YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds

In this paper, we present YunMa, an exoplanet cloud simulation and retrieval package, which enables the study of cloud microphysics and radiative properties in exoplanetary atmospheres. YunMa simulates the vertical distribution and sizes of cloud particles and their corresponding scattering signature in transit spectra. We validated YunMa against results from the literature. When coupled to the TauREx 3 platform, an open Bayesian framework for spectral retrievals, YunMa enables the retrieval of the cloud properties and parameters from transit spectra of exoplanets. The sedimentation efficiency (f sed), which controls the cloud microphysics, is set as a free parameter in retrievals. We assess the retrieval performances of YunMa through 28 instances of a K2-18 b-like atmosphere with different fractions of H2/He and N2, and assuming water clouds. Our results show a substantial improvement in retrieval performances when using YunMa instead of a simple opaque cloud model and highlight the need to include cloud radiative transfer and microphysics to interpret the next-generation data for exoplanet atmospheres. This work also inspires instrumental development for future flagships by demonstrating retrieval performances with different data quality.

1. INTRODUCTION Thousands of exoplanets have been detected since the late 20 th century.During the past decade, transit spectroscopy has become one of the most powerful tech-niques for studying exoplanets' atmospheres in-depth (e.g., reviews by Tinetti et al. 2013;Burrows 2014;Madhusudhan 2019).Data recorded from space-borne instruments (e.g., Hubble, Spitzer and James Webb Space Telescopes) or from the ground have revealed important information about exoplanet atmospheric chemistry and dynamics (e.g., Sing et al. 2016;Tsiaras et al. 2018;Welbanks et al. 2019;Changeat et al. 2022;Edwards et al. 2022; JWST Transiting Exoplanet Community Early Release Science Team 2022; Venot et al. 2020;Roudier et al. 2021) and may provide insight into planetary interior composition and formation (Madhusudhan et al. Ma et al. 2020;Yu et al. 2021;Tsai et al. 2021;Charnay et al. 2022).
Due to the -currently limited -observational constraints and computational resources available to simulate the complexity of clouds, retrieval studies of cloudy atmospheres are still in their infancy (see e.g., Fortney et al. 2021).For instance, many studies have adopted wavelength-independent opaque clouds, where all the radiation beneath the cloud top is blocked from reaching the telescope, and retrieve the vertical location of clouds (Boucher et al. 2021;Brogi & Line 2019).Wakeford et al. (2018) used a grey, uniform cloud in the ATMO Retrieval Code (ARC) (Goyal et al. 2017;Drummond, B. et al. 2016;Tremblin et al. 2015).Other models constrain from radiative transfer the uniform cloud particle sizes without being estimated through cloud mi-crophysics models.For instance, Benneke et al. (2019 a) have initially estimated the particle sizes in the atmosphere of GJ 3470 b using Mie-scattering theory.Extended from this highly parametric approach, cloud scattering parameters and inhomogeneous coverage were also retrieved: NEMESIS was used by Barstow (2020) and Wang et al. (2022) to retrieve the cloud's opacity, scattering index, top and base pressures, particle sizes and shape factor.Pinhas et al. (2019) run POSEIDON to constrain the cloud's top pressure and coverage fraction.Wang et al. (2022) adopted PICASO to extract the cloud's base pressure, optical thickness, single scattering albedo, scattering asymmetry and coverage.Lueber et al. (2022) extended the use of Helios-r2 to retrieve non-grey clouds, with extinction efficiencies estimated from Mie theory calculations.The model Aurora (Welbanks & Madhusudhan 2021) presents inhomogeneities in cloud and haze distributions by separating the atmosphere horizontally into four distinct areas.
The data provided by the next-generation telescopes will be greatly superior in quality and quantity, allowing us to obtain more stringent constraints to our understanding of clouds in exoplanetary atmospheres.Transit spectra of exoplanets recorded from space by the James Webb Space Telescope (JWST, 0.6-28.3µm, Bean et al. 2018;Greene et al. 2016;Gardner et al. 2006), Ariel (0.5-7.8 µm, Tinetti et al. 2018;Tinetti et al. 2021) and Twinkle (0.5-4.5 µm, Edwards et al. 2019) at relatively high spectral resolution and/or broad wavelength coverage will open the possibility of integrating selfconsistent, cloud microphysics approaches into atmospheric retrieval codes.A good example of such models is ARCiS (ARtful modelling Code for exoplanet Science, Min et al. 2020;Ormel & Min 2019), which simulates cloud formation from diffusion processes and parametric coagulation.ARCiS also generates cloudy transit spectra from Mie theory (Fleck & Canfield 1984) and Distribution of Hollow Spheres (DHS, Min, M. et al. 2005;Mollière, P. et al. 2019), and can be used to retrieve the cloud diffusivity and nuclei injection from transit spectra.
In this work, we present a new optimised model to study cloud microphysical processes directly integrated into a spectral retrieval framework.We consider clouds as a thermochemical product, i.e. the aggregation of condensates in the atmosphere, while hazes form photochemically (Kawashima & Ikoma 2018).The cloud distribution depends on the atmospheric conditions.Being generated thermochemically, clouds form and diffuse depending on the atmospheric thermal structure and, in return, contribute to it.They also depend on the mixing profiles of the condensable gases in the atmosphere.
Clouds act as absorbers and/or scatterers and therefore may dampen the atomic and molecular spectroscopic features and change the continuum.
Based on studies of the Earth and Solar System's planetary atmospheres, Lewis (1969) published a 1-D cloud model optimised to describe tropospheric clouds in giant planets.This model assumes that the fall speeds of all condensates are equivalent to the updraft velocities, and only vapour is transported upward.Lunine et al. (1989) included a correlation between cloud particle sizes, downward sedimentation and upward turbulent mixing.Based on previous models by Lewis (1969), Carlson et al. (1988), Lunine et al. (1989) and Marley et al. (1999), Ackerman & Marley (2001) proposed a new method to estimate the mixing ratio and vertical size distribution of cloud particles (A-M model hereafter).In the A-M model, the sedimentation timescale is estimated through cloud microphysics, taking into account the atmospheric gas kinetics and dynamical viscosity.The model assumes an equilibrium between upward turbulent mixing and sedimentation, where the turbulent mixing is derived from the eddy diffusion in the atmosphere.The key assumptions of the A-M model are as follows: 1. Clouds are distributed uniformly in the horizontal direction.
2. Condensable particles rain out at (super)saturation while maintaining a balance of the upward and downward drafts.
3. It does not consider the cloud cover variations caused by precipitation or the microphysics between different types of clouds.
The A-M model was originally proposed for giant exoplanets and brown dwarfs and was tested on Jupiter's ammonia clouds, demonstrating that this approach is applicable to a broad range of temperatures and planetary types.
Another popular 1-D cloud microphysics model is the Community Aerosol and Radiation Model for Atmospheres (CARMA), initially developed for the Earth's stratospheric sulfate aerosols (Turco et al. 1979;Toon et al. 1979).CARMA is a time-dependent cloud microphysics model which solves the discretised continuity equations for aerosol particles starting from nucleation.Gao et al. (2018) extended the use of CARMA to simulate clouds on giant exoplanets and brown dwarfs by including additional condensates predicted to form in hot atmospheres and compared the results with the A-M model.The A-M model, while able to provide the cloud particle sizes and number density distributions, is of intermediate numerical complexity and, therefore, potentially adaptable to be included in retrieval codes.In addition to the original implementation by Ackerman & Marley (2001), Virga (Rooney et al. 2022) simulates the cloud's particle size distribution from the A-M approach and estimates separately the sedimentation efficiency.PICASO (Robbins-Blanch et al. 2022;Batalha et al. 2019) adopts Virga to simulate cloudy exoplanetary atmospheres.Adams et al. (2022) couples MIT GCM and Virga to include clouds in 3-D models.The above are forward simulations only.One step further, Xuan et al. (2022) present retrieval studies on HD 4747 B with clouds using petitRADTRANS (Mollière, P. et al. 2020), where the cloud simulation in retrieval is motivated by the A-M approach.To estimate the cloud mixing ratio in the retrieval iterations, this model does not solve the full ODE (see Equation 2).Instead, it adopts an approximation of the A-M approach, assuming the mixing ratio of condensable gas above the cloud base is negligible.
To simulate inhomogeneities for cloud formation in the horizontal direction, we would need to consider global circulation atmospheric effects, such as those modelled in Cho et al. (2021).An example of a 3-D atmospheric model with clouds is Aura-3D (Welbanks & Madhusudhan 2021;Nixon & Madhusudhan 2022).The retrieval part for Aura-3D is highly parametrised, both for the atmospheric and cloud parameters.Helling et al. (2023Helling et al. ( , 2019) ) have simulated global cloud distributions by generating inputs to their kinetic cloud model from pre-calculated 3-D Global Circulation Models (GCMs).Unfortunately, these complex models require excessive computing time.In addition, the data expected to be observed in the near future are unlikely to constrain the large number of parameters needed in a 3-D model.Therefore, while theoretical studies with 3-D models are very important to progress in our understanding of clouds in exoplanetary atmospheres and as benchmarks, they are currently less useful for interpreting available data.
In this paper, we present a new cloud retrieval model, YunMa, optimised for transit spectroscopy.In YunMa, we built the cloud model based on Ackerman & Marley (2001) and simulated the cloud contribution to transit spectra using extinction coefficients as calculated by the open-source BH-Mie code (Bohren & Huffman 2008).
YunMa is fully integrated into the TauREx 3 retrieval platform (Al-Refaie et al. 2022a;Al-Refaie et al. 2021) and, for the first time, provides cloud microphysics capabilities into a retrieval model.We describe the model in Section 2. In Section 3, we detail the experimental setups.In Section 4, we validate particle size dis-Ma et al.
tributions and spectroscopic simulations against previous studies published in the literature.After validation, we show new spectral and retrieval simulations obtained with YunMa.In Section 5, we discuss our results and assumptions and identify possible model improvements to be considered in future developments.

YUNMA DESCRIPTION
YunMa estimates the vertical distribution of the cloud particle sizes (VDCP hereafter, see, e.g.Fig. 3 b and 11 a, b) based on A-M model and their contribution to the radiative transfer calculations.The YunMa module has been integrated into the TauREx 3 retrieval platform: the combined YunMa-TauREx model is able to constrain the VDCP from observed atmospheric spectra, as described in detail below.
2.1.Modelling the cloud particle size distribution YunMa model contains a numerical realisation of the A-M microphysical approach to simulate the VDCP.We show in Fig. 1 a pictorial representation of the A-M approach: it assumes that clouds form with different VDCP to maintain the balance between the upward turbulent mixing and downward sedimentation of the condensable species.Depending on the atmospheric T -p profile, multiple cloud layers may form.

Cloud mixing profile
Cloud particles start forming when the partial pressure of a certain gas exceeds the saturation vapour pressure (SVP): the formation strongly depends on the atmospheric thermal structure.The condensation process, occurring when the partial pressure exceeds the SVP, is estimated by comparing the molecular mixing ratio of the gas phase with its saturation vapour mixing ratio: where q c is the mixing ratio of the condensed species, z is the altitude, q s is the mixing ratio where the condensable gas saturates, q t is the total mixing ratio of a condensable chemical species, including both the condensate and gas phases, and S c is the supersaturation factor which persists after condensation.q s can be estimated from the ratio between the SVP of a certain chemical species and the atmospheric pressure at the same altitude.Note that in this paper, the mixing ratio refers to the volume fraction of a chemical species in the atmosphere.
In the A-M approach, the turbulent mixing of the condensate and vapour is assumed to be in equilibrium with the sedimentation of the condensate: Figure 1.Sketch of the A-M micophysical approach adopted in YunMa.Cloud particles may form when the mixing ratio of the condensable gas exceeds the saturation mixing ratio, which is derived from its saturation vapour pressure (SVP).The vertical distribution of the cloud particle sizes (VDCP) is derived from the balance between the sedimentation of the cloud particles and the atmospheric turbulent mixing.
where K (m 2 s −1 ) represents the vertical eddy diffusion coefficient, and w * (m s −1 ) is the convective velocity.f sed is the ratio between the mass-weighted droplet sedimentation velocity and w * , defined as: here ρ a is the atmospheric mass density, which can be estimated through the Ideal Gas Law, ρ p is the mass density of a condensed particle, ε is the ratio between the molecular weights of the condensates and the atmosphere, and v f is the sedimentation velocity which will be explained later.The first term in equation (2) describes the upward vertical draft derived from the macroscopic eddy diffusion equation.The second term describes the downward sedimentation, which is in equilibrium with the first term.The eddy diffusion coefficient (K) is one of the key parameters affecting cloud formation.In free convection (Gierasch & Conrath 1985), it can be estimated as: where H, µ and c p are, respectively, the atmospheric scale height, mean molecular weight and specific heat capacity.F = σT 4 eff is the approximated radiative flux.The turbulent mixing length (L) is the scale height of the local stability in eddy diffusion, as opposed to the atmospheric scale height (H).YunMa has an application programming interface for K and it can use the values provided by disequilibrium chemistry models plugged by the users into TauREx 3 -e.g. the kinetic model plugin of TauREx 3 (Al-Refaie et al. 2022b), and derive L accordingly, from equation (4).The convective velocity scale (w ⋆ ) mentioned above can also be estimated as a ratio between K and L. There are different ways to constrain K from the atmospheric chemical and vertical advective time scales (e.g.Baeyens et al. 2021;Komacek et al. 2019;Parmentier et al. 2013;Zhang & Showman 2018).However, the current estimations of K in the literature lack validation from observations.While YunMa is designed to be self-consistent with these approaches, this paper uses constant K in the experiments as a firstorder estimation.Free convection is justified by assuming the cloud forms in the deep convective layer of the atmosphere and by neglecting 3-D effects.These approximations in retrieval studies will need to be revisited with the improved quality of the data available and computing facilities.
The sedimentation velocity, denoted by v f , is the speed at which a cloud particle settles within a heterogeneous mixture due to the force of gravity.v f can be estimated through viscous fluid physics: where ∆ρ is the difference between ρ p and ρ a , β is the Cunningham slip factor, and η is the atmospheric dynamical viscosity (See Appendix A for more details of SVP, β and η).

Particle size and number density
Following the A-M approach, we assume spherical cloud particles with radii r.The particle radius at w ⋆ , denoted as r w , can be obtained using these relationships between v f and w ⋆ : and where α corresponds to the sedimentation velocity decrease in viscous flows.In A-M, the particle size distribution was constrained by in-situ measurements of Californian stratocumulus clouds, which followed a broad lognormal distribution.The assumption of lognormal distribution allows estimating the geometric mean radius (r g ), the effective radius (r eff ) and the total cloud particle number density (N ), using the detailed definitions and derivations listed in Appendix B.

Cloud contribution in transit spectra
To estimate the wavelength-dependent cloud contribution to transit spectra, YunMa adopts the scattering theory and absorption cross sections as described in Bohren & Huffman (2008, BH-Mie hereafter), assuming spherical cloud particles.The cross-section of the cloud particles (k λ ) at each wavelength (λ) and particle size are estimated through the extinction coefficient (Q ext ) of the corresponding wavelength and particle size, derived by BH-Mie from the refractive indices of the cloud particles: We used the water ice refractive indices reported in Warren & Brandt (2008) for our simulations of the temperate super-Earth.We show some examples of atmospheres with water ice clouds in Section 4. Postexperimental tests were conducted to avoid the contamination of liquid water particles.We simulate the cloud optical depths from the particle sizes and number densities along each optical path, which passes the terminator at altitude z ter , with a path length s zter of each atmospheric layer: where n is the accumulated number density of particles with a radius smaller than r and z top is the altitude at the top of the atmosphere.The contribution of the clouds to the transit spectra, ∆F c , can be estimated as: where z bottom is the altitude at R p .While YunMa has the capability to include any customised cloud particle size distribution in the spectral simulations, in this paper, we aim at model testing and, for simplicity, we use a single radius bin, i.e. uniform cloud particle size r c = r g (see equation B5 in Appendix B) for each atmospheric layer in the radiative-transfer simulation.
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Cloud simulation validation
We validated our implementation of A-M model against Ackerman & Marley (2001) by comparing the condensate mixing ratios of Jovian ammonia clouds (q c ) with different values of f sed , as shown in Fig. 2. The two sets of results are consistent and the small differences in q c translate into ∼ 10 −2 ppm in transit depth, which is completely negligible compared to typical observational noise.We further validated our implementation against the results from Gao et al. ( 2018) by comparing the KCl cloud molecular mixing ratio and particle-size profile (Fig. 3).Gao et al. (2018) used CARMA to simulate cloud microphysics in exoplanets and brown dwarfs with T eff = 400 K and log g = 3.25, 4.25 and 5.25 (in cgs units), corresponding to planetary masses of 0.72 M J , 8.47 M J , and 44.54 M J .In the best fit between CARMA to A-M model, the f sed = 0.125, 0.093 and 0.025 for the cases with K = 10 2 , 10 3 and 10 4 m 2 s −1 , respectively.The mixing length is derived from constant eddy diffusion, as described in equation ( 4).The results agree with each other within 8%, i.e. ∼ 10 −3 ppm difference in the transit depth.
We compared simulations from YunMa with the results from Charnay et al. (2021 b), which include the horizontal effects generated by global circulation.While very precise comparison and validation are not possible in this case due to the two approaches' very different natures and assumptions, it is useful to test whether we can reproduce similar results when we use consistent assumptions.In the comparison shown in Fig. 4, we used  bined to YunMa allow us to perform retrievals which include cloud microphysical processes and cloud scattering properties.Parameters estimated by TauREx 3 include atmospheric T -p and chemical profiles, planetary (e.g., mass and radius) and stellar (e.g., temperature and metallicity) parameters.The radiative transfer calculations executed by TauREx 3 consider molecular and atomic absorptions, Rayleigh scattering and collisionally induced absorptions (CIA) of H 2 -H 2 and H 2 -He pairs from Cox (2015).
YunMa uses as initial condition the gas mixing ratio profiles provided by TauREx 3 chemistry models (q t , equation 1, 2).In this paper, for simplicity, we assume the baseline chemical abundances are constant with altitude instead of a more complex chemical structure.YunMa then adjusts the gas phase mixing ratios, atmospheric mean molecular weight and atmospheric density in the TauREx 3 chemistry models as a result of the formation of clouds.To simulate transit spectra and perform retrievals, we use the atmospheric grids and optical paths defined in TauREx 3 and add the cloud opacities as estimated by YunMa BH-Mie to the absorptions caused by the chemical species, using the methods explained in Section 2.2.The retrievals were tested on 80 Intel(R) Xeon(R) Gold 6248 CPU @ 2.50GHz.

METHODOLOGY
Table 1.Priors for spectral retrieval experiments using YunMa of all the cases listed in Table 3.In this paper, we use YunMa to perform retrieval simulations of small temperate planets, where we expect a considerable amount of H 2 O to be present in the atmosphere.For simplicity, we consider only water clouds forming in the atmosphere and we do not consider supersaturation cases.The planetary parameters are inspired by K2-18 b (Tsiaras et al. 2019;Charnay et al. 2021 b;Yu et al. 2021), which is a suitable candidate for cloud model testing.We list all the priors of our experiment in Table 1.In this work, we estimate η using the approximation proposed by Rosner (2012, equation A4).We include both scattering and absorption due to water clouds based on Bohren & Huffman (2008), Rayleigh scattering of all the gas species and CIA of H 2 -H 2 and H 2 -He pairs, which are enabled by TauREx 3. We use N 2 as a representative inactive gas undetectable spectroscopically but that contributes to the increase of the atmospheric mean molecular weight, m, and decrease of scale heights H = k B T /mg.H 2 and He act as the filling gases.We use the POKAZATEL dataset for1  For the numerical parameter settings, after a number of tests, we decided to use the explicit Runge-Kutta method of order 8 (DOP853, Hairer et al. 1993) with relative tolerance (rtol) of 1 × 10 −13 and absolute tolerance (atol) of 1 × 10 −16 to solve the partial differential equation ( 2) for all the experiments presented in  (Sumlin et al. 2018).The extinction coefficients for cloud particles with different sizes are estimated from the theory of Bohren & Huffman (2008).To address the computational limitations of retrievals, we pre-calculated the extinction coefficients used in equation ( 8) to estimate the cross-sections of the cloud particles.The pre-calculated list includes values for particle radii from 1 × 10 −7 to 1 × 10 −2 µm, equally spaced in the logarithm space.Here we show only six examples in the top panel.Bottom panels: residuals obtained by subtracting the extinction coefficients as estimated by the two codes, YunMa and PyMieScatt.The maximum discrepancy, corresponding to the largest particle radius simulated here (r = 1 × 10 2 ), is negligible, i.e. ±0.002.
Section 4. We have opted for a logarithm sampling to retrieve most of the atmospheric parameters, e.g., f sed , X H2O and p c .We have used a linear sampling, instead, for N 2 to obtain a better numerical performance.The priors are sufficiently unconstrained to avoid biases generated by excessive pre-knowledge, as discussed, e.g. in Changeat et al. (2021a).After a number of tests, we have chosen to use 400 live points for 3-dimensional retrievals and 1000 for more dimensions.
To begin with, we run a sensitivity study with YunMa about the planetary and instrumental parameters.We set the planetary radius (R p ), f sed and X H2O as free parameters in our 3-dimensional retrieval tests.We list in Table 1 the planetary parameters adopted in the simulations, the prior ranges and the sampling modes.The simulations are conducted with 80 atmospheric layers, from 10 bar to 10 −6 bar, which encompass the typical observable atmospheric range for super-Earths.We select Case 2 in Table 3 as the nominal case, and test the model sensitivity to the key parameters in the retrievals.In Case 2, clouds dampen the gas spectroscopic features but do not obscure them entirely (see Fig. 6).The nominal f sed refers to the value adopted in A-M.The water SVP (both liquid and ice) used are taken from Appendix A in Ackerman & Marley (2001).In this experiment, we will perform sets of retrievals with the aim of:

T-P PROFILE
Isothermal T -p profiles, as commonly used in transit retrieval studies, are too simplistic for cloud studies.In our experiments, we first assume T -p profiles with a dry adiabatic lapse rate in the troposphere, a moist adiabatic lapse rate in the cloud-forming region and a colder isothermal profile above the tropopause.However, to be compatible with the computing requirements in retrieval, we simplify it to a TauREx "two-point" profile (e.g., Fig. 8 in Section 4, "N-point" performances evaluated in Changeat et al. 2021a).We define as T c and p c the temperature and pressure at the tropopause and T surf the temperature at 10 bar.The two-point profile is a fit to the points (T c , p c ) and (T surf , p surf ).One condition of cloud formation is that the atmospheric pressure exceeds the SVP, which is influenced by the thermal gradient in the lower atmosphere controlled by T surf and T c .These factors determine the location where the pressure exceeds the SVP and therefore the cloud formation.

INSTRUMENTAL PERFORMANCE
The new generation of space-based facilities, such as JWST and Ariel, will deliver unprecedentedly highquality data in terms of wavelength coverage, signal-tonoise ratio and spectral resolution.We select as nominal case transit spectra covering 0.4-14 µm, at a spectral resolution of 100, with 10 ppm uncertainty across wavelengths.We chose 0.4 µm as the blue cut-off to maximise the information content about Rayleigh scattering and 14 µm as the red cut-off to maximise the information content about water vapour and atmospheric temperature for the type of planets considered here.The choice of wavelength coverage and precision are inspired by current and planned instrumentation, while not trying to reproduce a specific observatory with its own limitations.The focus of this paper is on the retrievability of clouds and not on the performance of a specific facility.

Simulated transit spectra with YunMa
We present here the transmission spectra generated with YunMa of a cloudy super-Earth.Fig. 6 shows five examples at resolving power of 100 which corresponds to the ground truths of some of the retrieval cases in Table 3: p c = 2 × 10 −3 bar with X N2 = 0 (blue, Case 2, 10, 11, 14 and 15) and = 0.5 (purple, Case 20-25); p c = 1 × 10 −2 bar with X N2 = 0 (green, Case 3 and 16) and = 0.5 (yellow, Case 27 and 28) and one case of opaque cloud with p c = 1 × 10 −2 bar in the H 2 /He dominated atmosphere (red, Case 4 and 17).The planetary and atmospheric parameters are listed in Table 1 and 3.All the simulations contain baseline 10% H 2 O abundance across the atmosphere, which is then altered by the cloud formation.The rest of the atmosphere is N 2 and H 2 /He.We select f sed = 3 for all the scenarios.The simulation results are summarised in Table 2.
In the experiment without N 2 and p c set to 2 × 10 −3 bar, clouds form at high altitude, where the atmospheric density (ρ a ) is low compared to the cases where p c = 5 × 10 −3 bar and = 1 × 10 −2 bar.Here the sedimentation velocity (v f ) is small with small cloud particle radii and number density.The cloud contribution (blue dash-dotted line) has a mean transit depth of 2660 ppm Ma et al.In atmospheres with relatively high mean molecular weight -and therefore small scale height -for the same value of p c , the transit depth is smaller, as expected.In the case with X N2 = 0.5 and p c = 2 × 10 −2 and = 5 × 10 −2 bar, the mean value of the transit depths are ∼ 200 ppm smaller than them in a H 2 /He dominated atmosphere.Here the cloud particles form at higher ρ a and therefore have larger particle size and larger number density compared to those formed in the H 2 /He dominated atmospheres.The spectrum has σ spec of 2.58 and 0.62 ppm, which are negligible compared to the observational uncertainty.
Besides p c and X N2 , we also have tested different f sed to understand how this parameter controls the cloud microphysics.The particle radii, r c , number density and transit spectra across all the cloud pressure levels and obtained with different f sed are shown in Fig. 11.Here we note that, from the results, the cloud particle sizes increase with f sed while the number densities at each layer behave reversely.Also, which is easy to understand, the more atmospheric layers with clouds, the larger the optical depth.

Retrieval results
We show in this section how YunMa performs with different model assumptions and ground truth parameters (GTPs), following the approach described in Section 3. GTPs and priors are listed in Table 1.The retrieved values and one standard deviation (1σ) of the poste- pc 2 × 10 −3 (bar) 5 × 10 −3 (bar) 1 × 10 −2 (bar) 2 × 10 −3 (bar) 5 × 10 −3 (bar) Note-MMR and MMW: molecular mixing ratio and mean molecular weight.The pressure at the bottom of the atmosphere is assumed to be 10 bar.
rior distributions obtained for all the simulated cases are summarised in Table 3.

SENSITIVITY STUDIES TO KEY ATMOSPHERIC PARAMETERS
We have performed sensitivity studies to test how the model behaves when changing the key atmospheric parameters, including p c , f sed , X H2O , X N2 (Case 1-9).Cases 1-5 test the effects of different p c to the transit spectra and retrievals.Tuning p c alters the cloud altitude and the optical thickness: in these cases, the larger is p c , the more opaque meanwhile, the lower altitude becomes the clouds.The significance of spectroscopic features owns to both factors.Generally speaking, the more significant features are, the easier it is to retrieve the atmospheric parameters.Here we test p c from 10 −3 to 10 −1 bar, which is a much broader range than the one considered in previous literature about K2-18 b.The transit spectra of Cases 2 and 4 are shown in Fig. 6 (blue and red solid lines, respectively).We choose p c = 2 × 10 −3 bar as nominal case and show the corresponding posterior distributions in Fig. 7.
Cases 2, 6 and 7 test the impact on the transit spectra and retrievals of the sedimentation efficiency (f sed ), which controls the cloud microphysics in the model.In Case 6, we set the sedimentation efficiency f sed to 0.01, i.e. the downward sedimentation of the cloud particles is relatively slow compared to the net upward molecular mixing of the condensable species.By contrast, in Case 7 where f sed = 10, we have a larger downward draft velocity scale compared to the upward one.In  3).The sedimentation efficiency (f sed ), which is the main parameter controlling the cloud microphysics, is well recovered together with the vapour water mixing ratio.
analysing the results, we utilize the term "accuracy" to indicate that our retrieved result is in a certain range of the ground truth and "precision" to the 1σ of the posteriors.In the simple cases, when f sed = 0.01 and 3 (Case 2 and 6), the accuracy levels of f sed , X H2O and R p are > 90%.In comparison, in the scenario where f sed = 10 (Case 7), the accuracy level is > 70%.This result is not unexpected, as high f sed scenarios tend to have negligi-

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ble impact on the planetary transit depth due to thinner cloud layers and smaller number density (N ) compared to a low f sed scenario, e.g., the f sed = 10 and 100 cases in Fig. 11.
In Case 8 and 9, we modulate the amount of condensable gas, here represented by the water vapour mixing ratio (X H2O ).In the cases of our experiment, the clouds start to form when X H2O reaches the significance of 1 × 10 −3 .When X H2O = 0.01 (Case 8), a thin cloud with low opacity may form, the water vapour spectral features are visible and the retrieved values of log(X H2O ) and log(f sed ) have > 90% accuracy.By contrast, a higher mixing ratio of the condensable gas increases the partial pressure and contributes to the condensation process.This is, for instance, the case of X H2O = 0.5 (Case 9) where the cloud is thick and largely blocks the spectral features, making the retrieval of the atmospheric parameters difficult.

SENSITIVITY STUDIES TO DATA QUALITY
We then test how YunMa's performances degrade when we compromise with the data quality, for instance: uncertainties of 30 ppm in Case 11 and spectral resolution of 10 in Case 12, which should have similar effects.We move the blue cut-off at longer wavelengths in Case 13.From the experiments on observational data quality, Case 10's Bayesian evidence (4568.52)compared to the ones calculated for Case 2 (3755.80)and Case 11 (3368.29)showcases how the retrieval performance improves when the observational uncertainties are small.Case 2's performance surpasses Case 12, as the spectral resolution of the transit spectrum used as input to the retrieval is higher.In Case 13, we omitted the information in the optical wavelengths, which means we have less information about the cloud scattering properties.The retrieval performances are degraded compared to Case 2, which includes the optical wavelengths.

RETRIEVALS OF ATMOSPHERIC THERMAL PROFILES
In Case 14-17, we retrieved the T -p profiles as free parameters for different p c to test how YunMa performs with increasingly complex model assumptions and which parameters may be problematic in these retrievals.Our results show that both the T -p profiles and cloud parameters can be constrained, although the retrieved gas phase mixing ratio and f sed distribution may have large standard deviations in some of the cases.

ADDITION OF N2 IN RETRIEVALS
Inactive, featureless gases, such as N 2 , inject much uncertainty in the retrieval.In Cases 18-20, we include different amounts of the inert and featureless gas N 2 in the atmosphere.N 2 may exist in super-Earths' atmospheres, as it happens for Solar System planets at similar temperatures.Being N 2 heavier than H 2 O, we adjust the mean molecular weight and the scale heights accordingly by modulating X N2 .Heavier atmospheres have smaller scale heights: the spectral features are less prominent and harder to detect.We first try to retrieve only R p , f sed , X H2O and X N2 .The results show that despite the minimal spectral features of the cloudy heavy atmospheres due to N 2 injection, YunMa is still able to retrieve these atmospheric parameters in simple cases.

DEGENERACY BETWEEN CLOUDS AND HEAVY ATMOSPHERE
In Cases 21-25, we conduct more complex retrievals to test the degeneracy between clouds and X N2 .The corresponding transit spectra are shown in Fig. 6 (purple lines).Similar to N 2 , the existence of clouds mitigates the spectral features, and the difference between these two scenarios may be difficult to distinguish from the current data quality.Case 21 retrieves these parameters except X N2 fixed to 50% for comparison with Case 22 to investigate the degeneracy imposed by the uncertainty of X N2 .The result of Case 21 shows how the atmospheric parameters, with the exception of X N2 , can be retrieved in a heavy atmosphere with X N2 fixed to the ground truth.When we include the uncertainty of X N2 (Case 22), the GTPs for R p , X H2O , f sed , T surf and T top still fall into the 2σ confidence range, where R p , X H2O and f sed have accuracy levels > 60% and T surf and T top > 45%.An illustration of the retrieved T -p profile is shown in Fig. 8. f sed is significantly less constrained in Case 22 than Case 21 due to the uncertainty of X N2 .
One hypothesis is that if we do not include N 2 among the priors, the model will add clouds to compensate for the missing N 2 .In Case 23, we test this hypothesis by forcing X N2 to zero and then monitor the cloud parameters in the posteriors obtained.The results suggest that potential degeneracy could happen, plotted in Fig. 9: when we omit N 2 among the priors, the retrieval tries to compensate for the missing radiative-inactive gas by decreasing R p and T surf , while increasing X H2O , and the Bayesian evidence of Case 23 (3757.08) is close to Case 22 (3757.40).

COMPARISON OF CLOUD RETRIEVAL MODELS
In the experiment of model comparison, Case 24 simulates the forward spectra with YunMa cloud microphysics and then retrieves the atmospheric parameters with another simplified cloud retrieval framework.The simplified clouds are described as an opaque cloud deck across wavelengths in TauREx 3, which is commonly used to retrieve data from the last decades with nar-YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds  3. The isothermal temperature (Tc), surface temperature (T surf ) and the pressure where the isothermal profile starts (pc) are retrieved using YunMa and indicated by squares in the plot.The solid purple line indicates the two-point profile fitted from the retrieved Tc, T surf and pc values.The shaded area indicates the standard deviation of the posterior distribution.
We also plot a T -p profile with an isothermal upper atmosphere, DALR in the lower atmosphere and MALR around the cloud-forming region to show how the two-point profile can deviate from the ALR when similar cloud cover forms.The two-point profile is a useful approximation to estimate cloud formation while reducing the retrieval computing time.
row wavelength coverage, e.g., HST/WFC3.In a nonopaque case, we compare the retrieved results from the simple opaque cloud retrieval model in TauREx 3 (Case 24) and YunMa.The posteriors using the two differ-ent cloud models are compared with each other in Fig. 9.The former shows relatively flat posterior distributions of the cloud and atmospheric parameters than using YunMa, and in this case, lower accuracy and precision of the retrieved results.In Case 25, we deliberately omit cloud parameters in the retrieval priors and learn if and how other parameters can compensate for those missing.Without the cloud in the prior (Case 25), the results show a lack of constraints on the T -p profile while comparable performance on other parameters with Case 22.

RETRIEVALS OF FEATURELESS SPECTRA
In Cases 26-28, we retrieve the atmospheric parameters from spectra with minimal spectroscopic features, where spectral standard deviation (σ spec ) is less than 1 ppm.The spectra are featureless due to heavy atmospheres and cloud contribution (p c = 5 × 10 −2 bar).The transit spectrum for Case 27 and 28 is shown in Fig. 6, yellow line: the spectral signal is very small compared to the observational uncertainty (10 ppm).In Case 28, we unrealistically decrease the uncertainty to 1 ppm at resolving power 100 to evaluate YunMa's performance with idealised data quality.As expected, the retrieval performance greatly improves when the observational uncertainty unrealistically decreases to 1 ppm (Fig. 10).3. Blue crosses indicate the ground truth parameters and the vertical dashed lines in histograms indicate the 1σ and 2σ confidence ranges of the posterior distribution.These three retrievals use the same transit spectrum as input (thin cloud with XN 2 = 0.5 in Fig. 6) but different retrieval assumptions.Case 22 (purple): H2O and N2 are included as priors and the cloud formation is simulated by YunMa.Case 23 (orange): same as Case 22 except that N2 is not included among the priors.Case 24 (green): same as Case 22 except that the cloud is simulated by a simpler model from TauREx, in which the atmosphere becomes opaque below the cloud deck.The only retrieved cloud parameter for the simpler model is the cloud deck pressure, which is not shown here for simplicity.

Ma et al.
Table 3. YunMa retrieval experimental results.The ground truth parameters (GTPs) assumed in the simulations are listed in the left columns of the table and the retrieved posteriors on the right.The retrieval priors are listed in Table 1.In each case, we test the model sensitivity of the atmospheric parameters of f sed , pc, XH 2 O, XN 2 , and the observational parameters of the error, wavelength coverage (λ) and the spectral resolution.We choose the nominal value of pc as 2 × 10 −3 bar, where the cloud is not either too thin to be detected or too thick to block the spectroscopic features.-1.10 +0.00 −0.00 -0.99 +0.17To understand the performances of YunMa in detail, we performed retrieval experiments for over a hundred cases, with different chemistry models, atmospheric and cloud scenarios for super-Earths/sub-Neptunes, hot-Jupiters and brown dwarfs.A variety of cloud species were modelled and analysed.This paper presents a selection of representative examples.We have validated the A-M cloud size distribution in YunMa against previous literature simulating NH 3 clouds in Jupiter's atmosphere, KCl clouds on artificial large exoplanets and brown dwarfs, and H 2 O clouds on K2-18 b.We have also tested a number of numerical settings, including fitting methods, tolerances and retrieval samplings.
In YunMa, the mixing ratios of the condensable gas and the condensate are strongly correlated, therefore when cloud forms, the condensable species in the gas phase decreases.Also, a balance is imposed between the upward turbulent mixing and the downward sedimentation velocity from the A-M approach (equation 2).In the current YunMa, at each pressure level, the particle number density represents the total number density of particles with different radii.The Earth measurements shown in Fig. 4 of Ackerman & Marley (2001) suggest a bimodal distribution of the particle sizes at the same pressure level.YunMa is able to use radius bins with their respective number densities to represent more precisely the cloud distribution in the spectral simulation.For the retrieval calculations, however, we had to simplify this information to reduce the computing time.
The cloud opacity is determined by the cloud particle size and number density; different particle sizes have absorption and scattering peaks at different wavelengths.Optically thick clouds cause transit depths with negligible or no modulations as a function of wavelength: the atmosphere below the cloud deck is, in fact, undetectable while the atmosphere sounded above the cloud deck is more rarefied.Retrieving information about the atmospheric composition and structure is very difficult in the most extreme cases.For an atmosphere with optically thin clouds, the absorption features due to radiative-active gases (water vapour here) are detectable but less prominent compared with a clear atmosphere.The abundances of these gases can be retrieved largely from their absorption features.If these are condensable species, their abundances further constrain the cloud microphysics.Both the wavelength-dependent features and the overall transit depth help the retrieval performance.
f sed is the ratio between the sedimentation velocity and the turbulent convective velocity.From the definition, a higher f sed means a shorter sedimentation timescale, as we fixed K to a constant value.A higher sedimentation efficiency leads to larger offsets to the downward draft, constraining the upward supplement of water vapour and cloud formation.On the contrary, for small f sed , the sedimentation timescale is much longer compared to the diffusion timescale, so the condensation continues at lower pressure to balance the downward sedimentation and upward turbulent mixing; therefore, the cloud region expands.f sed is sensitive to the cloud particles' nucleation rate (Gao et al. 2018).It has a close relationship with the condensate particle size and can be expressed by the particle radius in the lognormal distribution power-law approximation: which indicates that small f sed encourages small cloud particle formation.The upward transport is stronger than the sedimentation when the cloud particles are small and vice-versa, which is in line with the experimental results in Figure 11 (a, b).The spectrum is sensitive to f sed when this parameter has values between 10 −1 and 10 −3 , as shown in Fig. 11 (c), indicating the detectability of f sed in this interval.
In our experiments, the particle radii are typically 1-10 µm, so they do not block the Rayleigh scattering slope caused by H 2 and H 2 O at the optical wavelengths.For smaller radii (e.g., cyan line in Fig. 5), they are expected to contribute more to the optical spectrum, although theoretically it would be hard to form particles at very small sizes according to the nucleation theory in cloud formation (Gao et al. 2018).For any particle radii even smaller presented in this paper, it is just for model test purposes in extreme cases, and we make no efforts to show their detailed analysis here.
Radiative-inactive gases such as N 2 , if present, can change the atmospheric scale height.The increase of scale height decreases the transit depth and dampens its spectroscopic features, as illustrated in Fig. 6.As mentioned in previous sections, the presence of radiativeinactive gases cannot be detected directly through spectroscopic signatures.Opaque clouds may behave similarly to inactive gases in mitigating spectroscopic features, leading to potential degeneracy in retrieval experiments.Illustrated in Fig. 9, the comparison between Case 22 and 23 in Table 3 suggests the potential degeneracy between R p , the baseline condensable gas abundance, T -p profile, and the N 2 abundance.In Case 23, we force X N2 = 0 in the retrieval to monitor how YunMa  When simulating transit observations, YunMa uses a 1-D approach to estimate the cloud formation at ter-minators according to the thermal profiles present at those locations.If separate observations of the morning and evening terminators are available, these will help us to understand the impact of atmospheric dynamics and horizontal effects.Similarly, phase curves or eclipse ingress and egress observations will be pivotal to completing the 3-D picture of the planet.

Cloud formation with the next-generation facilities' data
With the high-quality transit spectra offered by the next-generation facilities, the uncertainties, wavelength coverage and spectral resolution will be significantly improved compared to most current data, and a simple opaque cloud model is insufficient for the transit study of next-generation data according to our results; the modelling of cloud radiative transfer and microphysics such as YunMa is needed.In our experiments, we chose a nominal 10 ppm as the observational uncertainty and found that clouds can be well-characterised in most experiments.With the next-generation data and YunMa, there are still limits in retrieving the featureless spectra.A smaller uncertainly may help in retrieving cloud parameters in the most difficult cases: we therefore adopted an unrealistic 1 ppm to test their detectability in an ideal case (Case 10, 15 and 10).With 1 ppm, the atmospheres were constrained well, although the spectra were featureless.The results of Case 2 and 13 suggest that broad wavelength coverage is paramount to characterising clouds well: here, optical wavelengths play a critical role when combined with infrared spectral coverage.

Numerical instabilities in cloud microphysics simulations
We have selected the explicit Runge-Kutta method of order 8 (DOP853, Hairer et al. 1993) to solve equation (2) as it delivered the most stable numerical performance for our experiments.However, numerical instabilities may occur when a large relative (rtol) and absolute (atol) tolerances are chosen.Sometimes the ODE solver cannot converge for q t and indicates "no clouds" as a solution due to the numerical instability.Caution is also needed in estimating the cloud mixing ratio (q c ), as q t might be two orders of magnitude larger than q c and numerical errors could be injected when q s is subtracted from q t .When solving the ODE with too large tolerance, q t might converge to a certain value, but q c would be estimated as negligible according to equation (2 and 1).In other words, even though when the ODE solution for the gas phase seems numerically stable, it might not be precise and accurate enough.In those cases, the retrieval performances are affected, as shown in Fig. 12, where multiple islands of solutions in the posteriors are visible; these are caused by numerical instabilities, and the issue is more obvious for larger tolerance values.After many tests, we have decided to use "DOP853" in solving the ODE with rtol = 1 × 10 −13 and atol = 10 −16 , which guarantee numerically stable results.

CONCLUSIONS
YunMa is a state-of-art cloud simulation and retrieval package optimised for the interpretation of the next generation of exoplanetary atmospheric data, as provided by e.g., JWST, Roman, Twinkle and Ariel.These facilities will provide an unprecedented amount of highquality data, where the cloud formation process and cloud scattering properties can no longer be ignored.
YunMa cloud microphysics is based on the model published by Ackerman & Marley (2001), while the scattering properties of clouds are calculated through the open-source BH-Mie code.When coupled to the Tau-REx framework (Al-Refaie et al. 2021), YunMa becomes a very versatile model that can simulate transit and eclipse spectra for a variety of cloudy exoplanets with different masses, atmospheric compositions and temperatures.Most importantly, YunMa+TauREx can be used as a spectral retrieval framework optimised for cloudy atmospheres.
We have validated YunMa against previous work that adopted the A-M approach and compared our results with a 3-D model simulation when consistent assumptions are adopted to produce the vertical profile.We have validated the radiative transfer calculations in YunMa, including cloud scattering, against PyMieScatt.
We have run over one hundred retrieval experiments with YunMa, with different cloud compositions (e.g., KCl, MgSiO 3 , Fe clouds).This paper presents and discusses 28 cases of water clouds in the atmosphere of a temperate super-Earth, K2-18 b-like.Through these experiments, we have learnt that YunMa is capable of retrieving cloud formation and atmospheric parameters when clouds are not so opaque to mask all the atmospheric features at most wavelengths.More specifically, if we assume spectroscopic data covering the 0.4-14 µm range, uncertainties at a level of 10 ppm and spectral resolution R = 100, we can retrieve the sedimentation coefficient, the baseline condensable gases and the T -p profile points with accuracy levels > 60% in our respective experiments.This is not the case using the simple opaque cloud model, which shows more degeneracy in the posterior distributions of the atmospheric parameters retrieved.An extension of YunMa to interpret phase-curve observations will be a valuable next step.2-D cloud models, which include horizontal convection, will soon be within reach of computing speed and might be considered in future versions of YunMa.While we are aware of the limitations of the specific cloud microphysics model embedded in YunMa, our results advocate for the need to include more realistic cloud models in spectral retrievals to interpret correctly the results of the nextgeneration facilities.
where ℓ is the latent heat of evaporation, R SV is the specific gas constant for the vapour, T is the atmospheric temperature, and T 0 is the temperature at vapour pressure e 0 .We use the results from laboratory measure-

Figure 2 .
Figure 2. Validation of the YunMa cloud microphysics model against the Jovian ammonia clouds in Ackerman & Marley (2001).qc corresponding to different sedimentation efficiencies (f sed ) are shown.Solid lines: results from YunMa.Dashed lines: results from Fig. 1 in Ackerman & Marley (2001).Dotted line: T -p profile.

Figure 4 .
Figure 4. Comparison of YunMa's results against simulations by Charnay et al. (2021 b) for the 100 × solar metallicity scenario at the substellar point (see Fig. 6 a in the original paper).The condensate (blue) and vapour (cyan) MMRs are simulated by YunMa using the A-M approach with K = 10 2 m 2 s −1 .The cross-section of the two shaded areas indicates the range of cloud MMR and location simulated by Laboratoire de Météorologie Dynamique Generic GCM (LMDG) in Charnay et al. (2021 b).LMDG is derived from the LMDZ Earth (Hourdin et al. 2006) and Mars (Forget et al. 1999) GCMs.

Figure 5 .
Figure 5. Validation of radiative transfer simulations obtained with YunMa against the open source code PyMieScatt(Sumlin et al. 2018).The extinction coefficients for cloud particles with different sizes are estimated from the theory ofBohren & Huffman (2008).To address the computational limitations of retrievals, we pre-calculated the extinction coefficients used in equation (8) to estimate the cross-sections of the cloud particles.The pre-calculated list includes values for particle radii from 1 × 10 −7 to 1 × 10 −2 µm, equally spaced in the logarithm space.Here we show only six examples in the top panel.Bottom panels: residuals obtained by subtracting the extinction coefficients as estimated by the two codes, YunMa and PyMieScatt.The maximum discrepancy, corresponding to the largest particle radius simulated here (r = 1 × 10 2 ), is negligible, i.e. ±0.002.

Figure 6 .
Figure 6.Simulated transit spectra of cloudy super-Earths using YunMa.Solid line: total transit depth with all contributions included; dot-dashed line: water ice clouds; faint-dotted line: water vapour.Blue, green and red lines: H2/He dominated, cloudy atmospheres with different pc (see legend).Purple and yellow lines: heavier cloudy atmospheres with 50% N2..

Figure 7 .
Figure 7. Example of cloud retrieval using YunMa integrated in TauREx 3 (Case 2 in Table3).The sedimentation efficiency (f sed ), which is the main parameter controlling the cloud microphysics, is well recovered together with the vapour water mixing ratio.

Figure 8 .
Figure 8. Retrieved T -p profile of Case 22 in Table3.The isothermal temperature (Tc), surface temperature (T surf ) and the pressure where the isothermal profile starts (pc) are retrieved using YunMa and indicated by squares in the plot.The solid purple line indicates the two-point profile fitted from the retrieved Tc, T surf and pc values.The shaded area indicates the standard deviation of the posterior distribution.We also plot a T -p profile with an isothermal upper atmosphere, DALR in the lower atmosphere and MALR around the cloud-forming region to show how the two-point profile can deviate from the ALR when similar cloud cover forms.The two-point profile is a useful approximation to estimate cloud formation while reducing the retrieval computing time.

Figure 9 .
Figure 9. Retrieval posteriors for Case 22 (purple), 23 (orange) and 24 (green) in Table3.Blue crosses indicate the ground truth parameters and the vertical dashed lines in histograms indicate the 1σ and 2σ confidence ranges of the posterior distribution.These three retrievals use the same transit spectrum as input (thin cloud with XN 2 = 0.5 in Fig.6) but different retrieval assumptions.Case 22 (purple): H2O and N2 are included as priors and the cloud formation is simulated by YunMa.Case 23 (orange): same as Case 22 except that N2 is not included among the priors.Case 24 (green): same as Case 22 except that the cloud is simulated by a simpler model from TauREx, in which the atmosphere becomes opaque below the cloud deck.The only retrieved cloud parameter for the simpler model is the cloud deck pressure, which is not shown here for simplicity.

Figure 11 .
Figure 11.Test results for rc (top left), N (top right) and the transit spectrum (bottom) with different sedimentation efficiency f sed .The cloud layer shrinks with the increasing of f sed .

APPENDIXA.
SUPPLEMENTARY EQUATIONS TO ESTIMATE THE CLOUD MIXING PROFILEIn Section 2.1.1,the SVP can be estimated with the Clausius-Clapeyron equation: Note-XH 2 O represents the water vapour mixing ratio

Table 2 .
Atmospheric and cloud parameters included in YunMa simulations.The corresponding transit depths are also reported.