# Longitudinally asymmetric stratospheric oscillation on a tidally locked
exoplanet

Maureen Cohen School of GeoSciences, The University of Edinburgh
Edinburgh, EH9 3FF, UK  Centre for Exoplanet Science, The University of
Edinburgh, UK  Massimo A. Bollasina School of GeoSciences, The University of
Edinburgh
Edinburgh, EH9 3FF, UK  Paul I. Palmer School of GeoSciences, The University
of Edinburgh
Edinburgh, EH9 3FF, UK  Centre for Exoplanet Science, The University of
Edinburgh, UK  Denis E. Sergeev Department of Mathematics
College of Engineering, Mathematics, and Physical Sciences, University of
Exeter
Exeter, EX4 4QF, UK  Ian A. Boutle Department of Astrophysics
College of Engineering, Mathematics, and Physical Sciences, University of
Exeter
Exeter, EX4 4QF, UK  Met Office, Fitzroy Road, Exeter EX1 3PB, UK  Nathan J.
Mayne Department of Astrophysics
College of Engineering, Mathematics, and Physical Sciences, University of
Exeter
Exeter, EX4 4QF, UK  James Manners Met Office, Fitzroy Road, Exeter EX1 3PB,
UK  Global Systems Institute,
University of Exeter
Exeter, EX4 4QF, UK Maureen Cohen<EMAIL_ADDRESS>
###### Abstract

Using a three-dimensional general circulation model, we show that the
atmospheric dynamics on a tidally locked Earth-like exoplanet, simulated with
the planetary and orbital parameters of Proxima Centauri b, support a
longitudinally asymmetric stratospheric wind oscillation (LASO), analogous to
Earth’s quasi-biennial oscillation (QBO). In our simulations, the LASO has a
vertical extent of 35–55 km, a period of 5–6.5 months, and a peak-to-peak wind
speed amplitude of -70 to +130 ms-1 with a maximum at an altitude of 41 km.
Unlike the QBO, the LASO displays longitudinal asymmetries related to the
asymmetric thermal forcing of the planet and to interactions with the
resulting stationary Rossby waves. The equatorial gravity wave sources driving
the LASO are localised in the deep convection region at the substellar point
and in a jet exit region near the western terminator, unlike the QBO, for
which these sources are distributed uniformly around the planet.
Longitudinally, the western terminator experiences the highest wind speeds and
undergoes reversals earlier than other longitudes. The antistellar point only
experiences a weak oscillation with a very brief, low-speed westward phase.
The QBO on Earth is associated with fluctuations in the abundances of water
vapour and trace gases such as ozone which are also likely to occur on
exoplanets if these gases are present. Strong fluctuations in temperature and
the abundances of atmospheric species at the terminators will need to be
considered when interpreting atmospheric observations of tidally locked
exoplanets.

Exoplanets (498) — Exoplanet atmospheres (487)

††software: We used the Iris python package to manage and analyse model output
data (MetOffice, 2010). We used the windspharm python library to perform
Helmholtz decompositions to characterise the mean circulation (Dawson, 2016)
and NASA’s Planetary Spectrum Generator to simulate transit spectra
(Villanueva et al., 2018).

##  1 Introduction

Rocky planets orbiting M-class stars have become a prime target for exoplanet
research due to their prevalence and potential habitability (Yang et al.,
2020; Schwieterman et al., 2018; Kopparapu, 2013; Heath et al., 1999; Scalo et
al., 2007; Wandel, 2018). As the habitable zones (HZ) of cool M stars are
found close to the host star, planets within the HZ have a high chance of
being tidally locked (Barnes, 2017; Pierrehumbert & Hammond, 2019). These
planets receive stellar irradiation on only one hemisphere, an orbital
configuration not found for planets in the solar system, which has a
substantial impact on the planet’s atmospheric dynamics and chemistry.

An accurate understanding of atmospheric dynamics is important both to
assessing the theoretical habitability of tidally locked rocky planets and
interpreting observations. Theoretical studies of tidally locked rocky
exoplanets have used 3-D atmospheric general circulation models (GCMs) to
explore their possible atmospheric dynamics and climate states (Koll & Abbot,
2016; Boutle et al., 2017; Wolf, 2017; Komacek & Abbot, 2019). A key finding
of GCM studies is that tidally locked planets within the HZ of M-class stars
can retain temperate climates in spite of their uneven thermal forcing as a
result of winds redistributing heat from their hot daysides to their cold
nightsides (Merlis & Schneider, 2010; Pierrehumbert & Hammond, 2019;
Wordsworth, 2015; Joshi et al., 1997). This day-night temperature gradient of
a tidally locked exoplanet has already been observed in some cases, even for a
rocky planet (Demory et al., 2016). However, most past studies have examined
mean climate states, masking time-dependent phenomena which may impact
observations by upcoming instruments such as the James Webb Space Telescope’s
Near Infrared Spectrograph (NIRSpec) (Mollière et al., 2017; Morley et al.,
2017) and the ESA Atmospheric Remote-sensing Infrared Exoplanet Large-survey
mission (Venot et al., 2018; Tinetti et al., 2018). As observations are
performed in a narrow time window, modes of variability with long periods such
as analogues of the Earth’s quasi-biennial oscillation (QBO) may result in
observed temperatures or abundances that differ by an order of magnitude or
more from predictions based on mean GCM states. Other studies have used GCMs,
coupled with synthetic planetary spectrum generators, to simulate observations
of transmission spectroscopy for transiting exoplanets (Suissa et al., 2020;
Arney et al., 2017; May et al., 2021; Boutle et al., 2020; Lines et al.,
2018), enhancing the value of these models as an aid to interpreting
observational data (Brogi et al., 2016; Louden & Wheatley, 2015). However,
gaps in our understanding of the physics driving atmospheric phenomena on
tidally locked planets, partly driven by differences in model structure and
parameterisations (Sergeev et al., 2020; Fauchez et al., 2021; Turbet et al.,
2021; Sergeev et al., 2021), may cause misinterpretations of atmospheric
observations.

Global 3-D models of tidally locked planets exhibit an area of deep convection
around the substellar point (Hammond & Lewis, 2021; Yang et al., 2013;
Kopparapu et al., 2016), leading to permanent cloud cover and heavy
precipitation in this region (Boutle et al., 2017; Sergeev et al., 2020;
Labonté & Merlis, 2020; Yang et al., 2013). The tropospheric and stratospheric
circulation is characterised by stationary Rossby waves of zonal wavenumber 1
induced by the day-night thermal forcing (Merlis & Schneider, 2010; Showman &
Polvani, 2011; Hammond & Lewis, 2021). These waves have been identified as a
driving source of the superrotating equatorial jets that develop in tidally
locked simulations of both Earth-like planets and hot Jupiters (Showman &
Polvani, 2010, 2011; Tsai et al., 2014; Debras et al., 2019, 2020). The jets
are associated with two pairs of cyclones-anticyclones: one pair forms in the
northern hemisphere, rotating clockwise on the dayside and anticlockwise on
the nightside, while the other pair forms in the southern hemisphere and
rotates in the opposite directions. The cyclones contribute a zonal wind
vector component near the equator that points westward in the substellar
region and eastward in the antistellar region. This results in a background
flow which is more eastward on the nightside than on the dayside, a pattern
that holds true in both the troposphere and the stratosphere. The day-night
thermal forcing additionally creates a region of rising air on the dayside and
subsiding air on the nightside (Hammond & Lewis, 2021; Joshi et al., 1997).

On Earth, the background flow determines the zonal structure of the QBO. In
the QBO, the stratospheric zonal wind forms a pattern of vertically stacked
jets flowing in opposite directions. The pattern propagates downwards from
roughly $20-40$ km at a speed of $1$ km per month such that the wind direction
at a given height reverses over time with a mean periodicity of $26-28$
months, with substantial variation from cycle to cycle (Dunkerton et al.,
2015; Braesicke, 2015). The oscillation of the zonal wind is accompanied by
oscillations in air temperature and trace gases. The basic mechanism driving
the QBO was described by Lindzen & Holton (1968) and Plumb (1977) and
reproduced in a laboratory setting by Plumb & McEwan (1978). The minimum
requirements for the QBO to occur are a background flow that is a function of
altitude and at least two gravity waves propagating in opposite directions
horizontally, as well as upwards vertically (Plumb, 1977). An eastward
propagating gravity wave will travel upwards until it encounters an eastward
perturbation in the background flow, where it is absorbed, accelerating the
flow in the eastward direction. Westward propagating waves pass through this
area of eastward flow until they reach a westward perturbation and are
absorbed in turn. The resulting pattern of stacked jets propagates downwards
over time as the bottom of each shear zone is accelerated and the lowest jet
dissipates due to viscous diffusion (Plumb, 1977; Dunkerton et al., 2015).
Although equatorially trapped Rossby and Kelvin waves contribute some momentum
flux, it is believed that up to $80\%$ of the momentum flux driving the QBO is
provided by convectively generated gravity waves (Dunkerton, 1997; Lane,
2015). Gravity waves are generated when the forces of gravity or buoyancy
react against vertical perturbations in a fluid medium. The waves travel
internally within the fluid until they are absorbed at critical levels where
the velocity of the background flow is comparable to the wave’s horizontal
phase speed (Booker & Bretherton, 1967), making the planet’s mean flow an
important element in the phenomenon.

Here, we describe the first reported simulation of a stratospheric
oscillation, analogous to the QBO on Earth, on a tidally locked planet with an
Earth-like atmospheric composition. This new stratospheric oscillation
consists of periodic reversals in the direction of the zonal wind, an
associated cooling and warming of the equatorial atmosphere, and fluctuations
in the stratospheric water vapour content. While analogous to the QBO in its
causal mechanism, the oscillation displays longitudinal asymmetries arising
directly from the planet’s tidally locked state. Accordingly, we refer to the
oscillation on a tidally locked planet as a longitudinally asymmetric
stratospheric oscillation (LASO) and reserve the term QBO for the phenomenon
on Earth. To put our results into context, throughout the paper we compare and
contrast key features of the tidally locked LASO with the QBO on Earth
(Baldwin et al., 2001).

Section 2 describes the global 3-D model we use to describe atmospheric
dynamics on a tidally locked planet, the metrics we use to characterize the
LASO, and the radiative transfer model we use to simulate JWST observations of
the fluctuations in atmospheric water vapour content caused by the LASO.
Section 3 describes the mechanism underpinning the LASO and characteristics of
the LASO that differ from those of the QBO. We also consider the effect of
LASO-related water vapour fluctuations on transit observations. In Section 4
we discuss the implications of the LASO for future spectral atmospheric
observations and the sensitivity of the LASO to differences in sub-grid scale
model parameterisations. We conclude the paper in Section 5.

##  2 Methods

###  2.1 Model description

We use the Global Atmosphere 7.0 configuration of the Met Office Unified Model
to simulate the atmosphere of a tidally locked Earth-like exoplanet, nominally
Proxima Centauri b (Walters et al., 2019). For brevity, we provide a basic
description of the model and refer the reader to Mayne et al. (2014), Boutle
et al. (2017), and Sergeev et al. (2020) for more details.

We run the model at a horizontal resolution of $2^{\circ}$ latitude by
$2.5^{\circ}$ longitude, and with 60 vertical levels from the surface to up to
85 km (Table 1). The atmospheric levels are identical to those in Boutle et
al. (2017) up to $38$ km. The remaining $47$ km are accounted for by an
additional $22$ levels. Table 2 shows the orbital and planetary parameters
used in this study, following Boutle et al. (2017), with data taken from
Anglada-Escudé et al. (2016) and Turbet et al. (2016). Proxima Centauri b is
simulated as an aquaplanet with a slab ocean (Frierson et al., 2006), with a
heat capacity of $10^{7}$ J K-1 m-2 representing a mixing layer of 2.4 m. We
specify an atmospheric composition of 100 % nitrogen with a trace fixed
$\mathrm{CO_{2}}$ abundance of $5.941\times 10^{-04}\mathrm{kgkg^{-1}}$. The
model includes interactive water vapour and a full water cycle with unlimited
evaporation from the slab ocean, precipitation in the liquid and ice phases,
and a cloud scheme. The stellar spectrum for Proxima Centauri was taken from
BT-Settl (Rajpurohit et al., 2013) with $T_{eff}=3000$ K, g=
$1000\mathrm{ms^{-2}}$, and metallicity $=0.3$ dex. The star is treated as
quiescent. We spin up the model for $900$ Earth days from an equilibrium state
taken from a previous run of the Proxima Centauri b model. The simulation was
determined to be in equilibrium when the incoming and outgoing radiation at
top-of-atmosphere were balanced, evaporation balanced precipitation in the
global mean, and the surface temperature no longer showed a long-term upward
or downward trend. The experiments we report were run for $1800$ Earth days in
total, and we analysed a sampling period of $900$ days after equilibrium,
which is long enough to include multiple oscillations. We define the
substellar point to be $0^{\circ}$ longitude and latitude. The antistellar
point is located at $180^{\circ}$ longitude and the eastern and western
terminators at $90^{\circ}$E and $90^{\circ}$W, respectively. Hereafter, days
refer to Earth days.

Level | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
Height (m) | 10.0 | 49.9 | 130.0 | 249.99 | 410.0 | 610.0 | 849.9 | 1129.9 | 1449.9 | 1810.0
Level | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
Height (m) | 2209.9 | 2650.0 | 3129.9 | 3650.0 | 4210.0 | 4810.0 | 5449.9 | 6129.9 | 6849.9 | 7609.9
Level | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30
Height (m) | 8409.9 | 9249.9 | 10130.0 | 11050.0 | 12001.0 | 13001.0 | 14050.0 | 15130.0 | 16250.0 | 17410.0
Level | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40
Height (m) | 18590.0 | 19770.1 | 20950.3 | 22131.3 | 23313.9 | 24499.5 | 25690.3 | 26889.2 | 28100.2 | 29328.5
Level | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50
Height (m) | 30580.3 | 31863.4 | 33186.9 | 34562.1 | 36001.5 | 37520.0 | 39134.3 | 40863.5 | 42729.0 | 44754.9
Level | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60
Height (m) | 46967.7 | 49396.9 | 52074.8 | 55036.7 | 58321.5 | 61971.1 | 66030.9 | 70550.2 | 75581.7 | 81182.4
Table 1: Vertical levels and heights (layer centres) used in the Unified Model to describe the atmospheric dynamics of an Earth-like tidally locked Proxima Centauri b. Parameter | Value
---|---
Semi-major axis (AU) | 0.0485
Stellar irradiance (W m-2) | 881.7
Orbital period (Earth days) | 11.186
Rotation speed (rad s-1) | 6.501$\times$10-6
Eccentricity ($\cdot$) | 0
Obliquity ($\cdot$) | 0
Radius (km) | 7160
Acceleration due to gravity (m s-2) | 10.9
Table 2: Simulation parameters used to describe an Earth-like tidally locked
Proxima Centauri b.

Earth GCMs typically require both resolved and parameterised waves to generate
a QBO (Garfinkel et al., 2021). Past studies have shown that the Unified Model
can reproduce a realistic oscillation (Scaife et al., 2000, 2002). Our model
can sustain resolved long wavelength gravity waves and inertio-gravity waves
generated by physical and numerical imbalances in the flow (as discussed in
Section 3). The model also includes a sub-grid scale non-orographic gravity
wave parametrisation (Warner & McIntyre, 1996, 2001; Bushell et al., 2015).
Within this parameterisation, an unsaturated spectrum of gravity waves is
launched near the surface and carries a vertical flux of horizontal wave
pseudo-momentum upwards in equal amounts towards the north, south, east, and
west. Upward-propagating flux is diminished by Doppler modification and
density amplification. Wave pseudo-momentum is conserved by an opaque model
lid which sets the outgoing flux to zero (Bushell, 2021). The scheme includes
a scale factor for the vertical flux and a characteristic (spectrum peak)
wavelength which must be tuned to the vertical resolution of the model to
achieve a QBO. We use a factor enhancement for the vertical flux of horizontal
pseudo-momentum of 1.2 and a characteristic wavelength of 4.3 km with model
level heights shown in Table 1. These are the standard values used in the GA
7.0 for Earth with the model levels in Table 1 and we retain them in order to
simulate Proxima Centauri b as an Earth-like planet. The gravity wave scheme
can be run with two different gravity wave source locations: 1) a standard
scheme in which the gravity wave launch flux is globally invariant,
distributed uniformly around the model; and 2) an experimental scheme in which
the gravity wave launch flux is determined by local precipitation, developed
based on an empirical correlation found between gravity waves and
precipitation on Earth (Bushell et al., 2015). Our study is based on the
standard globally invariant scheme. To test whether our results are robust
against the choice of scheme, we ran the experimental precipitation-dependent
scheme and found that the results are comparable with the exception of the
amplitude of the longitudinal asymmetry of the mean flow, which we discuss in
Section 3.

In addition to the control experiment described below, we performed a
sensitivity run to understand the influence of horizontal resolution by
increasing it to $2^{\circ}$ latitude by $2^{\circ}$ longitude. We performed a
separate sensitivity test by decreasing the timestep from 20 to 2 minutes. We
found that these sensitivity runs did not result in major changes from the
control experiment.

###  2.2 Metrics used to characterise the LASO

#### Gravity wave-induced acceleration

To confirm that the reversal in the direction of the stratospheric jets is
caused by absorption of gravity waves, we calculate the acceleration of the
atmosphere in each model gridbox due to gravity waves, $F$, as:

$F=\frac{\partial{(u^{\prime}w^{\prime})}}{\partial{z}},$ (1)

where $u^{\prime}$ is the high-pass filtered zonal anomaly of the zonal wind,
$w^{\prime}$ is the time anomaly of the vertical wind, and _z_ is the height.
This definition is adapted from the formulation of the wave momentum flux
described by Plumb (1977) and used in its original form by Showman et al.
(2019) to identify shear transition zones between alternating jets.

We modified $u^{\prime}$ to better describe the characteristic flow on a
tidally locked planet. The zonal anomaly is usually defined as the deviation
of that quantity from its zonal mean and, in a zonally symmetric baseline
flow, identifies transient phenomena. However, tidally locked planets lack
zonal symmetry. The planet’s stationary Rossby waves skew the zonal anomaly so
that it represents the Rossby waves rather than transient flows. To adapt
$u^{\prime}$ to this asymmetrical background flow, we use a high-pass filter
to remove the influence of waves with zonal wavenumber less than 5 (i.e., long
Rossby waves), following Sugimoto et al. (2021). The remaining waves are
resolved long wavelength gravity and inertio-gravity waves. Additionally, the
zonal anomaly of the vertical wind _w_ would be impractical for identifying
waves because rising air on the dayside and subsiding air on the nightside
cancel each other out when a zonal mean is calculated. We instead use the time
anomaly to distinguish transient, wave-like fluctuations in _w_ from the
background. The resulting metric identifies wave-induced acceleration caused
by resolved gravity and inertio-gravity waves only.

#### Latitudinal extent

We expect that a QBO-like oscillation on an exoplanet may extend to higher
latitudes than Earth’s QBO, covering a larger portion of the terminators,
which may affect transit spectroscopy. To characterise the latitudinal extent
$L$ of the LASO, we use a formula derived by Haynes (1998) for the latitudinal
extent of the QBO:

$L=(\frac{\sigma}{\alpha})^{1/4}(\frac{ND}{\beta_{0}})^{1/2},$ (2)

where $\sigma$ is the frequency of time variation of the applied force,
$\alpha$ is the radiative damping rate, _N_ is the buoyancy frequency, _D_ is
the depth scale of the force, and $\beta_{0}$ is the meridional gradient of
the Coriolis parameter evaluated at the equator. We assume that all quantities
except $\beta_{0}$ are identical to those for Earth, and focus our analysis on
the impact of differences in the Coriolis factor.

#### Period

The period of the LASO is of interest because the duration of the temperature
and atmospheric composition oscillations may affect observations and in
particular could cause discrepancies between repeated observations of the same
target exoplanet. We determine the period $T$ of the LASO using a wavelet
analysis of the wind velocity time series at the altitude where the LASO has
its greatest amplitude. Following Plumb (1977), the period of the oscillation
is inversely proportional to the momentum flux:

$T=\eta\frac{\hat{k}\hat{c}^{3}}{\hat{N}\hat{\mu}\hat{F}},$ (3)

where $\hat{k}$, $\hat{c}$, $\hat{N}$, $\hat{\mu}$, and $\hat{F}$ are scaling
factors for the wavenumber and phase speed of the wave driving the
oscillation, the buoyancy frequency, the thermal dissipation rate, and the
momentum flux, respectively. The parameter $\eta$ is a dimensionless number
that depends on the specific details of the problem. In Plumb’s simple
numerical model, the wavenumber and phase speed of the waves can be specified,
while in the real atmosphere the oscillation is driven by a broad spectrum of
waves whose constituents vary with location and time. However, the inverse
relationship between the period and the momentum flux at the lower boundary of
the QBO shear zone is also accepted to exist for the real QBO (Dunkerton et
al., 2015). We assume all other values are identical to those for Earth and
focus on the impact of the wave momentum flux $\hat{F}$ on the period.

#### Lagrangian Rossby number

To understand how a tidally locked orbital configuration affects where gravity
waves are generated on the planet we use the Lagrangian Rossby number,
$Ro^{(L)}$, to identify resolved gravity wave sources. $Ro^{(L)}$ is not an
exclusive measure of gravity wave sources, but is often used to detect
unbalanced flows which give rise to gravity waves (Zhang et al., 2000; Lin,
2007; Sugimoto et al., 2021). It is defined as the ratio of the acceleration
of a parcel of air to the Coriolis acceleration and essentially measures the
local departure from geostrophy:

$Ro^{(L)}=\frac{\mid\frac{\partial{v_{h}}}{\partial{t}}+v_{h}\cdot\nabla
v_{h}\mid}{f\mid v_{h}\mid},$ (4)

where $v_{h}$ is the horizontal wind vector and _f_ is the Coriolis parameter.

We modify $Ro^{(L)}$ in several respects to adapt the metric to a slowly
rotating planet. Following Sugimoto et al. (2021), we disregard the local wind
tendency term, $\frac{\partial{v_{h}}}{\partial{t}}$, and fix the value of _f_
near the equator ($\leq\pm 10^{\circ}$) equal to _f_ at $10^{\circ}$ to extend
the metric to cover the equatorial region. As the slowly rotating Proxima
Centauri b is not in geostrophic balance, the resulting values of $Ro^{(L)}$
are very large. We normalise $Ro^{(L)}$ by dividing by the minimum value,
which we consider the baseline ageostrophy for our simulation. A larger value
indicates a relative local increase in imbalances in the flow. We also exclude
the top five model levels and latitudes above $\pm 40^{\circ}$ to avoid
interference from instability at the model top and poles.

###  2.3 NASA Planetary Spectrum Generator

To describe the spectrum of the atmospheric water vapour content that might be
observed by the James Webb Space Telescope’s NIRSpec instrument, we use the
NASA Planetary Spectrum Generator (Villanueva et al., 2018), publicly
available at https://psg.gsfc.nasa.gov/. The NIRSpec instrument has a range of
0.6 to 5.3 $\mathrm{\mu m}$, covering water vapour features in the infrared at
1.4, 1.8, and 2.7 $\mathrm{\mu m}$, with spectral resolutions (R) of 100,
1000, and 2700. As inputs to the PSG, we use the basic orbital, planetary, and
stellar parameters for the Proxima Centauri system, as well as pressure,
temperature, altitude, H2O, N2, and CO2 data from our UM simulations. We
generate a transit spectrum for each latitude at both the eastern and western
terminators (i.e., every $2^{\circ}$ of latitude, for a total of 180
individual spectra) and average the outputs to get a final spectrum, following
Suissa et al. (2020).

##  3 Results

###  3.1 Longitudinally asymmetric stratospheric oscillation

Figure 1 shows the wind speed and temperature oscillations in our simulation
of the atmosphere of the tidally locked Proxima Centauri b. The period of the
LASO is much shorter ($5-6.5$ months) than that of the QBO on Earth ($26-28$
months). There is also an asymmetry between the dayside and nightside.
Nightside zonal winds are more positive (eastward) than dayside winds and have
shorter westward phases. Consistent with the theory that describes the QBO,
the dayside-nightside asymmetries in the mean planetary-scale circulation, in
particular the dayside-nightside differences in wind speeds, are the primary
factor responsible for the asymmetry of the LASO. We find that the location of
gravity wave sources also affects the local flow. Figure 1c shows temperature
fluctuations with a peak-to-peak amplitude of $\sim 60-70$ K, more than an
order of magnitude larger than the variation of $\sim 4$ K associated with the
QBO (Dunkerton et al., 2015).

Figure 1: Hovmöller diagrams of a) dayside and b) nightside zonal mean
equatorial winds on the simulated planet and c) the absolute temperature
anomaly at 40 km. The data are sampled every six hours.

To confirm that the LASO is generated by the same physical mechanism as the
QBO, we calculated the wave-induced acceleration due to gravity waves and
compared it to the change in zonal wind direction over time. Where the
oscillation is driven by gravity waves, the direction of the acceleration
caused by gravity waves should match the direction of the change in the zonal
wind throughout the vertical extent of the LASO. Figure 2a shows the filtered
$u^{\prime}$ field with zonal wavenumbers smaller than 5 removed (Section
2.2). We find positive and negative fluctuations in wind speeds due to
resolved long wavelength gravity waves, with shear zones at 30 km, 41 km, and
50 km that correspond to regions where the flow changes direction. Waves with
a leftward tilt correspond to westward-propagating modes and those with a a
rightward tilt correspond to eastward-propagating modes. The band of reduced
$u^{\prime}$ intensity between $15$ km and $30$ km corresponds to the
permanent superrotating equatorial jet in the troposphere. Figure 2b shows the
zonal mean gravity wave-induced acceleration calculated using Equation 1 and
averaged over 90 days, corresponding to approximately half of a LASO cycle.
The contour lines show the change in zonal wind speed over the same 90 days.
Areas of positive (negative) wind direction change match well with areas of
gravity wave-induced eastward (westward) acceleration. Regions below 25$-$30
km show little change in zonal wind speeds and negligible gravity wave-induced
acceleration.

Figure 2: a) Zonal wind anomalies at the equator as a function of altitude,
with zonal wavenumbers $<$5 removed, corresponding to a six-hour snapshot from
simulation day 810 from the simulation sampling period. b) Zonal mean wave-
induced acceleration due to gravity waves, overlaid with the change in zonal
mean wind speeds. The values represent a 90-day mean from days 720 to 810 of
the simulation sampling period.

###  3.2 Latitudinal extent of LASO

The LASO extends to higher latitudes than the QBO on Earth. The latitudinal
extent of the QBO, defined here by the full width at half maximum of a
Gaussian fit to the horizontal cross section of the zonal wind at the height
of the QBO maximum (Schenzinger et al., 2017), is roughly 15∘S-15${}^{\circ}N$
(Braesicke, 2015). Figure 3 shows that the LASO extends from about 60∘S–60∘N
by this definition. As the LASO on a slowly rotating tidally locked planet
extends to higher latitudes, the oscillations in air temperature and in
abundances of atmospheric species will cover more of the terminators and
potentially have a larger impact on observations.

Previous work suggested that at higher latitudes the Coriolis force would
counteract the acceleration due to gravity waves, limiting the QBO to
equatorial regions on the Earth (Lindzen & Holton, 1968). Equation 2 gives a
value for Earth of about $L_{\rm Earth}$=10∘ (Baldwin et al., 2001), which
underestimates the actual extent of 15∘ (Dunkerton & Delisi, 1985) by $\simeq$
33%. The Coriolis force on Proxima Centauri b is an order of magnitude weaker
than on Earth due to the planet’s slower rotational period of 11.2 days
($\Omega$=6.5$\times$10-6 rad/s). If we assume the values of the other
quantities in Eqn. 2 for Proxima Centauri b are identical to those for Earth,
this gives a latitudinal extent of $\sqrt{10}L_{Earth}$ or $\simeq$47∘.
Scaling this value to account for the 33% underestimation gives the
latitudinal range simulated by the model. The broader extent of the LASO
compared to the equatorial QBO agrees with the description of slowly rotating
tidally locked planets as ‘all tropics’ (Showman et al., 2013). A LASO on a
slow rotator may therefore have a larger impact on transit spectroscopy than
one on a more rapidly rotating planet.

Figure 3: Latitudinal extent of the LASO, described by simulation data at a
resolution of six hours.

###  3.3 Period of LASO

Another major difference between the LASO and the QBO is the period of
oscillation. The period of the QBO has a large variability, with a mean value
of 26–28 months (Braesicke, 2015). Our calculations show that the LASO in our
simulation of Proxima Centauri b’s atmosphere has a period of 5–6.5 months.
The period of the QBO is not directly related to the planet’s rotation rate
(Plumb & McEwan (1978) and Eqn. 3). Because a tidally locked planet does not
rotate with respect to its star, the oscillation is also not related to
seasonal and annual variations. The shorter period of LASO instead implies
stronger wave momentum flux and wave-induced acceleration.

On Earth, up to $80\%$ of the QBO forcing originates from convectively
generated gravity waves (Lane, 2015). Figure 4 shows that deep convection in
our simulation of Proxima Centauri b occurs within an area of $30^{\circ}$ of
the substellar point. This area has the shape of a pointed oval and is
asymmetrical, with convection consistently more intense at the western edge of
the region. This region of deep convection is collocated with a gravity wave
source (as shown in Figure 7b). The accepted dependence of the QBO’s period on
the gravity wave momentum flux as expressed in Equation 3 supports the
conclusion that greater wave momentum flux, driven by the intense convection,
accounts for the shorter period of the LASO compared to the QBO on Earth.
Accordingly, we find higher wind speeds in the LASO than the QBO. The QBO has
maximum eastward winds of 15 ms-1 and maximum westward winds of -30 ms-1
(Dunkerton et al., 2015), while the LASO’s eastward wind speeds reach 100 ms-1
and westward winds reach -75 ms-1.

Figure 4: Deep convection indicator, averaged over a 30-day period (1:
convection is present in the gridbox; 0: not present). In the Unified Model,
convection is diagnosed by the surface buoyancy flux and an undilute parcel
ascent from grid points where this flux is positive. It is then categorised as
deep if the parcel reaches neutral buoyancy above $2.5\mathrm{km}$ or the
freezing level (whichever is higher), and if the vertical velocity is greater
than $0.02\mathrm{ms^{-1}}$ within a layer at least $1.5\mathrm{km}$ thick
above the level of neutral buoyancy (Walters et al., 2019).

###  3.4 Longitudinal asymmetry of the LASO

The LASO exhibits longitudinal asymmetries that do not have an analogy on
Earth. We show below that this is related to underlying asymmetries in the
steady-state atmospheric dynamics of tidally locked planets and ultimately to
the asymmetrical thermal forcing.

As described above, the emergence of the QBO is associated with gravity wave
absorption at critical layers in the atmosphere where the speed of the
background flow is similar to the phase speed of the wave, leading to wave
breaking and the deposition of momentum. Longitudes and heights where the
background flow is faster will absorb faster gravity waves and experience
stronger acceleration, perpetuating the asymmetry. This accounts for the
stronger nightside eastward winds seen in Figure 1b, as the nightside
experiences higher eastward winds due to the Rossby wave contribution to the
zonal wind, as well as for the height at which the LASO has its maximum
amplitude (41 km), which is also where the Rossby waves are most intense.

The zonal wind does not change direction simultaneously at every longitude.
Figure 6 shows the temporal variations of the 41 km zonal wind along the
equator at two phase transitions, one from eastward to westward and the other
from westward to eastward flow. In both transitions, longitudes close to the
western terminator transition first, in addition to exhibiting higher wind
speeds. Figure 6a shows that the westward winds begin to appear around 90∘W on
days 125–130, while other longitudes transition around days 135–140. The
converse is also true (Figure 6b). Wind speeds are consistently highest at
90∘–120∘W. Close inspection also shows that the antistellar point has a very
weak westward phase, with wind speeds of only 5–10 ms-1 in some places,
compared to 100–120 ms-1 during the eastward phase. Periods of westward flow
at the antistellar point are comparatively brief. The weakness of the westward
phase here implies lower westward acceleration, which may be due to the
absorption of westward-propagating gravity waves before they travel around the
planet to the nightside, a lack of critical levels available to absorb
westward-propagating modes, an asymmetry in the underlying resolved gravity
wave spectrum itself, or a combination of these mechanisms.

Figure 5: (b) Transition from westward to eastward at 41 km Figure 6:
Equatorial zonal wind speeds at 41 km over two 40-day periods, determined by
six-hourly simulation data. The top panel shows the transition from eastward
to westward winds and the bottom panel shows the transition from westward to
eastward winds. Plots are centred on the antistellar point to highlight the
tendency to resist westward flow in this region.

The amplitude of the LASO is largest at the western terminator and smallest at
the antistellar point (Figure 6). This pattern cannot be explained directly by
the presence of Rossby waves because the equatorial wind vector component
contributed by the Rossby waves is larger at the antistellar point than at the
western terminator. A possible explanation for the greater amplitude and early
phase switching of the oscillation at the western terminator is the presence
of a second gravity wave source in this region. The western terminator is a
jet exit region where both stratospheric and tropospheric jets decrease
rapidly in speed. The steep gradient in wind speeds is caused by the
convergence of the two opposing wind vectors contributed by the Rossby wave
cyclone-anticyclone pair. Jet exit regions are well-known to be sources of
gravity waves (Plougonven & Zhang, 2014). To investigate further, we
calculated the Lagrangian Rossby number (Eqn. 4).

Figure 7a shows that at $41$ km, where the LASO reaches its maximum amplitude,
a strong gravity wave source is present at the western terminator. In the
troposphere, unbalanced flows are found in the region of deep convection
(Figure 7b and Figure 4). The equatorial altitude-longitude cross-section
(Figure 7c) confirms that areas of instability that can generate gravity waves
occur preferentially westward of the substellar point in the troposphere and
around the western terminator in the lower stratosphere. To link wave
acceleration to the jet exit region, Figure 7d shows the gravity wave-induced
acceleration at the equator (as calculated in Figure 2), overlaid with the
zonal wind. The strongest acceleration is found at the jet exit region for
both the tropospheric and the lowest stratospheric jets between 20–45 km. The
western terminator both generates gravity waves due to the slowing of the jet
speeds in this region and also absorbs high-energy gravity waves because this
is where the critical levels exhibit the highest wind speeds. The result is an
unusually unstable region where zonal winds reverse earlier than elsewhere and
may oscillate back and forth in shorter time periods, as seen in Figure 6a
between 125–145 days and 120∘–90∘W .

Figure 7: The Lagrangian Rossby number, normalised by its minimum value in the
dataset and averaged over the same 30-day period shown in Figure 4, at a) 41
km and b) 8 km, and c) the corresponding height-longitude cross-section at the
equator. We exclude the top five model levels and the high latitudes to avoid
including model instability at the model top and poles. d) The height-
longitude cross-section of gravity wave-induced acceleration over the same
time period as Figure 2, overlaid with the mean zonal wind as contour lines.

We also ran the model using the experimental precipitation-dependent gravity
wave scheme to investigate the impact of moving the gravity wave source on the
characteristics of the LASO, as described in Section 2. In the precipitation-
dependent scheme, parameterised gravity waves are launched preferentially from
the convection zone around the substellar point on the dayside, creating an
asymmetry in the parameterised gravity wave source that does not exist in the
standard scheme. While the period, latitudinal extent, amplitude, resolved
gravity waves, and water vapour oscillation did not differ between the
precipitation-dependent and standard schemes, the longitudinal asymmetry in
wind speeds and phase transitions _decreased_ and the LASO became more zonally
symmetric. In the standard scheme, the wind velocity maximum at the western
terminator and the minimum at approximately 30∘E differ by $\sim$100 ms-1 at
41 km, while in the precipitation-dependent scheme they differ by only
$\sim$20 ms-1. The stronger gravity wave source at the substellar point
appears to play a similar role as the source at the jet exit region in
modifying the local flow at higher altitudes, counteracting the westward
component contributed by the dayside cyclone. Although the overall mean
circulation remains longitudinally asymmetrical, this experiment demonstrates
the non-linearity in outcomes resulting from interactions between Rossby waves
and gravity waves in the simulation.

In summary, as discussed above, the LASO is shaped by interactions between
multiple components of the circulation typical of tidally locked planets.
Figure 8 shows a schematic illustration of the interplay between Rossby waves,
the superrotating tropospheric jet, and the gravity wave sources in the
substellar region and at the western terminators. The diagram represents the
mean circulation generated with the standard, globally invariant gravity wave
source.

Figure 8: Schematic diagram of the location and interaction of Rossby waves,
equatorial jets, and gravity wave sources. The perspective is looking downward
from the north pole. The lighter side of the sphere is the dayside and the
darker is the nightside. Red (blue) arrows indicate eastward (westward) air
flow, with larger arrows representing faster wind speeds. Black spirals
represent Rossby waves. The star symbol indicates gravity wave sources.

###  3.5 Atmospheric water vapour oscillations

On Earth, the oscillation of the zonal wind direction is associated with
variations in the amount of atmospheric water vapour of the order of $\pm$10%
(Randel et al., 1998; Wang et al., 2020). An analogous oscillation on
exoplanets is of interest because large fluctuations in atmospheric water
vapour may be observable. In our simulations, specific humidity at 41 km
varies about $\pm$50% on average, with peak oscillations up to 200% of the
mean. This results in oscillations with minimum and maximum values that can
differ by an order of magnitude.

To test whether a fluctuation of this magnitude would be detectable, we
generated synthetic planetary spectra as observed by the NIRSpec instrument
aboard JWST for the above maximum and minimum water vapour concentrations. We
found the fluctuations are too small to be observable for an Earth-sized
planet like Proxima Centauri b, but as QBO-like phenomena have also been
detected on Jupiter and Saturn in the solar system (Showman et al., 2019;
Orton et al., 1991; Leovy et al., 1991), a LASO could potentially occur on a
larger planet with greater potential for observation. This could be
particularly important for short-period, potentially tidally locked, super-
Earth or mini-Neptune-type planets detected by missions such as the Transiting
Exoplanet Survey Satellite (see Bluhm et al. (2021) for a recent example)

##  4 Discussion

Using a global 3-D climate model we show that Earth-like tidally locked
planets can exhibit a longitudinally asymmetric stratospheric wind oscillation
(LASO), analogous to the quasi-biennial oscillation (QBO) on Earth. As QBO-
like oscillations are also known to exist on Jupiter and Saturn (Showman et
al., 2019; Orton et al., 1991; Leovy et al., 1991), our finding suggests a
similar conclusion for gas giants. The LASO reported in our study has
implications for the climatology and observations of tidally locked
exoplanets, as well as for our understanding of GCM results.

###  4.1 Oscillations in meteorological and atmospheric composition

It is well established that the QBO on Earth is responsible for oscillations
in the abundance of many atmospheric species, notably ozone and methane
(Bowman, 1989; Huang et al., 2008; Randel et al., 1998; Li et al., 2009). In
this initial study, we have focused on the properties and acceleration of the
LASO. However, by including atmospheric chemical kinetics, future work could
also explore LASO-induced variations in species important for observations.
For example, Earth’s total ozone column fluctuates by $\pm$1 ppm ($\sim$ 4%)
on the 26–28 month timescale of the QBO. A variation of this magnitude is
unlikely to be detectable with existing instrumentation, but an ozone column
oscillation may be larger in amplitude on a tidally locked planet than on
Earth, in line with our simulation results for water vapour. Studies of ozone
photochemistry on tidally locked planets (Yates et al., 2020) have also shown
that the planet’s nightside accumulates ozone in its cold traps, resulting in
a zonally asymmetrical ozone layer without analogy on Earth and high levels of
ozone in one hemisphere. Future work could explore how this tidally locked
ozone chemistry would interact with oscillations in dayside production of
ozone through stratospheric photochemistry. In addition, LASO-related
temperature anomalies in our simulation reach 40–50 K ($\sim$ 25% deviation
from the mean), while the QBO of temperature on Earth is limited to 1–2 K
(Tegtmeier et al., 2020). Large variations in air temperature will also
potentially affect measurements of thermal emission spectra from the
atmosphere.

###  4.2 Sensitivity to model parameterisations

Gravity wave parametrisation schemes and differences in ability to generate a
QBO-like phenomenon may account for some of the differences found in modelling
studies of the same exoplanet. For example, predicted nighttime surface
temperatures for Proxima Centauri b diverge by 50 K in different studies (e.g.
Turbet et al. (2016); Boutle et al. (2017)), a discrepancy that can only be
partly explained by differences in stellar irradiation used by these studies.
Boutle et al. (2017) instead proposed that these temperature discrepancies
could be explained by differences in convection parameterizations and model
resolutions, which affect the water vapour profile and consequently the
dayside-nightside heat transport. Gravity wave parameterisations likewise vary
between models and Watkins & Cho (2010) have shown that gravity waves can have
a significant impact on the development of the background flow, temperature,
and heat redistribution from dayside to nightside on tidally locked planets.
Gravity waves can become trapped within jets and travel longitudinally from
the gravity wave source on the dayside until the jet speed develops a critical
level where the wave can be absorbed (Pitteway & Hines, 1965; Nappo, 2012).
Differing gravity wave schemes may therefore result in different predictions
for heat redistribution and dayside-nightside temperature contrast (Watkins &
Cho, 2010).

###  4.3 The importance of the western terminator for atmospheric dynamics

Our results identify the western terminator as a region of particular
complexity. At the equatorial western terminator, the zonal wind components
contributed by the standing Rossby waves on the dayside and nightside meet and
oppose each other. We have also shown in Figure 7 that the western terminator
is a jet exit region where the flow is unbalanced and internal gravity waves
are generated and locally reabsorbed. These factors lead to significant
turbulence and variability due to the interplay between these elements. As
transit spectroscopy is expected to be a key tool in constraining the
composition of exoplanet atmospheres, understanding the dynamics of the
terminator regions of tidally locked planets is important to interpreting the
results of observational studies. If the time scale of an oscillation is
longer than the observing window, observers may view the planet in a low-
temperature/high-temperature or low-abundance/high-abundance phase of an
oscillation. As we noted above, a LASO on a hot Jupiter could be intense
enough to be detectable. The high amplitude of the oscillation at the western
terminator, a critical region for transit spectroscopy, suggests that the LASO
should be studied on tidally locked hot Jupiters to determine its possible
impact on observations.

##  5 Conclusion

Using a global 3-D general circulation model, we reported the discovery of a
phenomenon analogous to Earth’s quasi-biennial oscillation (QBO) in
simulations of the atmosphere of an Earth-like tidally locked planet,
nominally Proxima Centauri b. As the phenomenon exhibits asymmetries related
to the tidal locking, we refer to it as the longitudinally asymmetric
stratospheric oscillation (LASO). The LASO begins above the tropospheric
equatorial superrotating jet and extends vertically from roughly 35–55 km. It
consists of a periodic reversal in the direction of the zonal winds within
this region, accompanied by oscillations in temperature. We found the LASO in
our simulation to have a latitudinal extent of $\pm$ 60∘ and a period of 5–6.5
Earth months. The shorter period compared to its Earth counterpart is
explained to a first order by stronger wave driving from gravity wave
generation in the deep convection zone at the substellar point. We also
identified a secondary gravity wave source in the jet exit region at the
western terminator. Our analysis found that wind speeds in the LASO reach a
maximum at the western terminator and that reversals of direction at the
western terminator precede those at other longitudes. We further found that
the antistellar point experiences a very weak oscillation: zonal winds are
preferentially eastward-flowing, with only brief and low-speed westward
phases.

Our results highlight the significance of atmospheric variability in global
3-D simulations of exoplanets and point towards a need for further studies to
investigate this largely uncharted territory. In this initial study, we
focused on the characteristics and mechanism of the LASO and did not explore
the potential effect of oscillations in abundances of atmospheric species
other than water vapour on exoplanet observations. Future work in this area
could use a 3-D model with online atmospheric chemistry such as presented in
Yates et al. (2020) to explore such effects. While we performed sensitivity
tests by varying the horizontal and temporal resolution, we expect that the
LASO’s fundamental dependence on the planet’s mean circulation will make it
sensitive to factors that alter the planet’s atmospheric dynamics, such as
different atmospheric compositions, the presence of land, and the planet’s
rotation rate. On the Earth, the QBO is also known to interact with other
dynamical phenomena such as the semi-annual oscillation, Madden-Julian
oscillation (Martin et al., 2021), and polar vortex. We leave such potential
interactions between the LASO and similar sources of variability in the
atmosphere for future work.

As the body of observational data from exoplanets grows in the coming decades,
our understanding of the parameter space for planetary climate states must
grow with it. This parameter space includes time-dependent atmospheric
phenomena such as the one we describe in this paper which, even if they are
familiar features on Earth, will inevitably take new forms on tidally locked
planets. Characterisation of the sources and properties of atmospheric
variability on tidally locked exoplanets promises to refine our
interpretations of observations and, in the case of terrestrial planets,
deepen our understanding of the Earth in the context of the larger population
of planets inside and outside the solar system.

_Acknowledgments:_ Material produced using Met Office Software. We gratefully
acknowledge the funding and support provided by the Edinburgh Earth, Ecology,
and Environmental Doctoral Training Partnership and the Natural Environmental
Research Council [grant number NE/S007407/1]. We also kindly acknowledge our
use of the Monsoon2 system, a collaborative facility supplied under the Joint
Weather and Climate Research Programme, a strategic partnership between the
Met Office and the Natural Environment Research Council. Our research was
performed as part of the project ‘Using UKCA to investigate atmospheric
composition on extra-solar planets (ExoChem).’ This work was partly funded by
the Leverhulme Trust through a research project grant [RPG-2020-82], a Science
and Technology Facilities Council Consolidated Grant [ST/R000395/1] and a UKRI
Future Leaders Fellowship [grant number MR/T040866/1]. JM and IAB acknowledge
the support of a Met Office Academic Partnership secondment.

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