon $\Delta\Omega$: i.e. as the difference between the rotational and orbital
periods increases, the time since the night-side was last irradiated gets
smaller, leading to increased residual heating and a hot-spot that appears to
trail the irradiation. Not only does this have the effect of reducing the
strength of the day/night forcing, it also prevents the formation of the
standing-wave structures which drive much of the dynamics of hot Jupiters,
including the super-rotating jet and hence meridional circulation and vertical
enthalpy flux. This can also shift the transition between the rotational-
dominated and divergence-dominated dynamical regimes to faster planetary
rotation rates, which will lead to additional changes in the strength of the
vertical advection, since divergence-dominated winds are less efficient at
vertical transport.
Overall, this suggests that hot/warm Jupiters with large differences between
their rotational and orbital periods should be observationally distinct from a
synchronously rotating planet at the same orbital radius. This includes both a
reduced phase curve amplitude as well as a reduced hot-spot offset, the latter
of which could be either to the east or the west of the sub-stellar point
depending upon the sign of the difference between the rotational and orbital
periods. We might also expect to see a reduced level of radius inflation for
such objects thanks to the reduced vertical heat flux, although this
conclusion is more tenuous since newly formed Jupiter-like objects which have
recently undergone planetary migration are likely still dissipating excess
energy, associated with for example gravitational collapse, from the formation
process, meaning that their envelopes are likely puffed up (Szulágyi et al.
2016). Such a scenario could be confirmed by comparing observed radii with
planetary formation models, such as those of Fortney et al. (2007).

In this work we have discussed how rotation alone can significantly impact and
alter the dynamics of Hot Jupiters, ranging from the outer atmosphere wind and
temperature structure, to the vertical transport of enthalpy. The latter
effect may be of particular importance when trying to explain variations in
hot Jupiter radii found at fixed surface irradiation, in particular the lack
of inflation observed for some very-short-orbit hot Jupiters and hot Brown
Dwarfs.
One interesting discovery which might have significant implications for
atmospheric dynamics and chemistry is the formation of a night-side hot-spot
(via divergent day-night winds) in the mid-atmosphere of our slowly and
synchronously rotating regime models. If such a night-side hot-spot is a
robust feature of more slowly rotating hot/warm Jupiters, something that we
wish to investigate using a next-generation GCM (such as expeRT/MITgcm) that
robustly models radiative heating and cooling at all pressure levels, the
associated temperature inversion may significantly alter atmospheric chemistry
by acting, for example, as a cold trap and hence reshaping the vertical
distribution of key observable models. Understanding such an effect would
likely require modelling the atmospheres of slowly rotating hot/warm Jupiters
with a GCM that includes both robust radiative transport and dis-equilibrium
chemistry, something that is currently far too computationally expensive to
robustly investigate.

###### Acknowledgements.

F. Sainsbury-Martinez and P. Tremblin would like to acknowledge and thank the
ERC for funding this work under the Horizon 2020 program project ATMO (ID:
757858). F. Sainsbury-Martinez would also like to thank UK Research and
Innovation for additional support under grant number MR/T040726/1.
The authors also wish to thank Idris, CNRS, University Paris-Salcay, and Mdls
for access to the supercomputers Poincare, and Ruche, without which the long
time-scale calculations featured in this work would not have been possible.
Additionally this work was granted access to the HPC resources of IDRIS (Jean-
Zay) and CEA-TGCC (Irene/Joliot-Curie) under the 2021/2022 allocation -
A0100410870 made as part of the GENCI Dari A10 call.

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