# The emergence of relaxation-oscillator convection on Earth and Titan

Francisco E. Spaulding-Astudillo Department of Earth, Planetary, and Space
Sciences, University of California, Los Angeles
595 Charles E Young Dr E, Los Angeles, CA 90095, USA  Jonathan L. Mitchell
Department of Earth, Planetary, and Space Sciences, University of California,
Los Angeles
595 Charles E Young Dr E, Los Angeles, CA 90095, USA  Department of
Atmospheric and Oceanic Sciences, University of California, Los Angeles
520 Portola Plaza, Los Angeles, CA 90095, USA

###### Abstract

In relaxation-oscillator (RO) climate states, short-lived convective storms
with torrential rainfall form and dissipate at regular, periodic intervals. RO
states have been demonstrated in two- and three-dimensional simulations of
radiative-convective equilibrium (RCE), and it has been argued that the
existence of the RO state requires explicitly resolving moist convective
processes. However, the exact nature and emergence mechanism of the RO state
have yet to be determined. Here, we show that (1) RO states exist in single-
column-model simulations of RCE with parameterized convection, and (2) the RO
state can be understood as one that has no steady-state solutions of an
analytical model of RCE. As with model simulations with resolved convection,
these simpler, one-dimensional models of RCE clearly demonstrate RO states
emerge at high surface temperatures and/or very moist atmospheres. Emergence
occurs when atmospheric instability quantified by the convective available
potential energy can no longer support the latent heat release of deep,
entraining convective plumes. The proposed mechanism for RO emergence is
general to all moist planetary atmospheres, is agnostic of the condensing
component, and naturally leads to an understanding of Titan’s bursty methane
weather.

Atmospheric dynamics (2300) — Planetary climates (2184) — Atmospheric clouds
(2180)

##  1 Introduction

Earth’s tropics are characterized by a steady balance between heating by moist
convection and radiative cooling, which is referred to as quasi-equilibrium
(QE) convection. QE states exhibit continuous, deep convection and small,
random fluctuations in clouds and precipitation about the mean. A new mode of
convection with quasi-periodic oscillations was recently discovered for high
surface temperatures in three-dimensional, cloud-resolving model simulations
of radiative-convective equilibrium (3D-RCE; Seeley & Wordsworth, 2021a,
henceforth SW21). In this new relaxation-oscillator (RO) mode of convection,
storms erupt at regular intervals with vigorous, short-lived rainfall,
separated by dry, cloud-free spells. Although the RO state is not a feature of
the modern climate, significant time-variation in convective storms have been
observed or simulated in a variety of planetary environments: from Jupiter
(Sugiyama et al., 2014) and Saturn (Li & Ingersoll, 2015; Fischer et al.,
2011) to Titan (Schaller et al., 2009; Turtle et al., 2011) and the hothouse
Earth (SW21).

SW21 propose that the physical ingredients required for RO states are (i)
realistic radiation, (ii) clouds, (iii) convection, and (iv) large-scale
condensation and re-evaporation; they further speculate that RO states require
model simulations with resolved convection, i.e., a model of 3D-RCE. These
four ingredients, however, are not unique to 3D-RCE – all of them can be
parameterized in simpler models of RCE. In this paper, we demonstrate the
existence of RO states in a single-column model of RCE, i.e., 1D-RCE. We then
develop a hypothesis for the emergence of RO states by exploring the limits of
an even simpler, QE model of RCE, and test our hypothesis against our 1D-RCE
simulations and the 3D-RCE simulations of SW21. To illustrate the flexibility
of our approach, we test it against a completely different moist-convective
system: methane on Titan.

##  2 Methods

We use a version of the ECHAM6 GCM in single-column mode that has been
modified to allow water vapor to comprise a significant fraction of the
atmospheric mass (Popp et al., 2015, henceforth P15) and the SVP of water to
be multiplicatively altered (Spaulding-Astudillo & Mitchell, 2023). The
single-column model has separate schemes for radiation, convection, and
clouds. Shortwave and longwave radiation is resolved into 14 and 16 spectral
bands by the Rapid Radiative Transfer Model for General Circulation Models
(RRTMG - Iacono et al., 2008). All phases of water are prognostic and
accounted for in the radiation calculation. As in P15, we use an exponential
extrapolation of all molecular absorption coefficients in the longwave and the
H2O self-broadened absorption coefficients in the shortwave for temperatures
above which no data in the original model exists. The effect of pressure
broadening by water vapor is neglected. Convection is represented by a bulk-
plume scheme (Tiedtke, 1989; Giorgetta et al., 2013) that parameterizes
turbulent entrainment and detrainment of air between updrafts, downdrafts, and
the environment. Clouds are formed by a large-scale condensation and re-
evaporation scheme (Sundqvist et al., 1989) as a function of the relative
humidity of the environment.

In our minimal-recipe experiment, the insolation is set 10% higher than the
present-day value and is temporally fixed (no diurnal or seasonal cycle). We
set the column latitude to 38∘N, where the globally-averaged insolation is the
same as the local value. Clouds are the only source of time-varying planetary
albedo. The atmosphere is composed only of nitrogen, oxygen, and water vapor.
We use a time step of 60 s and run the simulations for approximately 200
years. The sea surface temperature (SST or $T_{s}$) of a 1 m mixed-layer ocean
with an albedo of 0.07 is fixed at every time step through the use of an
artificial surface heat sink. We use fixed $T_{s}$ and vary it in 1-5 K
increments from 290 to 370 K.

##  3 Results

Figure 1: Displays (left to right) the minimal recipe, no-h2osw, and no-h2orad
single column experiments with fixed SSTs between 300-360 K. The top row is
net radiative temperature tendency in K/day. The bottom row is a time-series
of rainfall rates in cm/day over a 30 year period. Rainfall rates at each SST
are vertically offset by 5 cm/day and a zero rainfall contour is given for
reference as a dashed black line.

###  3.1 Description of RO states

Figure 1d clearly shows that our single column, minimal-recipe simulations
transition into RO states at a surface temperature around $T_{s}=350$ K. At
cooler temperatures, precipitation is steady and continuous and hovers around
a mean value of 2-6 mm/day; these simulations are in the QE regime. RO states
exhibit large bursts of precipitation with intensities up to 5 cm/day that
repeat every $O(100)$-$O(1000)$ days and that are relatively short-lived
(10-30 days).111The storm duration is calculated as the number of contiguous
days where the precipitation rate is above the mean value. Consistent with 2D
(Sugiyama et al., 2014) and 3D (SW21) simulations of oscillatory climate
states, 1D-RCE RO states cycle through 3 phases: a rapid convective phase
(RCP), a slow cooling phase (SCP), and a triggering phase (TP).

In the RCP, deep convection rapidly heats and moistens the atmosphere above
the boundary layer at $O(1)$-$O(10)$ K/day (Figure 2b). The storm generates a
spike in rainfall resembling a delta function that is 10-20 times stronger
than the mean background rate (Figure 2d). Strong warming disrupts cloud
formation by reducing the column relative humidity, contributing to a sharp
drop in vertically-integrated cloud water of 1-2 orders of magnitude (Figure
2a). Low-level temperature inversions in our warmest simulations are a clear
source of convective inhibition that form during the RCP and are slowly eroded
during the subsequent SCP; such inversions are common to the so-called moist
greenhouse state (Wolf & Toon, 2015; Popp et al., 2015).

Figure 2: Time series of the minimal-recipe experiment over 5 years at
$T_{s}=355$ K. Depicts (a) cloud water mixing ratio in color and vertically-
integrated cloud water in black. (b) Temperature tendency from the convection
scheme in K/day. (c) Difference in temperature between a parcel lifted from
the surface along the moist pseudo-adiabat and the environment. This is a
proxy for the atmospheric instability. (d) Rainfall rate and convective
available potential energy (CAPE).

During the SCP, the atmosphere cools radiatively from its upper layers. The
lower layers are water-dominated and thus are unable to cool directly to space
because of their high infrared opacity. Top-of-atmosphere radiative cooling is
communicated to the lower layers through the re-evaporation of precipitation;
re-evaporation destablizes the atmosphere from the top down, building CAPE in
the process (Figure 2c,d). The increase in CAPE is evident in a time series of
the difference in temperature between the surface-defined moist pseudo-adiabat
and the environment (Figure 2c). Total cloud cover steadily increases over the
SCP (Figure 2a).

The TP begins when the troposphere is everywhere unstable with respect to
surface-based, pseudo-adiabatic moist convection (Figure 2c). A striking
feature of the TP is the rapid descent of the cloud base (Figure 2a), which
has been observed in other numerical studies of oscillatory climate states
(SW21, Sugiyama et al., 2014). Vertically-integrated cloud water quadruples
during this period (Figure 2a), which could be related to the troposphere
nearing saturation. Eventually, the built-up CAPE is released (Figure 2d) and
the cycle repeats from the RCP.

The inhibitive cap that exists during the SCP is slowly eroded away by the TP
(Figure 2c). Paradoxically, CAPE continues to grow during the TP with no
apparent convective inhibition mechanism (Figure 2d). In fact, the TP
typically represents over 25% of the quiescent period. This means that there
is a significant delay between when the troposphere becomes globally unstable
and when the next storm begins. This apparent paradox suggests that we have
thus far overlooked an emergent constraint on deep convection, which we will
return to in Section 3.3.

###  3.2 RO states occur in the absence of lower-tropospheric radiative
heating

SW21 hypothesize that lower-tropospheric radiative heating (LTRH) that emerges
in hot climate states is essential for the emergence of RO convection; LTRH is
a consequence of strong shortwave absorption by water vapor. By enforcing LTRH
in fixed SST 3D-CRM simulations, SW21 observed that RO states emerged at SSTs
over 20 K less than their default experiment.

To test the LTRH hypothesis, we ran two separate mechanism-denial experiments
in which we zero out the contribution of water vapor in the SW radiative
calculation (“no-h2osw” experiment) and one in which it is zeroed out in the
SW and LW radiative calculation (“no-h2orad” experiment). The lower
troposphere remains opaque to infrared radiation in the no-h2osw experiment
and consequently net cooling rates there remain close to zero (Figure 1b). In
contrast, the whole troposphere radiatively cools in the no-h2orad experiment
(Figure 1c). In both the no-h2osw and no-h2orad experiments, we found that RO-
like states still occur (Figures 1e,f). From these mechanism-denial
experiments, we conclude that the radiative effects of water vapor influence
RO states, but they do not tell the whole story. We suspect that a previously-
unrecognized constraint on convection produces the RO phenomenon over warm
SSTs, which we now detail.

###  3.3 The mechanism for the emergence of RO states

The QE state is characterized by a steady balance between the generation of
CAPE by radiation and its destruction by convection. The RO state is clearly
not steady, however we posit that exploring the conditions in which a steady,
QE state is valid can illuminate the mechanisms that lead to the transition
from QE to RO convection. We carry this out quantitatively in Section 3.4, but
first consider heuristically how the dynamics of RO convection would differ
from the QE state. Essential to the QE state is the ability of radiative
cooling to keep pace with the more rapid convective heating. In warm and/or
humid conditions, however, the fast convective process releases an amount of
latent energy that the slower radiation cannot keep pace with. The sudden
release of this heat, $\langle r\rangle$, in convection would naturally
stabilize the atmosphere and shut off convection until radiation can restore
buoyancy. CAPE would necessarily be time-dependent, undergoing large
oscillations bounded above by $\langle r\rangle$. In this picture, the RO
state emerges when $\langle r\rangle$ exceeds the value of CAPE that would be
present in the QE regime. Thus, a predictive model of RO emergence requires a
plausible model of quasi-steady RCE.

Just such a steady-state model of QE convection has gained traction in recent
years (Singh & O’Gorman, 2013; Romps, 2014, 2016; Seeley & Wordsworth, 2023).
Romps (2016, henceforth R16) conceives of a saturated, convective plume
interacting with a sub-saturated environment with which it has zero buoyancy –
the so-called “zero-buoyancy model” (ZBM) – a condition that is supported by
observations in the tropics and in cloud-resolving models simulations (see
Section 3.4 for a derivation of the ZBM.). The ZBM predicts the steady value
of CAPE that would be present in QE for a given surface temperature and a set
of convective parameters. It’s important to note that the CAPE predicted by
the ZBM represents a steady-state storage of buoyancy, not the rate of CAPE
generation and destruction by radiation and convection. Furthermore, the ZBM
predicts a maximum in CAPE at intermediate, warm surface temperatures, which
are explored quantitatively in the following sections. Concurrently,
increasing the SST would naturally increase water vapor, and we will show that
the latent heat release by convection in the ZBM, $\langle r\rangle$,
increases monotonically with surface temperature. Thus, $\langle r\rangle$
inevitably exceeds the steady value of CAPE above a certain SST; above this
SST, the QE state is no longer viable. Our hypothesis is that the RO state is
preferred in this SST regime.

In what follows, we calculate CAPE and $\langle r\rangle$ of the ZBM as a
function of SST, and convective and thermodynamic parameters. We then test our
hypothesis for the emergence of RO states by seeking the answers to the
questions: (1) Under what conditions does $\langle r\rangle$ exceed CAPE in
the ZBM?; and (2) Is the point of divergence between $\langle r\rangle$ and
CAPE predictive of the emergence of RO states in simulations?

###  3.4 CAPE and $\langle r\rangle$ in an analytical model of an atmosphere
in RCE

Parameter | Definition | Earth-like | Titan-like
---|---|---|---
$c_{p,d}$ | Specific heat of dry air at constant pressure (Jkg-1K-1) | 1004 | 1040
$R_{d}$ | Specific gas constant of dry air (Jkg-1K-1) | 287 | 290
$R_{v}$ | Specific gas constant of the condensable (Jkg-1K-1) | 462 | 518
$L$ | Latent heat of vaporization of the condensable (Jkg-1) | $2.26\times 10^{6}$ | $5.5\times 10^{5}$
$g$ | Gravitational acceleration (ms-2) | 9.81 | 1.35
$T_{trp}$ | Tropopause temperature (K) | 200 | 70
$p_{s,d}$ | Partial pressure of dry air at the surface (Pa) | $10^{5}$ | $1.42\times 10^{5}$
Table 1: Parameters used to evaluate the analytical QE model at Earth-like and
Titan-like conditions. These are not necessarily the values used by ECHAM6
(Stevens et al., 2013) or the Titan Atmospheric Model (TAM - Lora et al.,
2015).

The system of equations for the analytical QE model of RCE derived are
presented below, to which we have contributed two innovations. First, we
introduce an artificial multiplier on the saturation vapor pressure (SVP) of
water, $\xi$, following Spaulding-Astudillo & Mitchell (2023) and McKinney et
al. (2022).

$e^{*}_{\xi}=\xi e^{*}$ (1)

where $e^{*}_{\xi}$ and $e^{*}$ are the modified and true SVP of water,
respectively. Moisture variables with a subscript of $\xi$ should be
understood to have an implicit $\xi$-dependence. Second, and most importantly,
we extend the analytical model to solve for $\langle r\rangle$. We refer the
reader to Appendix A for the extended derivation.

Under the assumption that the convective mass flux, the relative humidity of
the environment, and precipitation efficiency (PE - defined as the fraction of
condensates generated in updrafts at each height that are not re-evaporated)
are constant with height, the system of equations for an RCE atmosphere are

$\displaystyle q^{*}_{\xi}$ $\displaystyle=$
$\displaystyle\frac{R_{a}}{R_{v}}\frac{e^{*}_{\xi}}{p},$ (2)
$\displaystyle-\gamma$ $\displaystyle=$
$\displaystyle\frac{g}{R_{a}T}-\frac{L\Gamma}{R_{v}T^{2}},$ (3)
$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle\frac{\gamma
a}{\textrm{PE}},$ (4) RH $\displaystyle=$
$\displaystyle\frac{1+a-\textrm{PE}}{1+a},$ (5) $\displaystyle c_{z}$
$\displaystyle=$ $\displaystyle\frac{\gamma q^{*}_{\xi}}{1+a},\hskip
3.61371pt\textrm{and}$ (6) $\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{g}{c_{p,d}}\Big{[}\frac{1+a+q^{*}_{\xi}L/(R_{a}T)}{1+a+q^{*}_{\xi}L^{2}/(c_{p}R_{v}T^{2})}\Big{]}.$
(7)

The thermodynamic constants and their units and values are given in Table 1
and are assumed to be constant with height. $q^{*}_{\xi}$ is saturation
specific humidity ($kg_{h2o}$kg-1), $\gamma$ is the water vapor lapse rate
(kgh2okg-1m-1), $\varepsilon$ is the fractional convective entrainment (m-1),
RH is the relative humidity, $c_{z}$ is the specific condensation per unit
distance traveled by an entraining plume (kgh2okg-1m-1), and $\Gamma$ is the
temperature lapse rate (Km-1). PE and $a$ are prescribed constants; by
Equation 4, their ratio is a constant equal to $\gamma/\varepsilon$ ($\gamma$
and $\varepsilon$ both increase with height but do so in lock-step). $a$
controls the dilution of convective plumes by environmental air (for an
extended discussion, see Seeley & Wordsworth, 2023). For undiluted convection
($a=0$) in a real atmosphere ($\xi=1$), $\Gamma$ equals the moist pseudo-
adiabatic lapse rate, $\Gamma_{m}$. As $a$ increases, the tighter coupling
between the plume and the environment forces $\Gamma$ apart from $\Gamma_{m}$,
permitting more convective instability in steady-state. We calculate the
instability in steady-state as

$\textrm{CAPE}=\int_{z_{s}}^{z_{trp}}g\frac{T^{\prime}-T}{T}dz,$ (8)

where $T$ is the environmental temperature and $T^{\prime}$ is the temperature
of a parcel lifted moist pseudo-adiabatically from the surface ($z_{s}$) to
the tropopause ($z_{trp}$). The net release of latent energy per unit mass of
an entraining plume traveling from the surface to the tropopause is

$\langle
r\rangle=L\frac{\textrm{PE}}{1+a}\Big{(}q^{*}_{\xi}(T_{s})-q^{*}_{\xi}(T_{trp})\Big{)}.$
(9)

Equation 9 clearly relates the moisture lost by a saturated plume during
ascent, $q^{*}_{\xi}(T_{s})-q^{*}_{\xi}(T_{trp})$, to the details of turbulent
mixing and evaporative microphysics, $\textrm{PE}/(1+a)$, and to the latent
heat, $L$. Since $\langle r\rangle$ and CAPE share the same physical units (J
kg-1), we are equipped mathematically to compare their magnitudes in steady-
state as a function of SST in Section 3.5. The sensitivity of CAPE and
$\langle r\rangle$ to $\xi$ allow us to test the mechanism for the emergence
of RO states in Section 3.6.

###  3.5 The physical relationship between CAPE and $\langle r\rangle$

How are $\langle r\rangle$ and CAPE related physically? CAPE (Equation 8) is
proportional to the height-integrated temperature difference between an
undiluted lifted parcel and its environment, $\Delta T$, which can be
approximated following R16 as

$\textrm{CAPE}\propto\int_{z_{s}}^{z_{trp}}\Delta
Tdz\approx\int_{z_{s}}^{z_{trp}}\frac{\Delta h_{\xi}^{*}}{\beta}dz.$ (10)

$\Delta h_{\xi}^{*}$ is the saturated moist static energy (MSE) difference
between the two air masses at height $z$ and
$\beta=c_{p,d}+L^{2}q^{*}_{\xi}/(R_{v}T^{2})$ is the atmospheric heat capacity
at height $z$. $\beta^{-1}$ gives the fraction of the saturated MSE difference
that is expressed as a temperature anomaly or, equivalently, as CAPE. $\Delta
h_{\xi}^{*}$ can also be expressed as (R16)

$\Delta
h^{*}_{\xi}=\frac{a}{1+a}L\Big{(}q^{*}_{\xi}(z_{s})-q^{*}_{\xi}(z)\Big{)}.$
(11)

This allows us to rewrite $\langle r\rangle$ (Equation 9) as

$\langle r\rangle=\frac{\textrm{PE}}{a}\Delta h_{\xi}^{*}(z_{trp}).$ (12)

The ratio $\textrm{PE}/a=\gamma/\varepsilon$ (a constant in this restricted
model of RCE, but in practice varies in real atmospheres) is the fraction of
the saturated MSE at a given height that is expressed as $\langle r\rangle$.
Given Equations 10 and 12, it’s clear that CAPE and $\langle r\rangle$ are
both closely related to $\Delta h_{\xi}^{*}$.222Technically, CAPE and $\langle
r\rangle$ in our steady-state model are related to the saturated MSE
difference subject to the assumption of zero buoyancy and
entrainment/detrainment with a dilute atmosphere. The fact that these
conditions are met in observations of the tropics and CRMs (Singh & O’Gorman,
2013) lends credence to the simple model of RCE. At lower SSTs, both $\langle
r\rangle$ and CAPE grow in proportion to $q_{\xi}^{*}$ by Equation 11. Unlike
$\langle r\rangle$, however, CAPE peaks at intermediate SSTs because of its
dependence on $1/\beta$, which decreases rapidly at high SSTs. CAPE begins
this decrease with increasing SSTs where $\beta>2c_{p,d}$ at the tropopause
(see Figure 9 from R16), or in other words where the effective heat capacity
of the entire troposphere becomes moisture-dominated. Thus, the magnitude of
CAPE is set by a competition between the increase in $\Delta h_{\xi}^{*}$ and
decrease in $1/\beta$, the latter of which wins out in the warmest (Figure 3)
and/or wettest (Figure 4) environments. Further increasing the SST beyond the
peak in CAPE result in a smaller fraction of $\Delta h_{\xi}^{*}$ that is
expressed as CAPE through $\Delta T$ and a constant fraction and therefore
growing amount of $\Delta h_{\xi}^{*}$ that is expressed as $\langle
r\rangle$. Any SSTs for which $\langle r\rangle>\textrm{CAPE}$ violate the
steady-state assumption of the RCE model, inevitably pushing convection from
the QE state into the RO state.

###  3.6 Testing the emergence mechanism for RO states

Figure 3: Analytical CAPE (dashed) and $\langle r\rangle$ (dotted) for an
Earth-like atmosphere with PE/a ratios of (a,c) 0.35 and (b,d) 0.15. Values of
$a$ ranging between 0.2 and 2.8 are denoted by in-line labels and specific
line colors. Like-colored open circles are placed at the theoretical location
of each QE-to-RO transition, i.e., where the dotted and dashed lines with the
same value (color) of $a$ intersect. CAPE from our minimal-recipe simulations
is given as black circles and CAPE from Seeley & Wordsworth (2021b) is given
as black triangles. Filled circles/triangles denote QE states and open
circles/triangles denote RO states.  Figure 4: Sensitivity experiments for an
Earth-like atmosphere with (a,d) $\xi=1$, (b,e) $\xi=1.5$, and (c,f)
$\xi=1.9$. (a-c) Analytical CAPE (dashed) and $\langle r\rangle$ (dotted) with
PE=25%. Values of $a$ ranging between 0.2 and 2.8 are denoted by in-line
labels and specific line colors. Like-colored open circles are placed at the
theoretical location of each QE-to-RO transition, i.e., where the dotted and
dashed lines with the same value (color) of $a$ intersect. CAPE from our
minimal-recipe simulations is given as black circles. Filled circles denote QE
states and open circles denote RO states. (d-f) Time series of rainfall rate
over a 30 year period for minimal-recipe simulations with fixed SSTs between
330-350 K. The profile from each simulation is vertically offset by 5 cm/day
and a zero rainfall contour is given for reference as a dashed black line.

We first specify two convective parameters which are assumed to be constant
with height: PE and $a$. PE is the fraction of condensates generated in
ascending plumes that are re-evaporated in the environment. $a$ quantifies the
dilution of plumes by the environment, and represents the combined effect of
entrainment and re-evaporation. The quantity $\textrm{PE}/a$ may be estimated
as the water vapor lapse rate, $\gamma$, divided by the convective entrainment
rate, $\varepsilon$, over the troposphere. Typical values of $\varepsilon$ and
$\gamma^{-1}$ in Earth’s atmosphere at 300 K are $0.5-2\times 10^{-3}$ km-1
(Romps, 2010) and 2-8 km (R16). Taking intermediate values gives an expected
value for Earth of $\textrm{PE}/a=0.25$. Figure 3 shows theoretical profiles
of CAPE and $\langle r\rangle$ evaluated with the thermodynamic constants in
Table 1 in the neighborhood of this $\textrm{PE}/a$ ratio. Reducing
$\textrm{PE}/a$ (between columns in Figure 3) has no effect on CAPE because
CAPE is a function of $a$ only.

Regardless of the convective parameter values, $\langle r\rangle$ inevitably
exceeds CAPE at some SST, as shown in Section 3.5. The SST at which this first
occurs can be found by identifying the intersection of the analytical profiles
of CAPE (dashed lines) and $\langle r\rangle$ (dotted lines) that possess the
same value (color) of $a$ (Figure 3).333The value of PE along each like-
colored curve is also constant. For example, the value of PE along the orange
curve is PE $=0.35\times 0.8=0.28$ in Figure 3a and is PE $=0.15\times
0.8=0.12$ in Figure 3b. The CAPE from our 1D-RCE simulations is in rough
agreement with the theoretical curve denoted by $a=2$ (not plotted); for the
range of $\textrm{PE}/a$, the intersection between CAPE and $\langle r\rangle$
occurs at SSTs $\sim 340-350$K (Figures 3a,b), which is also the SSTs where
our simulations transition to the RO state (Figure 1d).

Extrapolating to other values of PE and $a$, the analytical model of steady-
state convection predicts that decreasing $a$ and/or increasing PE shifts the
intersection of CAPE and $\langle r\rangle$ to lower SSTs. Decreasing $a$
brings the analytical model into closer agreement with the results of SW21
(Figures 3c,d), which had SSTs at emergence of RO states closer to 320K.444At
smaller $a$, the analytical model predicts $\langle r\rangle$ exceeds CAPE at
least 10K lower than in Figures 3a,b. A consistent interpretation is that the
3D-RCE simulations of SW21 have a lower value of $a$ and/or higher value of PE
than our 1D-RCE simulations. Differences between models in, for instance, the
tuning of evaporative microphysics would naturally lead to an offset with
respect to the SST at which RO states emerge.

The convection and re-evaporation schemes in our 1D-RCE model were not
designed in a way that facilitates sensitivity tests using $a$ or PE as
independent control parameters. However, we introduce a multiplier on SVP
(Spaulding-Astudillo & Mitchell, 2023) into the analytical model of R16
whereby increasing it accomplishes a similar effect as to increasing
$\textrm{PE}/a$ (this part increases $\langle r\rangle$ – see Section 3.5)
while simultaneously decreasing $a$ (this part decreases CAPE). The emergence
temperature in our 1D-RCE simulations dropped from 350 K to 335 K for a
doubling of $\xi$, in agreement with our analytical model (Figures 4a-c).
Furthermore, the analytical QE model predicts a decrease in the peak CAPE and
a shift to lower SSTs with increasing $\xi$, which is borne out by our
numerical experiments (Figures 4a-c). As in the other parameter regime (Figure
3), $\langle r\rangle$ and CAPE both grow quasi-exponentially with increasing
SSTs to a peak in CAPE, beyond which the latter falls off with increasing SSTs
while $\langle r\rangle$ continues to grow. Thus, we find that our hypothesis
that the emergence of RO states is sensitive to moisture concentration is
supported by the quantitative agreement between the analytical model
prediction and numerical sensitivity experiments with varying SVP.

##  4 Discussion

We conclude that relaxation-oscillator (RO) convection is an emergent
phenomenon in sufficiently warm and/or wet atmospheres. We obtained this
result in a 1D-RCE model with parameterized convection that clearly reproduces
the RO states that develop in 3D-RCE with resolved convection (SW21).
Switching off the radiative effects of water vapor in 1D-RCE, we demonstrated
that lower-tropospheric radiative heating is non-essential for RO convection.
Since RO states are clearly non-steady, we hypothesized that RO states should
emerge in conditions where the energetics of moist convection violate steady-
state quasi-equilibrium (QE). QE is violated when the latent energy released
by deep, entraining convection ($\langle r\rangle$) exceeds the convective
available potential energy (CAPE) in steady-state. The RO mode of convection
is preferred in these conditions in two ways. First, the surface temperature
of RO emergence in 1D- and 3D-RCE agrees with the prediction from an
analytical QE model evaluated with Earth-like convective parameters. Second,
doubling the SVP of water at all temperatures ($\xi=2$) causes the surface
temperature of emergence to decrease by 10-15 K in 1D-RCE, once again in
agreement with the analytical prediction. The agreement is reassuring, as the
bulk plume equations of convection of the analytical model are based on the
parameterized convection scheme in our 1D-RCE model (see Methods). The
consistency of the QE-to-RO convective regime transition across the modeling
hierarchy explored here lends confidence to the robustness of this transition,
and the insights gained from the simpler end of the modeling hierarchy
developed here illustrates their power in the exploration of the sensitivity
of these states to model parameters rooted in the physical world.

We note three reported commonalities between this and other studies of
oscillatory radiative-convective states in both nitrogen- and hydrogen-rich
atmospheres (e.g., SW21 and Sugiyama et al., 2014): (1) the slow erosion of a
stable layer during the quiescent, storm-free period (SCP) followed by the
descent of the cloud base (TP), (2) a significant temporal delay between the
development of global tropospheric instability (TP) and the start of the next
storm (RCP), and (3) an abrupt end to each storm when CAPE drops to zero
(e.g., Figure 2d). We speculate that (2) and (3) could be explained by the
requirement that CAPE build well above its expected steady-state value in
order to accommodate the energy released by deep convection, as represented by
$\langle r\rangle$. CAPE builds in time until it reaches values near $\langle
r\rangle$, at which point all of the CAPE would be converted to heat in a
short period of time, and the CAPE would begin building again by the slower
process of radiative cooling. In this sense and in analogy to an RO electric
circuit, radiative cooling and virga act as the voltage source, CAPE acts as
the capacitor, and $\langle r\rangle$ acts as the threshold device.

There are at least two major differences between simulated RO states in our
single column model and those in a 3D-CRM (SW21). The first difference is the
SST above which RO states emerge, which is 30 K higher in this study than in
SW21. In Section 3.6, we hypothesized this difference is related to different
model assumptions about the details of convective mixing and re-evaporation,
and we demonstrated a case in which altering the SVP of water gave rise to
10-15 K differences in the SST of RO onset. The second difference is the
recurrence interval (RI), or the typical duration of the quiescent period. In
SW21, the RI is $O(1)$ day and here $O(100)$-$O(1000)$ days.555We suspect this
discrepancy is due to a model assumption of the convective parameterization
that allows converted precipitation to “leak” from the updraft base near the
surface and bypass re-evaporation. Of course, the time-dependent behavior of
the RO state is beyond the scope of our steady-state theory, and the
simulations are telling us that the RO regime is where convection is non-
steady. We might better understand the relevant time scales, peak
precipitation rates, etc. of the RO state by deriving linearized non-steady
forms of the bulk-plume equations that appear in Section 3.4. These
timescales, for instance, could then be compared to historical observations of
giant storms on solar system planets. We leave this to future work.

In our derivation of the R16 model, we made several approximations and
omissions (virtual effects being one of them) that introduce error in the
limit of a moisture-dominated atmosphere. Revisiting these assumptions is a
necessary next step, but is beyond the scope of this paper. Consistent with a
recent study (Seeley & Wordsworth, 2023), the CAPE in our numerical
simulations peaks at intermediate SSTs and falls to zero in the limits of a
dry atmosphere or a moisture-dominated atmosphere. In Section 3.5, we
discussed the “shape of CAPE”, i.e., how the magnitude of CAPE changes with
SST in the steady-state model (R16). Including virtual effects in Equation 8
causes the magnitude of the CAPE from our simulations to increase at each
$T_{s}$ (up to 50% at the peak), but does not significantly change the
location of the peak nor the qualitative shape of CAPE vs. SST (not shown). We
chose to omit virtual effects in the CAPE calculation because it allows us to
directly compare our simulations to the analytical QE model.

##  5 RO convection on Titan

While present-day Earth does not exhibit RO behavior, it is clear that
episodic storms of potentially great magnitude occur on present-day Titan
(Schneider et al., 2012; Turtle et al., 2011; Faulk et al., 2017; Rafkin et
al., 2022; Charnay et al., 2015; Mitchell & Lora, 2016). A plausible
assumption is that the physics of moist convection should be similar on Titan
and Earth. Given what we know about Titan, is RO-like moist (methane)
convection predicted by the analytical theory at Titan-like conditions (see
Table 1)? Our theory of RO emergence requires the input parameters of our
analytical model, which are difficult to estimate from the data, but insights
can be gained by making reasonable estimates. We estimate the mean methane
scale height from Equation 3, $\gamma^{-1}\approx 20$ km, which aligns with an
earlier estimate (Lemmon et al., 2002). Following Griffith et al. (2000), we
use an entrainment rate of $\varepsilon=0.1$ km-1. Centering our search around
the expected $\textrm{PE}/a$ value of 0.5, we look for analytical solutions
for RCE on Titan (Figure 5). In most cases, the analytical model predicts that
$\textrm{CAPE}<\langle r\rangle$ at the present day surface temperature of
Titan of 90-95 K. Only present-day scenarios with $\textrm{PE}/a\leq 0.1$
favor QE over RO convection. These scenarios, however, imply a mean
tropospheric relative humidity (RH) greater than 90%666The value of RH along
each like-colored curve is constant (Equation 5), but increases left-to-right
between Figures 5a-c. For example, the value of RH along the orange curve is
RH $=(1+0.8-1.0\times 0.8)/(1+0.8)=55\%$ in Figure 5a and is RH
$=(1+0.8-0.1\times 0.8)/(1+0.8)=95\%$ in Figure 5c., which is inconsistent
with observations of episodic rainstorms over Titan’s arid low latitudes
(Turtle et al., 2011), the frequency of Titan’s storms (Battalio et al.,
2022), and the paucity of global cloud cover (Griffith et al., 2000). Our
theory thus predicts a transition from RO to QE behavior on Titan at high RH;
just such a transition from discrete to continuous precipitation was found in
3D simulations of Titan by raising the prescribed background RH above 85%
(Battalio et al., 2022).777The 3D Titan model used by Battalio et al. (2022)
employs a simplified Betts-Miller convection scheme. During convective
adjustments, the atmosphere relaxes to a prescribed, vertically-uniform RH.

Figure 5: Analytical CAPE (dashed) and $\langle r\rangle$ (dotted) for a
Titan-like atmosphere with a $\textrm{PE}/a$ ratio of (a) 1.0, (b) 0.5, and
(c) 0.25. Values of $a$ ranging between 0.2 and 2.0 are given by the in-line
labels and are associated with specific line colors. We place open circles at
the locations of each QE-to-RO transition, i.e., where the dotted and dashed
lines with the same value (color) of $a$ intersect. (d) Time series of global-
mean rainfall rate in cm per Earth day over a period of 5 Titan years for TAM
(Lora et al., 2015) aquaplanet simulations from Faulk et al. (2020).

Indeed, global aquaplanet simulations of Titan by Faulk et al. (2020) with the
Titan Atmospheric Model (Lora et al., 2015) depicted in Figure 5d resemble the
1D (Figure 1d) and 3D (see Figure 1 from SW21) simulated RO states on Earth,
where brief periods of intense rainfall give way to extended periods of little
to no rain. The largest deluges are capable of driving fluvial erosion and
sediment transport, and influence the distribution of alluvial fans on Titan’s
surface (Faulk et al., 2017; Lewis-Merrill et al., 2022). The example of Titan
highlights the many potential applications of our theory for RO states. For
instance, intermittent storms could be examined as a driver of surface
morphological evolution on different planets in the solar system that have or
may once have had active volatile cycles. This is the focus of ongoing work.
Thus we show our theory is (1) general to any moist planetary atmosphere and
(2) agnostic of the condensing component, and because of this generality, it
will find wide application across the solar system and beyond.

Titan’s episodic storms indicate RO-type convection, which may reflect the
fact that methane volatility888Volatility is here defined as the SVP of a
condensable at typical surface conditions of a particular planet. on Titan
greatly exceeds water volatility on Earth today. Our exploration with the
water volatility parameter, $\xi$, suggests the underlying cause of the QE-to-
RO state transition is the amount of water vapor rather than the surface
temperature, and the latter may just be a proxy for the former (Figures 3 and
4). But what is it about water volatility (abundance) that causes this
transition? The behavior of CAPE, $\langle r\rangle$, and the onset of the RO
state are closely related to the parameter $\beta$, which is an effective heat
capacity that takes into account how SVP shifts with changes in temperature.
The thermodynamic quantities that $\beta$ is comprised of – $c_{p}$, $R_{v}$,
$L$, and $q^{*}$ – are easily estimated for any conceivable atmospheric
composition with a condensable gas. It seems that with these few thermodynamic
ingredients, it might be possible to erect a very general regime diagram in
$\beta$ and the other convective parameters that could be used to classify all
planetary atmospheres. Such a regime diagram could be tested (perhaps even
calibrated) against moist convection throughout the solar system. If found to
be predictive of the occurrence of QE- and RO-style convection, this
convective regime diagram could be used to predict essentially any atmosphere
one could concoct, and then tested against time-varying signals of moist
convection on exoplanets that will soon be resolved by the James Webb Space
Telescope. In taking this comparative approach, modifications to the
analytical model may be required to generate solutions for atmospheres where
the non-condensable background air is lighter than the condensable gases,
which introduces competing latent and molecular weight effects on convection.
We are carrying out this generalization in ongoing work.

This work was supported by NSF Grant 1912673 and an Early-Career Fellowship
from the Center for Diverse Leadership in Science at UCLA. The authors thank
Jake Seeley, Bowen Fan, and Namrah Habib for helpful discussions. The data
from this study is archived at UCLA and is available upon request at such time
that this manuscript has passed peer review.

##  Appendix A Derivation of the analytical QE model

The system of equations for the analytical QE model of RCE are derived below.
We introduce an artificial multiplier on the saturation vapor pressure (SVP)
of water, $\xi$, following Spaulding-Astudillo & Mitchell (2023) and McKinney
et al. (2022).

$e^{*}_{\xi}=\xi e^{*}$ (A1)

where $e^{*}_{\xi}$ and $e^{*}$ are the modified and true SVP of water,
respectively. Moisture variables with a subscript of $\xi$ should be
understood to have an implicit $\xi$-dependence. The saturation specific
humidity is

$q^{*}_{\xi}=\frac{R_{a}}{R_{v}}\frac{e^{*}_{\xi}}{p},$ (A2)

where $R_{a}$ is the specific gas constant of environmental air (everywhere
assumed to be that of dry air) and $p$ is the total air pressure. Taking the
vertical derivative of the natural log of $e^{*}_{\xi}$ (and using
$\Gamma=-\partial_{z}T$), we obtain

$\displaystyle\partial_{z}e^{*}_{\xi}$
$\displaystyle=\partial_{T}e^{*}_{\xi}\partial_{z}T=-\frac{Le^{*}_{\xi}\Gamma}{R_{v}T^{2}},$
(A3) $\displaystyle\partial_{z}\ln e^{*}_{\xi}$
$\displaystyle=-\frac{L\Gamma}{R_{v}T^{2}}.$

The vertical derivative of the natural log of $p$ is obtained from hydrostatic
balance and the ideal gas law:

$\partial_{z}\ln p=-\frac{g}{R_{a}T},$ (A4)

where $g$ is gravity. Taking the vertical derivative of the natural log of
$q^{*}_{\xi}$ and plugging in Equations A3 and A4, we obtain

$\displaystyle\partial_{z}\ln q^{*}_{\xi}$ $\displaystyle=$
$\displaystyle\partial_{z}\ln e^{*}_{\xi}-\partial_{z}\ln p,$ (A5)
$\displaystyle=$
$\displaystyle\frac{g}{R_{a}T}-\frac{L\Gamma}{R_{v}T^{2}}=-\gamma.$

where $\gamma$ is the water vapor lapse rate.999$\gamma^{-1}$ is the scale
height of saturation specific humidity. The tropospheric water budget is
obtained from the bulk-plume equations for convection:

$\displaystyle\partial_{z}M$ $\displaystyle=$ $\displaystyle e-d-c,\hskip
3.61371pt\textrm{where}\hskip 3.61371pte=\varepsilon M\hskip
3.61371pt\textrm{and}\hskip 3.61371ptd=\delta M,$ (A6)
$\displaystyle\partial_{z}(Mq^{*}_{\xi})$ $\displaystyle=$ $\displaystyle
eq_{\xi}-dq^{*}_{\xi}-c,\hskip 3.61371pt\textrm{and}$ (A7)
$\displaystyle-\partial_{z}(Mq_{\xi})$ $\displaystyle=$ $\displaystyle
dq_{\xi}^{*}-eq_{\xi}+\alpha c.$ (A8)

$M$ is the convective mass flux (kg m-2s-1), $e$ and $d$ are the turbulent
entrainment and detrainment rates (kg m-3s-1) in which $\varepsilon$ and
$\delta$ are fractional mixing efficiencies (m-1), and $c$ is the condensation
rate (kgh2om-3s-1). $\alpha$ is defined as the ratio of gross evaporation to
gross condensation at each height, so the gross evaporation is $\alpha c/M$
and the condensation minus evaporation is $\textrm{PE}c/M$, where
$\textrm{PE}=(1-\alpha)$ is the precipitation efficiency. We make the
following assumptions. The condensates not re-evaporated at each level
($\textrm{PE}c/M$) are immediately removed from the convective plume. The
gross condensation represents a small fraction of the total updraft mass
($\partial_{z}M>>c)$. Invoking the latter assumption in Equation A6 gives

$\displaystyle\partial_{z}M$ $\displaystyle=$ $\displaystyle e-d$ (A9)
$\displaystyle=$ $\displaystyle(\varepsilon-\delta)M.$

Expanding Equation A7 with the chain-rule and solving for
$\partial_{z}q^{*}_{\xi}$ (using Equation A9), we obtain

$\partial_{z}q^{*}_{\xi}=\varepsilon(q_{\xi}-q^{*}_{\xi})-\frac{c}{M}.$ (A10)

Doing the same to Equation A8 to find $\partial_{z}q$,

$-\partial_{z}q_{\xi}=\delta(q^{*}_{\xi}-q_{\xi})+\alpha\frac{c}{M}.$ (A11)

The relative humidity is $\textrm{RH}=q_{\xi}/q^{*}_{\xi}$. Rearranging for
$q_{\xi}$,

$q_{\xi}=\textrm{RH}q^{*}_{\xi},$ (A12)

and taking the vertical derivative of both sides, we obtain

$\partial_{z}q_{\xi}=q^{*}_{\xi}\partial_{z}\textrm{RH}+\textrm{RH}\partial_{z}q^{*}_{\xi}.$
(A13)

We assume that vertical variations in RH are much smaller than those in
specific humidity ($\partial_{z}\textrm{RH}<<\partial_{z}q^{*}_{\xi}$), as is
generally the case in Earth’s troposphere. Invoking this assumption in
Equation A13 gives

$\partial_{z}q_{\xi}=\textrm{RH}\partial_{z}q^{*}_{\xi}.$ (A14)

Using Equations A5, A12, and A14 to re-write Equations A7 and A8, we obtain

$\displaystyle-\gamma q^{*}_{\xi}$ $\displaystyle=$
$\displaystyle\varepsilon(\textrm{RH}-1)q^{*}_{\xi}-\frac{c}{M}\hskip
3.61371pt\textrm{and}$ (A15) $\displaystyle\textrm{RH}\gamma q^{*}_{\xi}$
$\displaystyle=$ $\displaystyle\delta(1-RH)q^{*}_{\xi}+\alpha\frac{c}{M}.$
(A16)

To solve for RH, we substitute $\frac{c}{M}$ from Equation A15 into Equation
A16.

$\textrm{RH}=\frac{\delta+\alpha\gamma-\alpha\varepsilon}{\delta+\gamma-\alpha\varepsilon}$
(A17)

To solve for $\frac{c}{M}$, we back-substitute Equation A17 into Equation A15.

$c_{z}=\frac{c}{M}=\Big{(}\frac{\delta+\gamma-\varepsilon}{\delta+\gamma-\alpha\varepsilon}\Big{)}\gamma
q^{*}_{\xi}.$ (A18)

$c_{z}$ is the specific condensation per unit distance traveled by an
entraining plume (kgh2okg-1m-1). We invoke the zero-buoyancy assumption (ZBA -
Singh & O’Gorman, 2013) to define the moist static energy (MSE) of the
environment and the plume under the further assumption that virtual effects
can be neglected.

$\displaystyle h_{\xi}$ $\displaystyle=$ $\displaystyle c_{p}T+gz+Lq_{\xi}$
(A19) $\displaystyle h^{*}_{\xi}$ $\displaystyle=$ $\displaystyle
c_{p}T+gz+Lq^{*}_{\xi}$ (A20)

The ZBA is that convective plumes are neutrally buoyant with respect to their
environment. Strictly speaking, this means that their virtual temperatures are
the same. When virtual effects are neglected, the plume and the environment
possess the same temperature $T$ such that their MSEs differ only by the
differences in their specific humidities. $c_{p}$ is the specific heat of the
atmosphere (assumed to be dry air, $c_{p,d}$). Next, taking the vertical
derivative of $h^{*}_{\xi}$ (and using $\Gamma=-\partial_{z}T$,
$\partial_{z}q^{*}_{\xi}=-\gamma q^{*}_{\xi}$, and Equation A5),

$\displaystyle\partial_{z}h^{*}_{\xi}$ $\displaystyle=$ $\displaystyle-
c_{p}\Gamma+g-L\gamma q^{*}_{\xi}$ (A21) $\displaystyle=$ $\displaystyle
g\Big{(}1+\frac{Lq^{*}_{\xi}}{R_{a}T}\Big{)}-\Gamma\Big{(}c_{p}+\frac{L^{2}q^{*}_{\xi}}{R_{v}T^{2}}\Big{)}$

It follows from Equation A6 that the vertical change in MSE flux with height
for an entraining plume is

$\partial_{z}(Mh^{*}_{\xi})=(\varepsilon h_{\xi}-\delta h^{*}_{\xi})M.$ (A22)

Using the chain rule to solve for $\partial_{z}h^{*}_{\xi}$ (and substituting
Equations A9, A19, and A20):

$\displaystyle\partial_{z}h^{*}_{\xi}$ $\displaystyle=$
$\displaystyle\varepsilon(h_{\xi}-h^{*}_{\xi})$ (A23) $\displaystyle=$
$\displaystyle\varepsilon(q_{\xi}-q^{*}_{\xi})L$ $\displaystyle=$
$\displaystyle\varepsilon(\textrm{RH}-1)Lq^{*}_{\xi}.$

By convention (R16, Seeley & Wordsworth, 2023), the “bulk-plume parameter” is
defined:

$a=\textrm{PE}\frac{\varepsilon}{\gamma}.$ (A24)

We will assume that $M$, RH, and $\alpha$ are all constant with height. By
these assumptions, $\varepsilon=\delta$ (Equation A9),
$\varepsilon\propto\gamma$ (Equation A17), and $a$ is constant with height.
Equation A24 can be re-arranged to solve for $\varepsilon$, while RH (Equation
A17) and $c_{z}$ (Equation A18) simplify to

$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle\frac{\gamma
a}{\textrm{PE}},$ (A25) $\displaystyle RH$ $\displaystyle=$
$\displaystyle\frac{1+a-\textrm{PE}}{1+a},\hskip 3.61371pt\textrm{and}$ (A26)
$\displaystyle c_{z}$ $\displaystyle=$ $\displaystyle\frac{\gamma
q^{*}_{\xi}}{1+a}.$ (A27)

Substituting Equations A25 and A26 into Equation A23,

$\partial_{z}h^{*}_{\xi}=-\frac{a}{1+a}\gamma Lq^{*}_{\xi}.$ (A28)

Equating Equations A21 and A28 and solving for $\Gamma$,

$\Gamma=\frac{g}{c_{p}}\Big{[}\frac{1+a+q^{*}_{\xi}L/(R_{a}T)}{1+a+q^{*}_{\xi}L^{2}/(c_{p}R_{v}T^{2})}\Big{]}.$
(A29)

Equation A29 is the temperature lapse rate set by entraining convection. In a
real atmosphere ($\xi=1$), Equation A29 reduces to Equation (18) in Seeley &
Wordsworth (2023). Once again neglecting virtual effects, we quantify the
atmospheric instability in steady-state as

$\textrm{CAPE}=\int_{z_{s}}^{z_{trp}}g\frac{T^{\prime}-T}{T}dz,$ (A30)

where $T$ is the environmental temperature and $T^{\prime}$ is the temperature
of a parcel lifted moist pseudo-adiabatically from the surface ($z_{s}$) to
the tropopause ($z_{trp}$). We obtain $T^{\prime}$ and $T$ by integrating the
analytical model with $a=0$ and $a\geq 0$, respectively. Following R16, we
assume an invariant tropopause temperature of 200 K over the range of SSTs
considered here, as may be suggested by the fixed anvil temperature hypothesis
(Hartmann & Larson, 2002). Note that when $\xi\neq 1$, the anvil temperature
does not remain fixed and, in fact, decreases with increasing $\xi$ (e.g., see
Figure 3d from Spaulding-Astudillo & Mitchell, 2023); however, our main
results are not sensitive to the choice of tropopause temperature, so for
simplicity we retain the aforementioned value of 200 K even when $\xi\neq 1$.

Finally, the net release of latent energy per unit mass of an entraining plume
traveling from the surface to the tropopause is

$\displaystyle\langle r\rangle$ $\displaystyle=$ $\displaystyle
L\textrm{PE}\int_{z_{s}}^{z_{trp}}c_{z}dz$ (A31) $\displaystyle=$
$\displaystyle
L\frac{\textrm{PE}}{1+a}\Big{(}q^{*}_{\xi}(T_{s})-q^{*}_{\xi}(T_{trp})\Big{)}.$

Self-consistent solutions of CAPE and $\langle r\rangle$ as SSTs are varied
require the ratio $\textrm{PE}/a=\gamma/\varepsilon$ to be constant. This is
not strictly true in real atmospheres, but follows from the assumptions made
in the derivation of the QE model.

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Forum for Information Retrieval Evaluation, December 9-13, 2022, India

<EMAIL_ADDRESS>] [1]

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[1]Corresponding author.

# Overview of the HASOC Subtrack at FIRE 2022: Offensive Language
Identification in Marathi

Tharindu Ranasinghe  University of Wolverhampton, United Kingdom  George Mason
University, United States  Kai North  Damith Premasiri  Marcos Zampieri

(2022)

###### Abstract

The widespread of offensive content online has become a reason for great
concern in recent years, motivating researchers to develop robust systems
capable of identifying such content automatically. With the goal of carrying
out a fair evaluation of these systems, several international competitions
have been organized, providing the community with important benchmark data and
evaluation methods for various languages. Organized since 2019, the HASOC
(Hate Speech and Offensive Content Identification) shared task is one of these
initiatives. In its fourth iteration, HASOC 2022 included three subtracks for
English, Hindi, and Marathi. In this paper, we report the results of the HASOC
2022 Marathi subtrack which provided participants with a dataset containing
data from Twitter manually annotated using the popular OLID taxonomy. The
Marathi track featured three additional subtracks, each corresponding to one
level of the taxonomy: Task A - offensive content identification (offensive
vs. non-offensive); Task B - categorization of offensive types (targeted vs.
untargeted), and Task C - offensive target identification (individual vs.
group vs. others). Overall, 59 runs were submitted by 10 teams. The best
systems obtained an F1 of 0.9745 for Subtrack 3A, an F1 of 0.9207 for Subtrack
3B, and F1 of 0.9607 for Subtrack 3C. The best performing algorithms were a
mixture of traditional and deep learning approaches.

###### keywords:

offensive language hate speech text categorization Marathi

##  1 Introduction

Social media is the main source of information, current events, and public
opinion for many individuals [1]. Users typically spend on average 2 hours per
day on social media platforms, such as Twitter, Facebook, and Instagram [2].
The impact social media has on an individual’s perception of the world, as
well as their mental well-being, is substantial [3]. Social media often
contains hateful and offensive content, including but not limited to
pejorative language [4], cyber-bullying [5, 6, 7], online extremism [8], and
racism [9, 10]. Such content has been shown to cause anxiety or depression
[3], incite self-harm [11], or violence towards others [12].

With governments realizing the impact social media has on public health and
opinion, social media companies have invested in ways to cope with hateful and
offensive content [12, 13]. To achieve this, they are interested in the
development of machine learning (ML) models to automatically detect such
content on their social media platforms. These models have traditionally
consisted of Naive Bayes (NB) [14, 15, 9], support vector machines (SVMs) [16,
17, 18, 19], and random forests (RFs) [16, 20] while, more recently, neural
networks [21, 22, 23, 24], and transformer-based models [23, 25, 26, 27] have
been applied to this problem.

As evidenced by recent international competitions on this topic [28, 29, 30,
31, 32, 33, 34], a known challenge in detecting offensive and hateful content
is the intrinsic subjectivity of such phenomena. In highly multilingual
countries, such as India, this task becomes even more challenging. India has
22 official languages, many of which do not have large annotated datasets that
can be used for the training and evaluation of hate speech and offensive
language identification models. Marathi is an example of such an under-
resourced language, especially when in comparison to high-resourced languages,
such as Hindi or English [35].

This paper presents the results of Subtrack 3 at HASOC-2022 [36]: Offensive
Language Identification in Marathi. HASOC-2022 challenged participating teams
with the development of state-of-the-art hate speech and offensive language
identification models. Subtrack 3 at HASOC-2022 aimed to encourage the
development of such models for Marathi and thus encourage more research into
the future development of models for under-resourced languages.

This paper is structured as follows. Section 2 introduces previous hate speech
and offensive language shared-tasks. Section 3 introduces Subtrack 3. Section
4 describes the data used for this subtrack. Section 5 lists the approaches
adopted by each participating team. Section 6 provides each teams’
performance. Section 7 gives a conclusion on what is the current state-of-the-
art for identifying hate speech and offensive language in Marathi and makes
predictions regarding the direction of future research.

##  2 Related Work

Numerous shared-tasks have been organized with the goal of providing
benchmarks for the automatic identification of hateful and offensive content.
These shared-tasks have consisted of differing forms classification, being
either binary, multi-class, or multi-label classification, and contained
differing subtracks aimed at predicting the type or target of the offensive
content as well as dealing with offensive content in languages other than
English.

###  2.1 Related Tasks

#### TRAC 2018-2022

The first Trolling, Aggression and Cyberbullying (TRAC-2018) workshop hosted
at COLING [28], challenged participating teams to create multi-class
classifiers for predicting which of three levels of aggression is depicted by
a Facebook comment: overt aggression, covert aggression, or nonaggression.
Their dataset consisted of randomly sampled English and Hindi Facebook
comments. Several issues were reported to the organizers regarding dataset
quality: 1). some instances contained code-mixing between English and Hindi,
and 2). some instances were considered incorrectly labelled due to their
subjectivity. The second TRAC workshop (TRAC-2020) hosted at LREC [34],
introduced a new language, Bengali, as well as a new domain, Youtube comments,
to resolve the previous issues in data quality. TRAC-2020 also provided a new
subtrack in the form of binary classification of gender: gendered (GEN) or
non-gendered (NGEN) instances. The third TRAC workshop (TRAC-2022) [37]
expanded this subtrack further to include the recognition of other types of
offensive content, including racial bias, religious intolerance, and casteist
bias, as well as the identification of the discursive role of a given instance
in relation to a previous comment. It also introduced a new domain, “COVID-19
related conversation” [37]. Performances on TRAC-2018 varied with neural
networks performing on par with other approaches, such as SVMs and RFs [28].
TRAC-2020 and 2022, however, saw transformer-based models achieve the best
performances [34]; teams that used an ensemble of transformers performed
exceptionally well in comparison to other approaches.

#### HatEval 2019

Task 5 at SemEval-2019: Multilingual Detection of Hate Speech Against
Immigrants and Women in Twitter (HatEval) [31], contained two subtracks .
Subtrack A was a binary classification task. It saw English or Spanish tweets
be classified into hatespeech or non-hatespeech instances. Subtrack B was
evaluated as a multi-label classification task which required the identified
instances of hatespeech to be further grouped in regards to their “aggressive
attitude and the target harassed” [31]. The organizers suggested that
participating teams divide this task into two binary classifications tasks,
whereby each instance’s target is classified as being aggressive or non-
aggressive, and its attitude is classified as being either a targeted message
(individual) or a hateful message in general (generic). The first and second
top performing models for Subtrack A were both SVMs having outperformed neural
networks and other approaches. Performances on Subtrack B were noticeably
worst than Subtrack A. The organizers contributed this to its multi-labelled
nature.

#### OffensEval 2019-2020

Task 6 at SemEval-2019: Identifying and Categorizing Offensive Language in
Social Media (OffensEval-2019) [29], presented three subtracks to
participating teams ranging for binary to multi-class classification on the
Offensive Language Identification Dataset (OLID) consisting of English tweets.
Subtrack A saw the binary classification of tweets with non-offensive and
offensive labels. Subtrack B and C was concerned with identifying the target
of the offensive tweet. Subtrack B was the binary classification of instances
into either targeted or untargeted insults, whereas Subtrack C was the multi-
class classification of instances into three labels concerned with the target
type: “individual, group, or other” [29]. Task 12 at SemEval-2020:
Multilingual Offensive Language Identification in Social Media
(OffensEval-2020) [33] was an extension of the previous shared-task. It
included the same subtracks as SemEval-2019, but presented additional tracks
and data in Arabic, Danish, English, Greek, and Turkish. Both shared-tasks saw
transformer-based models achieve the best results, such as BERT, RoBERTa, and
mBERT. The multilingual nature of OffensEval-2020 also allowed for cross-
lingual transfer learning, which was adopted by the highest performing
systems.

###  2.2 HASOC

#### HASOC 2019-2020

The Hate Speech and Offensive Content Identification in Indo-European
Languages track (HASOC-2019) at FIRE [30], held three subtracks. Data was
collected from Twitter and Facebook in Hindi, German, and English. Subtrack A
was a binary classification task to distinguish between hateful and non-
hateful instances. Subtracks B and C were more fine-grained classification
tasks to identify the type (hate speech, offensive, and profanity) and target
(targeted or untargeted) of each instance respectively. HASOC-2020 [32] saw
the removal of Subtrack C and the creation of a new dataset with the aim of
minizing bias in the data collection process. This was achieved through the
adoption of a more randomized sampling technique that minimizied the selection
of types of hate-speech known to the dataset’s creators [32]. Neural networks
achieved the best performances at HASOC-2019, with an LSTM model achieving the
highest performance for English [21]. BERT-based models become more popular at
HASOC-2020 and achieved the best performance for German. However, LSTM-based
models again reported the best results for Hind and English. Overall
performances were considered to be worst for HASOC-2020 than they were for
HASOC-2019 [32]. Organizers’ reported this as being due to the more realistic
sampling technique used in HASOC-2020.

#### Marathi at HASOC-2021

HASOC-2021 [38] included the same two subtracks as HASOC-2020 for English and
Hindi, but was the first to apply these subtracks to Marathi. The data used
for Marathi was extracted from the MOLD dataset [35]. The MOLD dataset
consists of 2,547 hateful and non-hateful tweets extracted from Twitter’s API.
Its hateful tweets were obtained by searching for keywords “related to
politics, entertainment, and sports along with the hashtag #Marathi” [35]. All
instances were then labelled by six Marathi speakers. The best performing
models for Marathi used cross-lingual transfer learning [39]. Nene et al. [39]
trained an XLM-R model on pre-existing data in Hindi from HASOC 2019. They
then used the returned model’s updated weights and softmax activation layer to
initialize the weights of a new XLM-R model on the provided Marathi dataset.
The second best performing model adopted a similar approach. Glazkova et al.
[40] fine-tuned the monolingual LaBSE and multilingual XLM-R models on Marathi
and Hindi data and discovered than the multilingual XLM-R model achieved the
best results. The other top-10 performing models submitted to HASOC-2021’s
Marathi subtrack included feature engineering approaches using word and
character level n-grams [41, 42], an LSTM model [43], as well as other
transformer-based models, such as mBERT and DistilmBERT [44]. However, these
models failed to achieve performances on par with those models fine-tuned on
both Hindi and Marathi data.

##  3 HASOC-2022

###  3.1 Subtrack 3: Offensive Language Identification in Marathi

HASOC-2022 hosted three subtracks with subtracks 1 and 2 dealing with binary
and multi-class classification of German and the code-mixed language,
Hinglish, respectively. Subtrack 3 focused on the automatic identification of
offensive language in Marathi. Subtrack 3 was further divided into three
additional subtracks.

  * •

Subtrack 3A: A binary classification task classifying posts into two classes:
offensive (OFF), and non-offensive (NOT).

  * •

Subtrack 3B: A binary classification task identifying whether a post is either
targeted (TIN) or untargeted (UNT).

  * •

Subtrack 3C: A more fine-grained classification task, whereby the previously
identified TIN posts were further classified into individual (IND), group
(GRP), or other (OTH).

##  4 Data

The data used for Subtrack 3 consisted of extracts taken from MOLD 2.0 [45].
MOLD 2.0 is an expansion of the original MOLD dataset (v1.0) [35]. It contains
3,600 tweets with an additional 1053 instances on top of the original
instances provided by MOLD (v1.0) [35]. All instances were annotated using the
popular OLID taxonomy [46] with additional information being provided in
regards to the target (level B) and target type (level C) of the offensive or
non-offensive tweet. The annotation was carried out by three native speakers
of Marathi with an inter-annotator agreement of 0.79 Cohen’s kappa across all
instances.

The MOLD 2.0 dataset was split into an 80/20 split between train and test
sets, respectively. An unequal number of offensive (1067) and non-offensive
(2034) instances were provided to the participating teams resulting in the
label distribution shown in Table 1.

A | B | C | Training | Test | Total
---|---|---|---|---|---
OFF | TIN | IND | 503 | 51 | 554
OFF | TIN | OTH | 80 | 56 | 136
OFF | TIN | GRP | 157 | 51 | 208
OFF | UNT | — | 327 | 102 | 429
NOT | — | — | 2,034 | 250 | 2,284
All |  |  | 3,101 | 510 | 3,611

Table 1: MOLD v2.0 - Distribution of labels. Table taken from [45].

##  5 Approaches

In total, 8 out of the 10 participating teams submitted system reports for
Subtrack 3. 10 teams took part in Subtrack 3A, 7 in Subtrack 3B, and 6 in
Subtrack 3C. System descriptions are provided below.

###  5.1 Traditional

#### ssncse_nlp

Vani and Bharathi [47] used Scikit-learn’s CountVectorizer function [48] to
generate vectors of term/token counts. They fed these vectors as features into
four types of traditional classifier: RF, SVM, logistic regressor, and a KNN
classifier. They discovered that their RF model achieved the greatest
performance and submitted this model for all three subtracks.

#### satlab

Bestgen [49] submitted a simplified version of their previous model used for
HASOC-2021 [38]. They trained a logistic regressor taken from the LIBLinear
package [50] on several character-level n-gram features. These features
consisted of n-gram length, sentence position, and BM25 frequency. All n-grams
with a frequency of less than two and over five characters in length were not
considered. Feature scores were normalized per L2 regularization, and
parameter optimization was conducted using four-fold cross validation over
several random grid searches. This model was submitted to all three subtracks.

#### ml_ai_iiitranchi

Kumari and Singh [51] experimented with a logisitic regressor, SVM, and RF.
They lemmatized each instance and trained their models on token frequencies
obtained using Scikit-learn’s TF-IDF-Vectorizer function [48]. Their logistic
regressor surpassed the performances of the other three models for Subtrack
3A, whereas their SVM achieved the best performance for subtracks 3B and 3C.
They submitted their best performing models to each of the subtracks,
respectively.

###  5.2 Deep Learning

#### optimize_prime

Chavan et al. [52] compared the performances of various BERT-based models
taken from HuggingFace that have been pre-trained on Marathi data [45]: MuRIL
[53], MahaTweetBERT [54], MahaTweetBERT-Hateful [54], and MahaBERT [55]. They
cleaned each instance provided by the shared-task of placeholder texts, such
as “@USER”, and removed hastags, URLs, empty parentheses and new lines. Each
instance was then automatically encoded and then fed into each model. The
AdamW optimizer was used during training with a learning rate of 1e-5 and
batch size of 32. The training was conducted over 25 epochs. The MahaTweetBERT
[54] outperformed all of models when trained on a combination of instances
from HASOC-2021 [38] and HASOC-2022 [36]. This model was only submitted for
Subtrack 3A.

#### hate-busters

Ripoll et al. [56] treated Subtrack 3 as a multi-class classification problem
instead of a multi-label or several binary classification tasks, and attempted
to predict all possible labels at once (as shown in Table 1). They applied
five and ten-fold cross validation by splitting the training and test sets
into n independent sets and trained an ensemble of n separate models. The
final output label was derived from plurality voting. They tested two
different types of model: InfoXLM [57] and L3Cube [55]. The former being
trained on cross-lingual representations of 100 languages, whereas the later
being trained solely on Hindi and Marathi data. The more language-specific
L3Cube model achieved the greatest performance in their ensemble architecture.
This ensemble was submitted for all three subtracks.

#### sakshi_hasoc

Kalra et al. [58] fine-tuned mBERT [59], DistilmBERT [60], and MuRIL [53] on a
resampled version of the provided training set to compensate for the imbalance
between offensive and non-offensive instances. They oversampled the training
set by including duplicates of offensive instances that made for a 0.5 ratio
between the two labels. They achieved this using the RandomOverSampler
function in the imblearn library [61]. MuRIL was found to achieve the best
performance when trained over 4 epochs and set to a learning rate of 1e-4 and
a batch size of 4. They submitted this model to Subtrack 3A.

#### irlab@iitbhu

Chanda et al. [62] removed usernames, punctuation, URLS, and emojis. They then
concatenated the tweet, its comments, and responses and fed these concatenated
instances into Fast.ai [63], being an off-the-shelf deep learning package.
They submitted their Fast.ai model to all three subtracks.

#### citk_isi

Ghosh et al. [64] compared the performance of mBERT [59] and MahaBERT [55]
with the latter being pre-trained on the L3 Cube-MahaCorpus. Each model was
trained with a learning rate of 1e-5 and a batch size of 5 over 5 epochs.
MahaBERT was found to outperform mBERT and was submitted for all three
subtracks.

##  6 Results

Team performances are displayed within tables 2 to 4 and are discussed
throughout the following section.

Rank | Name | Number of Runs | Precision | Recall | F1
---|---|---|---|---|---
1 | ssncse_nlp [47] | 4 | 0.9758 | 0.9741 | 0.9745
2 | optimize_prime [52] | 2 | 0.9588 | 0.9589 | 0.9588
3 | hate-busters [56] | 5 | 0.9592 | 0.9562 | 0.9568
4 | sakshi_hasoc [58] | 3 | 0.9464 | 0.9446 | 0.9450
5 | satlab [49] | 2 | 0.9372 | 0.9373 | 0.9372
6 | irlab@iitbhu [Hasoc_2022_IRLab@IITBHU@IITBHU] | 5 | 0.9353 | 0.9355 | 0.9353
7 | ml_ai_iiitranchi [51] | 2 | 0.9292 | 0.9203 | 0.921
8 | citk_isi [Hasoc_2022_CITK_ISI_ISI] | 1 | 0.9021 | 0.9022 | 0.9020
9 | gunjan | 3 | 0.8395 | 0.8071 | 0.7997
10 | the shivi hunters | 1 | 0.8053 | 0.7775 | 0.7746

Table 2: Results of Subtrack 3A. Ordered from best to least performing system.

Rank | Name | Number of Runs | Precision | Recall | F1
---|---|---|---|---|---
1 | hate-busters [56] | 5 | 0.9114 | 0.9342 | 0.9207
2 | satlab [49] | 2 | 0.9060 | 0.9288 | 0.9149
3 | ssncse_nlp [47] | 4 | 0.7587 | 0.7067 | 0.6958
4 | irlab@iitbhu [Hasoc_2022_IRLab@IITBHU@IITBHU] | 1 | 0.4837 | 0.6107 | 0.5354
5 | ml_ai_iiitranchi [51] | 2 | 0.4516 | 0.4371 | 0.4429
6 | citk_isi [Hasoc_2022_CITK_ISI_ISI] | 1 | 0.3405 | 0.2869 | 0.3073
7 | gunjan | 1 | 0.1111 | 0.2073 | 0.0897

Table 3: Results of Subtrack 3B. Ordered from best to least performing system.

Rank | Name | Number of Runs | Precision | Recall | F1
---|---|---|---|---|---
1 | satlab [49] | 2 | 0.9365 | 0.9893 | 0.9607
2 | ssncse_nlp [47] | 4 | 0.7963 | 0.8663 | 0.7929
3 | ml_ai_iiitranchi [51] | 2 | 0.7927 | 0.7464 | 0.7423
4 | hate-busters [56] | 5 | 0.7999 | 0.7705 | 0.7255
5 | irlab@iitbhu [Hasoc_2022_IRLab@IITBHU@IITBHU] | 1 | 0.2858 | 0.4213 | 0.2891
6 | citk_isi [Hasoc_2022_CITK_ISI_ISI] | 1 | 0.2323 | 0.1961 | 0.2064

Table 4: Results of Subtrack 3C. Ordered from best to least performing system.

Team ssncse_nlp [47] achieved the highest F1 of 0.974 for Subtrack 3A. Their
RF model surpassed the performance of several transformer-based models,
optimize_prime [52] and hate-busters [56], that came 2nd and 3rd place
respectively. Other traditional approaches also performed on par with deep
learning approaches for this subtrack, with the logistic regressors submitted
by satlab [49] and ml_ai_iiitranchi [51] achieving similar performances to the
transformer-based models provided by irlab@iitbhu [62] and citk_isi [64]. The
size of the provided dataset is a possible explanation. Deep learning
approaches were likely unable to infer useful feature representations that are
able to differentiate between offensive and non-offensive instances, given
that the provided dataset only contains 1067 offensive and 2034 non-offensive
tweets. The submitted traditional approaches, on the other hand, were all
engineered with features, such as token frequency or character-level n-grams,
that have been proven to be useful for hate speech and offensive language
identification [30, 32, 38]. Those deep learning approaches that did perform
well, namely optimize_prime [52] and sakshi_hasoc [58], also increased the
amount of available training data by using augmented datasets. These augmented
datasets included data from HASOC-2021 [38] or oversampled instances [58] that
likely increased their models’ ability to infer distinguishing features.

Team hate-busters [56] came 1st place with an F1 of 0.921 for Subtrack 3B,
closely followed by satlab [49] having achieved an F1 of 0.915. Team hate-
busters was the only team that treated all three subtracks as one multi-class
classification problem and that adopted an ensemble architecture. Their
ensemble of L3Cude models in particular may be responsible for their high
performance, as their final target output labels were generated by averaging
those of multiple transformers trained on 10-folds. This is further supported
by the performance of the non-ensemble-based deep learning approaches
submitted by irlab@iitbhu [62] and citk_isi [64], with the latter team’s
submission also being pre-trained on L3Cude data. These teams scored
drastically worst F1s of 0.5354 and 0.3073 respectively. However, the
character-level n-gram features used by satlab appeared to have performed
equally as well as hate-busters’s ensemble. The use of such features as n-gram
length, sentence position, and BM25 frequency, were likely able to identify
character combinations or words, including pronouns, that are useful for the
identification of targeted instances [49]. In addition, it appears that satlab
did not remove meta data, such as “@USER”, that would likewise have been
beneficial; many of the other submissions removed such data during their pre-
processing [52, 62].

The 1st, 2nd, and 3rd leading teams of Subtrack 3C all used traditional
approaches. The logistic regressor of satlab [49] came first place with F1 of
0.9607, whilst the RF and SVM of ssncse_nlp [47] and ml_ai_iiitranchi [51],
came 2nd and 3rd place with F1s of 0.7929 and 0.7423 respectively. The deep
learning approaches submitted to this subtrack all achieved inferior
performances with the ensemble submitted by hate-busters [56] attaining the
best F1 of 0.7255 and the non-ensemble-based models submitted by irlab@iitbhu
[62] and citk_isi [64] getting noticeably worst F1s of 0.2891 and 0.2064
respectively. Traditional approaches were therefore superior for Subtrack 3C.
A highly likely explanation for this is once again the size of the provided
dataset. Subtrack 3C was a multi-class classification task, whereby models had
to differentiate between three types of target for each instance: individual
(IND), group (GRP), or other (OTH). However, only 503 instances with an
individual target, 157 instances with a group target, and 80 instances with a
target marked as other, were provided to the participating teams (Table 1). A
recommended approach for dealing with such small sample sizes is to apply
cross-lingual transfer learning for transformer-based models, such as training
transformers on large datasets in Hindi and applying said models to Marathi,
as shown by the winning system of HASOC-2021 [39].

##  7 Conclusion and Future Work

This paper has described Subtrack 3 of HASOC-2022: Offensive Language
Identification in Marathi. Subtrack 3 consisted of three additional subtracks
for Marathi concerned with first distinguishing between hateful and non-
hateful tweets, and then identifying whether a tweet had a target as well as
its target type. In total, 10 teams submitted systems to this task and those
that submitted system reports have been described in Section 5. Team
performances have been listed and discussed in Section 6. In general,
traditional approaches achieved the best performances, especially for Subtrack
3C. A RF (ssncse_nlp) came 1st, 3rd, and 2nd place for subtracks 3A, 3B and 3C
respectively [47], and a logistic regressor trained on n-gram features
(satlab) came 2nd place for Subtrack 3B and 1st place for Subtrack 3C [49].
Deep learning approaches performed less well. However, those that that
utilized an augmented dataset (optimize_prime and sakshi_hasoc) [52, 58], or
that had an ensemble architecture (hate-busters) [56] outperformed those that
used a singular pre-trained deep learning model (irlab@iitbhu and citk_isi)
[62, 64]. We believe this to be due to the small size of the provided dataset.

Marathi continues to be an under-resourced language for many NLP-related
tasks. Subtrack 3 aimed to encourage the development of ML models that were
capable of performing well, regardless of having a limited amount of available
data. The results of Subtrack 3 indicate that traditional approaches can still
be effective when given a small training set. Nevertheless, ensemble
architectures and cross-lingual transfer learning would likely surpass the
performance of traditional approaches if said training sets were supplemented
with additional instances in Marathi or Hindi. We suspect these to become
common approaches within future iterations of this shared-task.

###### Acknowledgements.

We would like to thank the teams referenced in this report for participating
in the competition. We further thank the organizers of the other HASOC tracks
for their support.

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