Tristan Guillot
email: tristan.guillot@oca.eu

David J Stevenson

Sushil K Atreya

Scott J Bolton

Heidi N Becker

Storms and the Depletion of Ammonia in Jupiter: I. Microphysics of "Mushballs"

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Ammonia condenses in Jupiter's atmosphere at pressures lower than about 0.8 bar and would be expected to be uniformly mixed below that level [START_REF] Atreya | A comparison of the atmospheres of Jupiter and Saturn: deep atmospheric composition, cloud structure, vertical mixing, and origin[END_REF]. Ground-based VLA radio-wave observations have shown that, in several regions of the atmosphere, ammonia is depleted down to at least several bars [START_REF] De Pater | Peering through Jupiter's clouds with radio spectral imaging[END_REF][START_REF] De Pater | Jupiter's ammonia distribution derived from VLA maps at 3-37 GHz[END_REF]. MWR (Microwave Radiometer) observations from Juno [START_REF] Bolton | Jupiter's interior and deep atmosphere: The initial pole-to-pole passes with the Juno spacecraft[END_REF][START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF] show that the depletion extends throughout the mid latitudes, is variable and is much more prevalent than previously reported, reaching very deep levels: At mid-latitudes, the volume mixing ratio of ammonia remains relatively low (between about 120 to 250 ppmv) until it increases to a value ∼ 360 ppmv at pressures greater than 20-30 bars. In the northern component of Jupiter's Equatorial Zone, at latitudes between 0 and 5 • N, the mixing ratio is relatively uniform vertically and equal to ∼360 ppmv. Such a global change in ammonia abundance cannot be explained solely by meridional circulation because it would violate mass balance [START_REF] Ingersoll | Implications of the ammonia distribution on Jupiter from 1 to 100 bars as measured by the Juno microwave radiometer[END_REF]. A local depletion of ammonia down to 4-6 bars may be explained by updrafts and compensating subsidence [START_REF] Showman | Dynamical implications of Jupiter's tropospheric ammonia abundance[END_REF], but this process cannot extend much deeper below the water cloud base and is thus unable to account for the Juno measurements.

We propose a scenario that can account for the observed vertical and latitudinal dependence of the ammonia concentration. In this paper, we show that during strong storms, ammonia in Jupiter can dissolve into water-ice crystals at temperatures around -90 • C, subsequently leading to the formation of partially melted hailstones that we call 'mushballs', and to their transport to great depths. In a second paper, we will apply this scenario to explain the Juno MWR measurements.

In Section 2 we first investigate the interaction between ammonia vapor and water-ice crystals. We then calculate in Section 3 the growth and transport of the 'mushballs' thus formed. We discuss in Section 4 how further downward transport of ammonia-and water-rich gas must result from evaporative cooling and subsequent downdrafts.

2 The interaction between ammonia vapor and water-ice crystals 2.1 The NH 3 -H 2 O phase diagram Ammonia is known to dissolve easily into liquid water, a consequence of similar dielectric properties of the two molecules. This has been recognized early on [START_REF] Lewis | The clouds of Jupiter and the NH 3 , H 2 O and NH 3 .H 2 S systems[END_REF][START_REF] Weidenschilling | Atmospheric and cloud structures of the Jovian planets[END_REF] and led to the current models of Jupiter's cloud structure which state that at pressures levels between 2 and 9 bars, depending on the H 2 O abundance, a water-cloud layer is formed, and some ammonia is dissolved into liquid water droplets forming a weak aqueous ammonia solution cloud [START_REF] Atreya | A comparison of the atmospheres of Jupiter and Saturn: deep atmospheric composition, cloud structure, vertical mixing, and origin[END_REF]. The amount dissolved is however small: At -20 • C (corresponding to a ∼ 4 bar pressure level in Jupiter) equilibrium chemistry predicts that a maximum of only 3% of ammonia can dissolve into supercooled liquid water droplets [START_REF] Ingersoll | Implications of the ammonia distribution on Jupiter from 1 to 100 bars as measured by the Juno microwave radiometer[END_REF]. Deeper in the atmosphere, at higher temperatures, ammonia solubility decreases while at higher elevations water freezes and should not include any significant amount of ammonia. Given the solar O/N ratio of 7.2 [START_REF] Lodders | Solar System Abundances and Condensation Temperatures of the Elements[END_REF] it is difficult to imagine how rainstorms could affect in any significant way the ammonia budget (Ingersoll et al., 2017).

However, in the same pioneering article about Jupiter clouds, John S. Lewis states: "It is not as commonly known that the freezing point of aqueous NH 3 can be depressed as low as -100.3 • C, and that the solid phases formed upon freezing of concentrated NH 3 solution can be NH 3 •H 2 O or 2NH 3 •H 2 O, not necessarily solid NH 3 or H 2 O".

In Fig. 1, we reproduce the NH 3 -H 2 O phase diagram of [START_REF] Weidenschilling | Atmospheric and cloud structures of the Jovian planets[END_REF], showing solid phases in grey (from left to right, NH 3 ice, 2NH [START_REF] Kargel | Ammonia-water volcanism on icy satellites: Phase relations at 1 atmosphere[END_REF] -is not included, but will not affect the results of the present work). The concentration of ammonia in the aqueous solution decreases from left to right from over 95% in the upper left to less than 1% in the lower right. Using the pressure temperature profile P (T ) measured in Jupiter by the Galileo probe [START_REF] Seiff | Thermal structure of Jupiter's atmosphere near the edge of a 5-µm hot spot in the north equatorial belt[END_REF] and a given volume mixing ratio x NH3 of ammonia, we can readily calculate the partial pressure of ammonia as a function of temperature in Jupiter, i.e. P NH3 (T ) = x NH3 P (T ). The result for x NH3 between 100 and 360 ppmv, the approximate range of ammonia mixing ratios measured by Juno [START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF] is shown as a red ribbon in Fig. 1.

Let us follow the upward motion of a water droplet formed below the 5-bar level in Jupiter's deep atmosphere by following the red ribbon in Fig. 1 from right to left. As liquid, it can dissolve a small fraction of ammonia -but this fraction remains smaller [START_REF] Weidenschilling | Atmospheric and cloud structures of the Jovian planets[END_REF] as a function of partial pressure of H2O and NH3. Solid phases are indicated in grey, otherwise, a liquid mixture forms with a concentration in ammonia indicated by the blue diagonal contour. The temperatures in Celsius are indicated as contour lines running from the bottom to the left of the plot. The red region labeled "Jupiter" corresponds to Jupiter's atmosphere assuming a minimum NH3 abundance of 100 ppmv and a maximum value of 360 ppmv [START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF].

than a percent in equilibrium conditions and reaches a few percent only by invoking large supercooling of the water droplets to -20 • C or so, as obtained by [START_REF] Ingersoll | Implications of the ammonia distribution on Jupiter from 1 to 100 bars as measured by the Juno microwave radiometer[END_REF]. When the droplet freezes to become an ice crystal, the equilibrium solution predicts the existence of pure water ice, implying that any ammonia must be expelled. However, when moving still higher up, in a region between 173 K and 188 K (i.e., -100 • C to -85 • C), equilibrium chemistry predicts that a liquid H 2 O • NH 3 mixture with a 30% -40% concentration of ammonia should form. Although this was recognized early on, this possibility was never really considered for Jupiter because of the fast rainout of water droplets and ice crystals [START_REF] Lewis | The clouds of Jupiter and the NH 3 , H 2 O and NH 3 .H 2 S systems[END_REF][START_REF] Weidenschilling | Atmospheric and cloud structures of the Jovian planets[END_REF][START_REF] Atreya | A comparison of the atmospheres of Jupiter and Saturn: deep atmospheric composition, cloud structure, vertical mixing, and origin[END_REF]. However models of water thunderstorms including detailed microphysics show that storms are able to loft 100 ppmv of water ice to the 1 bar level in the form of 10-to 100 -µm particles [START_REF] Yair | Microphysical processes and dynamics of a Jovian thundercloud[END_REF]. Storms so large that they can reach the stratosphere have been observed and modelled as extended water storms [START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF][START_REF] Sugiyama | Numerical simulations of Jupiter's moist convection layer: Structure and dynamics in statistically steady states[END_REF]. These storms can last for up to about ten days. They are believed to carry most of the intrinsic heat flux of the planet [START_REF] Gierasch | Observation of moist convection in Jupiter's atmosphere[END_REF].

Thus, although on average the abundance of water near the 1-bar level in Jupiter's atmosphere is extremely small, during large storms, conditions are met for the presence of a significant amount of ice in a region in which liquid NH 3 •H 2 O may form. On Earth, hail grows most rapidly in the presence of supercooled liquid water [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF] -it is thus possible that on Jupiter large storms lead to the formation of large NH 3 •H 2 O condensates and their fall to deeper levels. Because the concentration in ammonia can be large, up to 40%, this is a mechanism that can potentially deplete ammonia from the upper atmosphere more efficiently than it depletes water. Interestingly, at even higher levels (pressures lower than 1.2 bars), the equilibrium phase is a solid NH 3 •H 2 O condensate with an even higher ammonia concentration (up to 50%).

We name these condensates "mushballs" because we expect the presence of both solid and liquid phases containing variable amounts of ammonia and water and because the liquid phase thus formed is a highly viscous 'mush' [START_REF] Kargel | Rheological properties of ammonia-water liquids and crystal-liquid slurries: Planetological applications[END_REF]. Now let us examine whether they have time to form and grow.

Adsorption of ammonia into water-ice particles

Large thunderstorms on Jupiter can loft small-size (∼ 10 -100 µm) ice particles up to regions near a pressure of 1 bar [START_REF] Yair | Microphysical processes and dynamics of a Jovian thundercloud[END_REF]. These storms develop over timescales of hours to days [START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF]. Can ammonia be efficiently adsorbed into these water-ice particles on these timescales?

Let us consider an icy H 2 O particle that reached a level where equilibrium chemistry (Fig. 1) predicts the formation of a NH 3 •H 2 O liquid solution (e.g., ∼ 1.5 bar, T ∼ -85 • C for a vapor concentration of NH 3 x NH3 ∼ 300 ppmv). An estimate of the timescale to melt the particle is obtained by dividing the number of H 2 O molecules in the particle to the NH 3 vapor collision rate. Because the mean free path of ammonia vapor λ NH3 ∼ 3D NH3 /v th ∼ 0.1 µm (D NH3 ∼ 0.3 cm 2 /s is the diffusion coefficient of ammonia in hydrogen and v th ∼ 1.2 × 10 5 cm/s is the average gas velocity for this pressure level in Jupiter -see Table B1 in Appendix B) is much smaller than the size of the particles that we consider (10 -100 µm), the process is limited by diffusion effects. Given the small terminal velocity of the ice crystals (see Fig. 2 hereafter), they can be considered as co-moving with the gas. In this case, the timescale for the melting of an ice crystal by adsorption of ammonia vapor is [START_REF] Davidovits | Mass Accommodation and Chemical Reactions at Gas-Liquid Interfaces[END_REF])

τ ads = 1 36 r NH3•H2O a Kn ρH2O µ H2O µ NH3 µ RT x NH3 P d2 D NH3 ≈ 6 d 100 µm 2 s, (1)
where r NH3•H2O ∼ 1/2 is the ratio of NH 3 to H 2 O molecules of the equilibrium mixture, a Kn ∼ 0.75 results from an empirical fit [START_REF] Davidovits | Mass Accommodation and Chemical Reactions at Gas-Liquid Interfaces[END_REF], ρH2O is the physical density of ice grains, µ H2O and µ NH3 are the molar masses of H 2 O and NH 3 molecules respectively, µ ∼ 2.3 g/mol is the mean molar mass of the atmosphere, x NH3 ∼ 300 ppmv is the molar abundance of NH 3 , P is pressure (∼ 1.5 bar), T temperature (∼ 188 K), R the gas constant, and d is the ice grain diameter. Following measurements in Earth's clouds [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF], we adopt ρH2O ∼ 0.3 g/cm 3 but admittedly, this parameter is extremely uncertain.

While short, this timescale is longer by (a Kn /2)( d/λ NH3 ) ∼ 375( d/100 µm) compared to a kinetic timescale [START_REF] Davidovits | Mass Accommodation and Chemical Reactions at Gas-Liquid Interfaces[END_REF]. Experiments show that ammonia adsorption by ice crystal in vacuum is imperfect, i.e., the so-called uptake coefficient ranges between α ∼ 3×10 -4 to 4×10 -3 at temperatures between 170 K and 190 K [START_REF] Jin | Uptake of NH3 and NH3+ HOBr Reaction on Ice Surfaces at 190 K[END_REF][START_REF] Kasper | Uptake of ammonia gas by Jovian ices[END_REF]. This could lead to a timescale one to two orders of magnitude higher than the above one. However, our situation is different because of melting. Based on liquid-droplet-train experiments [START_REF] Davidovits | Mass Accommodation and Chemical Reactions at Gas-Liquid Interfaces[END_REF], we expect in that case values of α much closer to unity, implying that adsorption should be limited by diffusion.

Other limitations include the fact that only the partial pressure of NH 3 above saturation contributes to the adsorption, and the fact that molecules at the surface must diffuse into the interior. The first effect is estimated from the distance to the pure H 2 O ice curve in Fig. 1 to lead to a limited increase of timescale (decrease of partial vapor pressure) by a factor ∼ 2 across the mushball formation region. The latter is linked to the diffusion timescale inside the grain: τ diff ∼ d2 / DNH3 , where DNH3 is the diffusion coefficient for NH 3 inside the grain.

Let us consider diffusion of ammonia vapor through the liquid NH [START_REF] Kargel | Rheological properties of ammonia-water liquids and crystal-liquid slurries: Planetological applications[END_REF] compared to room temperature. Owing to the Einstein relation, we expect a comparable decrease of the diffusion coefficient, i.e., yielding Dliq NH3 ∼ 10 -8 cm 2 /s in our case. This implies that small ice crystals of 10 µm sizes can be melted in ∼100 seconds but that larger 100-µm crystals could take up to several hours to melt completely if they are compact. The melting time should be significantly shorter if the water-ice crystals are porous.

We thus expect adsorption in the mushball-formation region to be limited by diffusion effects so that τ ads ∼ 100 d/10 µm 2 s. Assuming a 50 m/s updraft, 100 s corresponds to the expected crossing-time of the ∼5 km mushball-formation region. The lifetime of storms (at least hours) and the residence time of small particles (about 1.5 hr for a 100 µm particle) indicate that ice crystals smaller than 10 to 100 µm should be entirely melted by the adsorption of NH 3 vapor.

We note that we did not consider the heat balance in the grain. Heat conduction takes place with a timescale τ cond ∼ d2 ρH2O cP,H2O / kH2O where cP,H2O ∼ 1.5 × 10 7 erg g -1 K -1 is the heat capacity of water ice at -80 • C and kH2O ∼ 3.2 × 10 5 erg s -1 K -1 cm -1 its thermal conductivity. Thus for the small grains considered heat conduction takes place on a timescale τ cond ∼ 10 -3 s, i.e., extremely fast compared to the other timescales. We note however that this ignores latent heat effects which should also be considered.

3 Growth and transport of mushballs

Fall velocities

Let us first examine how particles may be lofted by updrafts or fall because of a too large mass in Jupiter's atmosphere. The terminal velocity of particles falling in the atmosphere is obtained from the equilibrium between drag force and gravitational acceleration. It is conveniently expressed as:

v fall = 4 3C d ρg d ρ a 1/2 , ( 2
)
where d is the particle size, ρ its physical density, g the gravitational acceleration, ρ a the atmospheric density and C d is the dimensionless drag coefficient. For hard spheres, d is the diameter and C d is only a function of the Reynolds number of the particle, defined as N Re = dρ a v fall /η a , with η a being the dynamic viscosity of the atmosphere.

For large spheres (mm-size or more in our case), C d ∼ 0.47, but in the general case this is a function of N Re , and of the shape of the particle [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF]. We use the formulation of C d (N Re ) of [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF] based on studies of hailstones on Earth. (We correct a typo (a forgotten minus sign) in Eq. (B1) of [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF]: log 10 N Re = -1.7095 + 1.33438W -0.11591W 2 .)

We will see that large hailstones/mushballs in Jupiter can reach large Reynolds numbers. It is known experimentally that above a value N Re,crit ≈ 3 × 10 5 , the drag coefficient suddenly drops by a factor ∼ 5. While this is generally not the case on Earth for hailstones [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF][START_REF] Roos | A Giant Hailstone from Kansas in Free Fall[END_REF], it is of relevance to golf and tennis balls [START_REF] Kundu | Fluid Mechanics: Sixth Edition[END_REF] and probably of mushballs in Jupiter. We therefore include the effect by imposing that for N Re > 3 × 10 5 , C d = 0.1. (As we will see, this level of simplification is sufficient for our purposes). Fig. 2 shows how the terminal velocity of particles (assumed dense and spherical) varies with size at various levels in Jupiter atmosphere, and on Earth. Due to Jupiter's higher gravity and lower molecular weight of its atmosphere, terminal velocities are [START_REF] Nelson | The Influence of Storm Flow Structure on Hail Growth[END_REF]. The diamond corresponds to a giant hailstone collected on 1970/09/03 also in Oklahoma, weighting 766 g, with 15.5 cm of longest dimension and 11.8 cm of effective diameter [START_REF] Roos | A Giant Hailstone from Kansas in Free Fall[END_REF].

about 4 times larger than on Earth for the same pressure level. For sizes below 100 µm, we are in the Stokes regime, implying C d ∼ 24/N Re and v fall ∝ d2 . The fall velocities are slower than 1 m/s. At larger sizes, C d decreases to reach a value measured to be C d ∼ 0.6 for real hailstones [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF]. At larger sizes, when reaching the critical Reynolds number N Re,crit , the terminal velocity is expected to increase suddenly, which is represented by a kink in Fig. 2. A near-critical giant hailstone of 766 g was collected in Oklahoma and found to be slightly sub-critical [START_REF] Roos | A Giant Hailstone from Kansas in Free Fall[END_REF], with a terminal velocity measured in wind tunnels reaching 44 to 47 m/s, slightly below our theoretical curve (this can be attributed to its complex shape). On Jupiter, because of a higher kinematic viscosity, the critical Reynolds number is reached for particle sizes about 3 times smaller than on Earth, i.e., for particle diameters above 4 to 6 cm.

We may distinguish three types of condensed particles:

• Cloud droplets and ice crystals: On Earth, most have sizes between 10 and 50 µm [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF][START_REF] Rogers | A short course in cloud physics[END_REF]. Similar values are found in models of Jupiter water clouds using realistic microphysics, but with a tendency for a faster growth and thus slightly larger sizes ∼ 100 µm or more [START_REF] Yair | Microphysical processes and dynamics of a Jovian thundercloud[END_REF]). • Raindrops: Their maximum diameter is set by hydrodynamical stability considerations: d ∼ (γ/ρg) 1/2 , where γ is surface tension, ρ is density of the liquid. We expect surface tension to be only weakly affected by ammonia content and temperature, implying that since on Earth the maximum droplet diameter is about 5 mm, it should be of order 3 mm on Jupiter due to its larger gravity. These maximum droplet sizes should fall with a velocity ∼ 20 m/s at 5 bar. • Hailstones/mushballs: They can reach large sizes, provided that the updraft velocity balances their terminal velocity. Of course, this also requires fast growth, something that is obtained on Earth when supercooled water is present to allow an efficient sticking of droplets. The circle in Fig. 2 corresponds to the maximum hailstone diameter in a powerful hailstorm which occurred in Oklahoma in 1976 and for which the maximum updraft speed was measured to be 50m/s [START_REF] Nelson | The Influence of Storm Flow Structure on Hail Growth[END_REF]. This value corresponds to the terminal velocity of these largest hailstones, showing that balance between updraft speed and terminal velocity is key. Given storms with updraft speed ranging from 10 to 100 m/s in Jupiter [START_REF] Stoker | Moist convection: A mechanism for producing the vertical structure of the Jovian Equatorial Plumes[END_REF][START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF][START_REF] Sugiyama | Numerical simulations of Jupiter's moist convection layer: Structure and dynamics in statistically steady states[END_REF], we should expect hailstones in Jupiter to be able, in principle, to reach similar sizes as on Earth.

Growth of mushballs

We now examine how initially small (∼ 100 µm) water-ice crystals in a strong (∼ 50 m/s) updraft may adsorb ammonia, grow, collect more icy particles until they become too large to remain part of the updraft and begin to fall. Although this model is simple and may be considered naive in regard to the complexity of hail formation on Earth [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF], we believe that the framework presented here provides a useful insight into the Earth-like phenomena taking place in Jupiter's atmosphere and should help to explain Juno's observations. The adsorption of NH 3 vapor by ice particles is expected to heterogeneous, a consequence of the temperature gradients between the core of the updraft which should be warmer by up to ∼ 5 K compared to the outside. The ammonia adsorption and resulting melting of the ice particles should occur faster towards the edge of updrafts because of the entrainment of this colder surrounding atmosphere.

Prior to reaching the 1.5-bar level, the growth of ice particles in the updraft could be considered as essentially stalled: larger particles having rained out, only small-size particles (between 1 and 100 µm) remain and have a low collision probability [START_REF] Yair | Microphysical processes and dynamics of a Jovian thundercloud[END_REF]. Crossing the mushball-formation region suddenly has two effects: The adsorption of ammonia vapor increases particle mass by 30% . Melting also increases their density from low values (say ∼ 0.3 g/cm 3 ) [START_REF] Davidovits | Mass Accommodation and Chemical Reactions at Gas-Liquid Interfaces[END_REF] to that of the liquid ammonia-water mixture, i.e. 0.9 g/cm 3 . Both processes lead to an increase of the fall velocity for these particles. For example, at the 1.5-bar level, the terminal velocity of a 30-µm ice particle with a density of 0.3 g/cm 3 is about 2.5 m/s, it grows to 2.7 m/s due to mass increase and to 3.9 m/s due to melting, an overall 60% increase. It is natural to assume that because of cloud heterogeneity, the differential velocities of the particles will quickly increase.

In what follows we will use a simplified approach, by considering that, in an updraft of velocity v up , one particle (hereafter "mushball" ) of mass m, diameter d and terminal velocity v fall grows at the expense of other particles (hereafter "cloud droplets") with comparatively much smaller terminal velocities. The mass of the mushball, its altitude z and ammonia mixing ratio evolve with time according to the following relations:

d MH2O dt = E π 4 d2 µ H2O xH2O P RT v fall , (3)
d MNH3 dt = E π 4 d2 µ NH3 xNH3 P RT v fall , (4)
dz dt = v up -v fall , (5)
where MH2O and MNH3 are the mushball masses in water and ammonia, respectively, µ H2O and µ NH3 the molecular masses, xH2O and xNH3 their volume mixing ratios in the condensed phase (as cloud droplets), and E is the collection efficiency. Since we assume sphericity mass and diameter are related by M = MH2O + MNH3 = (π/6)ρ d3 where ρ is the physical density of the mushball.

The value of xH2O , the mixing ratio of condensed water, is set by the ability of the storm to loft small icy particles to the region considered. Because at the temperatures that we consider, the vapor pressure of water is extremely low (see fig. 1), we assume that xH2O = x H2O , the total mixing ratio of water. [START_REF] Yair | Microphysical processes and dynamics of a Jovian thundercloud[END_REF] find a mass mixing ratio of water at the 1 bar level that can reach 1 g/kg, corresponding to xH2O = 133 ppmv. This value is obtained for a solar-composition atmosphere and should increase for a higher deep abundance of water. We also note that higher values are likely due to a feedback mechanism not considered in that study: The formation of mushballs can increase updraft speed by decreasing condensate load at depth and by creating strong horizontal temperature gradients upon melting and evaporation. On the other hand, cloud-ensemble simulations [START_REF] Sugiyama | Numerical simulations of Jupiter's moist convection layer: Structure and dynamics in statistically steady states[END_REF] using the so-called Kessler parameterization of microphysical processes [START_REF] Kessler | On the Distribution and Continuity of Water Substance in Atmospheric Circulation[END_REF] impose a conversion rate from non-precipitating condensates to precipitating condensates that cannot be used to reliably predict the amount of small-size particles at high altitudes. We thus adopt three possible values of xH2O , 100, 600 and 1200 ppmv.

The value of xNH3 , the mixing ratio of condensed ammonia, is set by the abundance of ammonia vapor x NH3 , the value of xH2O and the location in the phase diagram set by the pressure and temperature conditions. We consider that xNH3 = 0 in the pure H 2 O ice region of the phase diagram. Mushballs start forming when liquid H 2 O•NH 3 forms, at pressures P 1.5 bar and temperatures T 188 K for x NH3 = 360 ppmv, corresponding to the global ammonia abundance of the north Equatorial Zone [START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF]. In order to calculate xNH3 , we determine for the temperature of the levels considered the intersections with the pure H 2 O ice phase and with the H 2 O•NH 3 ice phase. We derive the corresponding values of the ammonia vapor mixing ratio, x 1 and x 2 , respectively. If

x 1 < x NH3 ≤ x 2 the equilibrium is between H 2 O•NH 3 liquid and H 2 O ice. If x 2 < x NH3 , at temperatures T 170K, it is between H 2 O•NH 3 ice, H 2 O•NH
=    0 if x NH3 ≤ x 1 , min [x NH3 -x 1 , xH2O /2] if x 1 < x NH3 ≤ x 2 , min [x NH3 -x 1 , xH2O + (x NH3 -x 2 ) /2, xH2O ] if x NH3 > x 2 . (6)
Based on the values of MH2O and MNH3 , we can calculate the mass fraction of ammonia in the mushballs

fNH3 = MNH3 MNH3 + MH2O . ( 7
)
Conversely, the mass fraction of water is fH2O = 1 -fNH3 .

The collection efficiency depends on (1) how ice particles follow the flow around the mushball and ( 2) how effectively they remain bound upon collision. The first parameter is directly linked to the Stokes parameter of the ice particles, i.e., the ratio of their stopping time to the mushball-crossing time v fall / d. For the ice particles that we consider, we are in the Stokes regime, implying a Stokes number St ∼ ρparticle s2 particle v fall / (18η a d), where ρparticle and sparticle are the particle physical density and size, respectively [START_REF] Kundu | Fluid Mechanics: Sixth Edition[END_REF]. For ρparticle = 0.3 g.cm -3 and sparticle = 100 µm, and using the approximation that C d = 0.6 for mushballs in the 0.1 to 5 cm size range, we obtain St ∼ 100( d/1 cm) 1/2 implying that hydrodynamic effects should not decrease the collection efficiency [START_REF] Homann | Effect of turbulence on collisions of dust particles with planetesimals in protoplanetary disks[END_REF].

The second parameter, the collection efficiency E, is difficult to estimate. In the Earth's atmosphere, its value for collisions between ice particles ranges between unity to less than 0.1 [START_REF] Phillips | A Parameterization of Sticking Efficiency for Collisions of Snow and Graupel with Ice Crystals: Theory and Comparison with Observations*[END_REF]. Being at or close to the melting temperature is a key feature of the ability of particles to stick. Extrapolating these results to the Jupiter case, we thus expect E ∼ 1 when thermodynamic conditions predict the presence of liquid NH 3 •H 2 O and a smaller value away from that regime. For simplicity, we assume that E = 0.3 in the regime where the only condensates are made of H 2 O ice and E = 1 elsewhere.

Evaporation of mushballs

As mushballs fall into a high-enough temperature region, they will begin to melt and evaporate. In order to account for this process, we use the approach derived for the melting of hail on Earth [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF]. The rate at which hail melts is controlled by heat conduction from the atmosphere into the hailstone, the development of an interface between liquid water and solid ice inside the hail stone and the shedding of the water shell due to hydrodynamic instabilities. The hailstone is kept cooler than the surrounding atmosphere due to latent heat release by evaporation.

The evolution of the hailstone structure upon melting can be relatively complex: a water torus generally forms and shedding of either small or large drops can take place. Depending on the hailstone size, this can take place either continuously or intermittently. At mm sizes, an eccentric melting of the ice core takes place [START_REF] Rasmussen | A Wind Tunnel and Theoretical Study on the Melting Behavior of Atmospheric Ice Particles: III. Experiment and Theory for Spherical Ice Particles of Radius larger than 500 microns[END_REF].

Here, we use a simplified approach that considers that shedding takes place instantaneously. In that case the hailstone is kept near its melting temperature T0 ∼ 0 • C and its size is governed by the following equation [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF]:

da dt = 1 L m + cP,H2O ∆ T ρ i a   -k a T -T0 f h + D H2O µ v L v R   P sat T0 T0 -H a P sat (T ) T   f v   , (8)
where f h and f v are ventilation coefficients for heat and vapor, respectively, and are measured experimentally to be

f h = χ 2 N 1/2 Re ν a K a 1/3 f v = χ 2 N 1/2 Re ν a D H2O 1/3 (9)
The following quantities have been used: L m and L v are the latent heat of melting and vaporization, respectively (accounting for their temperature dependence, but assuming pure H 2 O), k a is the thermal conductivity of the atmosphere, ν a its kinematic viscosity, K a = k a / (ρc P ) its thermal diffusivity, D H2O the diffusivity of water vapor in the atmosphere, P sat the saturation pressure, H a the relative humidity of the atmosphere, N Re is the Reynolds number defined in the terminal velocity section, and χ is a mass transfer coefficient of order unity. Given the extended fall, we account for the internal temperature change of the hailstone, with cP,H2O being the specific heat and ∆ T = Ti -T0 the difference between an internal temperature Ti and that at the surface T0 .

Given the large Reynolds number (10 3 to 10 6 ) considered here, the ventilation coefficients are large and represent the largest effect governing the melting of the hailstone. The Prandtl and Schmidt numbers that enter these coefficients are close to unity: As seen from Table B1, (ν a /K a ) 1/3 ∼ 0.88 and (ν a /D H2O ) 1/3 ∼ 1.05 so that to first approximation f h ∼ f v . Experiments suggest that χ ∼ 0.76 [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF]. Overall, this yields f h ∼ f v ∼ 10 to 400. We note that this scaling law should change in the super-critical regime (N Re 3 × 10 5 ), but it is not even clear whether melting should be increased or decreased over this relation. Dedicated experiments should be conducted in order to determine more precisely the mushball evaporation level.

Although most of the complex processes observed during hailstone melting are not included, Appendix A shows that the approach does reproduce relatively well observations in the Earth's atmosphere and in wind tunnels. It also shows that additional heating due to viscous drag can be neglected.

The rise and fall of mushballs in Jupiter's atmosphere

We now apply our model for growth and evaporation of mushballs to the case of Jupiter. We are interested in situations where storms are able to reach the upper regions of the atmosphere (above the 1 bar pressure level). This corresponds to large storms. We thus assume an updraft velocity of 50 m/s generated from the cloud-base level and extending to the 0.4-bar level [START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF][START_REF] Sugiyama | Numerical simulations of Jupiter's moist convection layer: Structure and dynamics in statistically steady states[END_REF]. We assume that the upward velocity goes to zero away from that pressure range gradually with an error function: Based on terrestrial data showing that graupels and ice crystals have densities ranging from 0.05 to 0.9 g/cm 3 [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF], we adopt a physical density both for H 2 O ice and for H 2 O•NH 3 ice of 0.3 g/cm 3 . In the region where H 2 O•NH 3 liquid forms we assume that the collected ice have a density of 0.9 g/cm 3 . In that region, we also assume that the mushball melts partially to an overall density of 0.9 g/cm 3 .

v up = v 0    1 -erf   max   0, - log 10 P Ptop δ P           1 -erf   max   0,
We use the following values of the physical parameters, evaluated at 300K which corresponds approximately to the atmospheric temperature where mushballs melt: for water ice, c P = 2.0 × 10 7 erg g -1 K -1 , L m = 3.34 × 10 9 erg g -1 ; for water vapor, L v = 2.515 × 10 10 erg g -1 , D H2O = 0.17 cm 2 s -1 , µ v = 18; for hydrogen, k a = 1.85 × 10 4 erg s -1 cm -1 K -1 . We further assume that H a = 1, except below the cloud base, assumed to be at 5.8 bar, corresponding to an enrichment of water equal to 2.7 times the protosolar value [START_REF] Li | The water abundance in Jupiter's equatorial zone[END_REF]. (As shown in Appendix A3, this assumption has negligible consequences for the outcome of the model.)

We use a temperature profile that is based on the Galileo probe measurements [START_REF] Seiff | Thermal structure of Jupiter's atmosphere near the edge of a 5-µm hot spot in the north equatorial belt[END_REF] and extended below 22 bars using an adiabatic profile derived from an interior model of Jupiter [START_REF] Guillot | A suppression of differential rotation in Jupiter's deep interior[END_REF].

Figure 3 shows the resulting evolution of mushballs for three cases: Global abundances of water ice carried above the 1-bar level of 100 ppmv, 600 ppmv and 1200 ppmv, respectively. The simulation starts when water-ice particles generated at depth by the storm and carried in the updraft reach the 1.5-bar level. We start from an initial seed of 100 µm that melts due to NH 3 adsorption, starts collecting H 2 O•NH 3 liquid and, for the 600ppmv and 1200ppmv water-ice cases, H 2 O ice particles. Its terminal velocity is small compared to the updraft velocity. When reaching the 1-bar region, the particle accretes solid H 2 O•NH 3 and H 2 O ice. It continues to ascend until it has grown to a point where its terminal velocity equals the updraft velocity. At this point, it will start to fall, scavenging more particles on the way.

Between 1.1 and 1.5 bar, the mushball crosses again the liquid H 2 O•NH 3 region and partially melts. The density change (to about ∼ 0.9 g.cm -3 ) yields an increase of the Reynolds number. For the middle and high ice abundance case, it becomes supercritical which yields a very significant increase of the terminal velocity to about 300 m/s. (In the low-abundance case, the density change is not sufficient and the velocity stays confined to ∼100m/s).

In this same range of pressures, the scavenging of H 2 O ice leads to a progressive increase of the H 2 O mass in the mushball. The fraction of NH 3 decreases to a minimum of 3% in the high water-ice case to 20% in the low water-ice case. At that point, the temperature has reached 0 • C, the water-ice melting point, which leads to a progressive melting of the outer shell of the mushball. The NH 3 fraction thus increases up to the value it had after crossing the liquid H 2 O•NH 3 region. The last phase is a very quick melting and evaporation of the mushball, at pressures of 6.3 bar, 8.1bar, and 9.6 bar for the low, medium and high water-ice cases, respectively. If we decrease the ventilation factor by an order of magnitude (to account for possible changes of the empirical relation at high Reynolds number), the mushballs penetrate deeper, i.e., to 10, 17 and 24 bar, respectively (see Fig. 3). We find the depth at which mushballs evaporate to be insensitive to our choice of the drag coefficient C d for supercritical Reynolds numbers due to a balance between shorter timescales and larger ventilation coefficients. However, the time taken for mushballs to reach the evaporation level is proportional to √ C d and is thus correspondingly shorter due to the supercritical Reynolds number effect.

We can derive several important conclusions from this relatively simple model: The first one is that during strong storms, ammonia can be efficiently carried from the top of Jupiter's atmosphere down to levels below the water-cloud base. This is the case at least for the medium and high water-ice abundances. Equally significantly, for a number of cases, NH 3 is carried below the water cloud base more efficiently than H 2 O, i.e., fNH3 / fH2O > (N/O) = 0.135, or equivalently fNH3 > 0.117, where (N/O) is the protosolar nitrogen to oxygen mixing ratio [START_REF] Lodders | Solar System Abundances and Condensation Temperatures of the Elements[END_REF]. This implies that the downward transport of ammonia by water storms is efficient and can lead to a depletion of the upper atmosphere ammonia.

An exploration of the sensitivity to the parameters of the model is presented in Appendix A3. The depth of penetration of mushballs extends from 5.1 to 31 bar, depending on parameter values (quantity of water ice, updraft velocity, sticking efficiency...etc.). The cases for which water ice evaporate mostly above the water cloud base and the mushball core is deposited below appear most favorable. In that case, evaporated water may be recycled into storms while the aqueous ammonia mixture at the core of the mushballs would then be able to form evaporative downdrafts below the cloud base. Storms with updraft velocities between 10 and 50 m/s and between 100 and 600 ppmv of water ice carried to the 1.5 bar region therefore seem most promising to account for the Juno data.

Of course, we must add several important caveats. Compared to models of hail formation on Earth, this one is extremely simplified. In particular, it does not include complex geometrical effects inherent to hailstorm formation on Earth, the effect of turbulence within the cloud, the combined growth of a population of particles, and feedbacks due to evaporative cooling. On the other hand it does show that a simple model can already account for the formation of ∼10-cm mushball hail in Jupiter. When putting Earth and Jupiter in perspective, we note that Earth can form large hailstones (up to 0.77 kg -see e.g. [START_REF] Roos | A Giant Hailstone from Kansas in Free Fall[END_REF], that this requires strong updrafts (∼ 50 m/s) and the presence of liquid water droplets that are supercooled to around -15 • C [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF], a relatively rare occurrence. Jupiter has equivalently strong updrafts [START_REF] Stoker | Moist convection: A mechanism for producing the vertical structure of the Jovian Equatorial Plumes[END_REF][START_REF] Gierasch | Observation of moist convection in Jupiter's atmosphere[END_REF][START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF][START_REF] Sugiyama | Numerical simulations of Jupiter's moist convection layer: Structure and dynamics in statistically steady states[END_REF], and the presence of a liquid phase in contact with solids is guaranteed as long as ice particles are carried at least to the 1.5-bar level (which occurs only for storms with already large upward velocities). Two important differences are that on Jupiter large storms (characterized by large updraft velocities > 10 m/s at 2 bars) should always be able to loft ice particles to the 1.1-to 1.5-bar region where melting occurs, that the range of altitudes over which growth by scavenging can take place is vastly larger (∼ 50 km in Jupiter versus ∼ 3 km on Earth). This points to a hail formation mechanism on Jupiter that should be significantly more efficient than on Earth. The buildup of an H 2 O ice shell in phase 3 is critical because it isolates the liquid core of the mushball thermally and prevents NH 3 from diffusing out and be lost to the atmosphere. Even though the ice crystals collected should be very porous, the part of the H 2 O ice shell in contact with the H 2 O•NH 3 liquid is expected to be compact due to its interaction with the liquid.

Internal evolution of mushballs

Thermal equilibration within the mushball takes place with a characteristic time of order τ ≈ d 2 /α i where α i ∼ 2.2 × 10 -2 cm 2 s -1 is the thermal diffusivity of H 2 O ice. It is thus approximately only 45 seconds for a 1 cm mushball but 1.3 hours for a 10 cm one. For comparison, the examples shown in Fig. 3 correspond to falltimes of ∼ 40 min from 1.5 to 5 bar for the 100 ppmv H 2 O abundance case and only about 5 min from 1.5 to about 10 bar for the two other cases. Thermal equilibration will lead to a progressive melting of the H 2 O ice crust of the mushball from the inside-out and a decrease of the ammonia concentration in the mushball core. It could lead to a sudden break-up of the mushball when the H 2 O crust becomes too thin. This effect should be examined but should not affect our conclusions qualitatively.

Similarly, diffusion of ammonia through the solid-ice crust is expected to be slow. The diffusion coefficient for ammonia in water ice measured experimentally at 142 K fig1_mushball_interior.pdf is Dsol NH3 ∼ 4 × 10 -10 cm 2 /s [START_REF] Livingston | General Trends for Bulk Diffusion in Ice and Surface Diffusion on Ice[END_REF]. This may be extrapolated to be up to 2 orders of magnitude higher at ∼ 250 K, based on Na which has a similar behavior [START_REF] Livingston | General Trends for Bulk Diffusion in Ice and Surface Diffusion on Ice[END_REF]. Thus, using the same approach as in Section 2.2, we expect ammonia to diffuse outward only by about ∼100 µm in one hour, i.e., a negligible amount given that we expect ∼cm sizes for the mushballs.

Importantly, the highly concentrated ammonia-water mush at the center would be delivered last in Jupiter's deep atmosphere. Some of the water that is evaporated at higher levels can thus be recycled to power new storms and lead to the formation of more mushballs.

Importance of evaporative downdrafts

For our nominal ventilation coefficient, the evaporation of mushballs occurs near 10 bars, a pressure level that is not sufficiently deep to account for abundance increase inferred from the Juno MWR data [START_REF] Bolton | Jupiter's interior and deep atmosphere: The initial pole-to-pole passes with the Juno spacecraft[END_REF][START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF]. One possibility is that ventilation coefficients in the super-critical regime are decreased. However an other mechanism, the presence of evaporative downdrafts, must lead to further sinking of ammonia (and water).

On Earth, any rain, snow or hail accumulates on the surface. In Jupiter, the absence of such a surface implies that a pocket of gas with an increased concentration of ammonia and water must form. It is difficult to estimate precisely the concentration increase because it depends on geometrical factors and the time evolution of the storm. But we estimate that it may be substantial. Let us assume a storm surface area σ storm , an updraft velocity v up for a typical characteristic timescale ∆t. The mushballs evaporate lower than the cloud base, in an area σ down and down on to a depth H down . The typical densities are ρ storm ∼ 5×10 -4 g cm -3 around 5 bar and ρ down ∼ 2×10 -3 g/cm 3 around 30 bar. The enrichment (i.e. fractional increase of the mixing ratio of water and ammonia) is of order ∆x ∼ x(σ storm /σ down )(ρ storm /ρ down )(v up ∆t/H down ). The first term in parenthesis is of order unity, the second one is ∼ 1/6. The Voyager storm analyzed by [START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF] took about 10 days to develop and seemed relatively well fixed in latitude and longitude (on the local differential rotation frame). We hence estimate that H down ∼ 100 km, v up ∼ 50 m/s and ∆t ∼ 3 hrs. The last term in parenthesis is thus v up ∆t/H down ∼ 5. With an assumed mushball formation rate ∼ 0.3 we thus get ∆x/x ∼ 0.25. This is of course only an order of magnitude estimate and could vary significantly depending on the storm geometry and velocity. It is likely that localized bubbles that are highly enriched in water and ammonia will form and be only weakly affected by turbulence, thus effectively increasing ∆x much above that value.

In fact, even a modest enrichment can power strong downdrafts: For a perfect gas with a volume mixing ratio of vapor x, with ζ = µ v /µ d being the ratio of the mean molecular mass of vapor to that of dry gas

ρ = [1 + (ζ -1)x] µ d P RT . (11)
Evaporation of water (and ammonia) will result in an increase of mean molecular weight (due to the addition of vapor) and a cooling by evaporation, leading to a density increase ∆ρ ρ

≈ (ζ -1)∆x - ∆T T , (12)
where we assumed x 1. The change in temperature due to evaporation is ∆T = L v ζ∆x/c P , where L v is the latent heat of vaporization per unit mass of condensate (water) and c P is the heat capacity per unit mass of atmosphere. Thus we can rewrite the density change as a function of the increase in vapor mixing ratio Although the milk would be able to dissolve homogeneously in the glass, its slightly higher density resulting from its higher mean molecular weight and lower temperature yields strongly localized downdrafts. The final state is characterized by a gradient of increasing milk concentration with depth. Similarly, we expect strong storms in Jupiter to deliver to about 10 bar a cold and relatively highly concentrated water-and ammonia-rich gas leading to downdrafts able to reach the deeper levels of the planets. Individual storms should have horizontal extents of about ∼25 km [START_REF] Hueso | A model for large-scale convective storms in Jupiter[END_REF] and Juno measurements indicate that ammonia concentration increases on a vertical scale of at least 100 km. Although this is largely coincidental, we note that the geometry for that simple experiment is relatively similar to that in Jupiter.

∆ρ ρ ≈ ζ -1 + L v ζ c P T ∆x. ( 13
)
For our conditions we get ζ -1 ≈ 6.8 and L v ζ/c P T ≈ 4.7, i.e. the increase in mean molecular weight dominates slightly over the effect of evaporating cooling.

An estimate of the downdraft velocity can be obtained by calculating the work of the buoyancy force over a depth and by equating half of this work to the kinetic energy. This implies

v down ≈ g ∆ρ ρ 1/2 . ( 14
)
For a length equal to the pressure scale height ∼ H P ∼ 30 km, and Jupiter's gravity, we get v down ≈ 100 ∆x/10 -3 1/2 m/s. For comparison, [START_REF] Sugiyama | Numerical simulations of Jupiter's moist convection layer: Structure and dynamics in statistically steady states[END_REF] obtain downdrafts reaching about 50 m/s. We point out that downdrafts have been recognized to be an essential part of the Sun's convection [START_REF] Stein | Simulations of Solar Granulation. I. General Properties[END_REF]. In Jupiter, downdrafts are powered both by evaporative cooling and by molecular weight effects and should play an even more prominent role [START_REF] Ingersoll | Implications of the ammonia distribution on Jupiter from 1 to 100 bars as measured by the Juno microwave radiometer[END_REF].

Figure 5 illustrates what might be occurring in Jupiter with a simple experiment. Milk and water are fully miscible, like ammonia and water with hydrogen below the water cloud base in Jupiter. But when adding a spoonful of milk in the glass of water, instead of slowly diffusing in the glass, it rapidly sinks to the bottom through "milk plumes". These result from Rayleigh-Taylor instabilities which are of course well known in hydrodynamics (see e.g., [START_REF] Turner | Buoyant Plumes and Thermals[END_REF]). Here, our purpose is to illustrate the fact that this process, while of minor importance in the Earth atmosphere (moist air is slightly lighter than dry air at the same temperature), is likely to play a crucial role in Jupiter. (Of course, our water and milk experiment is strongly affected by wall effects and cannot be used to infer the depth of the plumes. In Jupiter, compression effects, turbulence and horizontal mixing are factors that should all be taken into account to infer the possible vertical extent of these downdrafts.) Unfortunately, its modelling and proper inclusion into global atmospheric models is notoriously difficult because of the variety of scales involved.

We also note that collective effects may play a role in leading to a further sinking of the condensates (water and ammonia) in Jupiter: In mushball regions, the temperatures should be locally cooler by δT /T ≈ -4.7∆x. We have estimated for the whole column that ∆x ≈ 0.25x, i.e., a quantity of order 10 -3 . But it is likely that in some regions, this value is much larger than that, in which case the evaporation would be delayed by the low temperature of the downwelling plume. For example if ∆x ∼ 10 -2 , at the 10 bar pressure level where the temperature should be 65 • C, it would be locally depressed to 50 • C, corresponding to an increased sinking by ∼ 8 km. Furthermore, the formation of a downdraft also means a faster downward transport of the mushballs with delayed evaporation. Detailed hydrodynamical simulations should be conducted in order to estimate the depth to which ammonia-and water-rich bubbles can be transported to.

Last but not least, we note that for the sinking to stop, the surrounding vapor mixing ratio must increase with depth so that the buoyancy force reverses. The location and magnitude of this increase will depend on local turbulence, entrainment of gas both in updrafts and downdrafts; on the radiative cooling of the plumes and on global horizontal mixing. This problem is beyond the scope of the present work, but it is likely to have deep consequences for our understanding of the interior structure of the planet.

Conclusion

The variability of ammonia's concentration as a function of latitude and to great depths in Jupiter's deep atmosphere [START_REF] Bolton | Jupiter's interior and deep atmosphere: The initial pole-to-pole passes with the Juno spacecraft[END_REF][START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF] is one of the most important surprises of the Juno mission and remains thus far unaccounted for. We have shown that thermoequilibrium chemical calculations predict the existence of a low-temperature region in which ammonia and water can form a liquid mixture with a high (∼1/3) concentration of ammonia. This region is located between 1.1 and 1.5 bar and temperatures between 173 K and 188 K. Jupiter's powerful storms can deliver water-ice crystals to that region. We have shown that ammonia vapor can dissolve into the ice crystals to form a high viscosity liquid ammonia-water 'mush', on timescales of minutes to tens of minutes. The increased mass and density of the particles thus formed increases differential velocities and the presence of liquid is expected to also lead to a high sticking efficiency, two factors which are crucial for the growth of hail-like particles that we call 'mushballs'. We have presented a simple model to account for their growth, their fall to the deep atmosphere and their evaporation. Depending on the amount of water-ice particles lofted by the storms, and depending on the poorly known ventilation coefficients governing heat conduction efficiency from the atmosphere to the mushballs, they should reach pressure levels of 5 bars and even as deep as 27 bars. Further sinking is warranted by the fact that the evaporated mushballs both have a high molecular weight and low temperature.

The fact that the cores of the mushballs contain a mixture that is highly concentrated in ammonia and the fact that this core is the last to be evaporated provides a potential mechanism to explain the ammonia depletion in a large fraction of Jupiter's atmosphere. Their evaporation deeper than the water cloud level and their further transport by downdrafts can potentially explain the great depth to which ammonia depletion is observed by Juno. We note (i) that since ammonia is at the center of the mushballs, it is delivered last, (ii) that H 2 O that evaporated on the way can be reused in other thunderstorms and therefore cycles further ammonia depletion, (iii) that the NH 3 /H 2 O concentration at the center of mushballs is ∼0.3, much greater than the solar N/O ratio of 0.1320, implying that the mechanism is efficient. We also note that the minimum in the derived NH 3 abundances [START_REF] Bolton | Jupiter's interior and deep atmosphere: The initial pole-to-pole passes with the Juno spacecraft[END_REF][START_REF] Li | The distribution of ammonia on Jupiter from a preliminary inversion of Juno microwave radiometer data[END_REF] is very close to the minimum NH 3 abundance below which the mushball mechanism cannot work (i.e., from Fig. 1, a partial pressure P NH3 ∼ 10 -4 bar, corresponding to a ∼100 ppmv NH 3 mole fraction in Jupiter). Finally, recent Juno observations in the optical show lightning flashes that are formed between 1 and 2 bar, consistent with the presence of liquid NH 3 •H 2 O and large particles in the mushball formation region [START_REF] Becker | Small lightning flashes from shallow electrical storms on Jupiter[END_REF]. In a subsequent paper, we develop a model of Jupiter's deep atmosphere to attempt to reproduce the dominant features of Juno's observations.

Appendix A Evaporation of hail A1 Application to the Earth case

We apply our simple model for the evaporation of hailstones and mushballs (Eqs. 8,9) to the case of the Earth atmosphere, based on the work of [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF]. We assume the Earth gravity, g = 981 cm/s 2 , and extremely simplified model reproducing the case of [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF], from altitude z = 0.8 km to z = 5.2 km, pressure from 0.9 to 0.6 bar, temperature from 24 • C to 0 • C and a relative humidity between 60% to 100% at the highest altitude where the hail originates. The mean molar weight of air is µ = 29, its thermal conductivity k a = 2570 erg s -1 cm -1 K -1 , its dynamic viscosity η = 1.8 × 10 -4 g cm -1 s -1 and the diffusivity of water in air, 0.3 cm 2 /s. The other parameters are the same as for Jupiter.

Figure A1 compares observational data and theoretical tracks [START_REF] Rasmussen | Melting and Shedding of Graupel and Hail. Part I: Model Physics[END_REF] to results of our model calculated with Eqs. ( 8) and ( 9). Some differences are visible but they are small compared to other uncertainties in the model.

A2 Effect of drag heating

In the case of Jupiter, the high fall speed of mushballs raises the question of whether drag friction (not included in Eq. 8) may lead to an even faster evaporation. This can be estimated as follows: Assuming an approximate constant terminal velocity, the energy dissipated per time ∆t by drag is ∆E ∼ M gv fall ∆t. Because the size considered is much smaller than the mean free path, this energy is dissipated in the gas and can then potentially heat the mushball. The part that is of interest to us is the fraction drag that is dissipated in the boundary layer around the mushball, which has a thickness ∼ K a d/v fall . With K a ∼ 0.3 cm 2 /s, d ∼ 10 cm and v fall ∼ 300 m/s, we obtain ∼ 0.01 cm. The gas in the boundary layer of volume V ∼ πd 2 is replaced at a rate ∆t ∼ d/v fall , implying a change of temperature in the gas

∆T ∼ drag ∆E c p,a ρ a V ∼ drag 1 6 ρice ρ a gd 2 c P,a .
With g = 2600 cm s -2 , ρice = 0.9 g/cm 3 , ρ a = 3 × 10 -4 g cm -3 and c P,a = 1.4 × 10 8 erg g -1 K -1 , we obtain ∆T ∼ drag × 30 K.

In order to estimate drag , let us consider the case of a human skydiver on Earth, falling at a terminal velocity around 50 m/s. With a weight of 75 kg and a density of ρ = 1 g/cm 3 , we consider that d ≈ 50 cm. Using parameters for the Earth at sea level, K a ∼ 0.19 cm 2 /s, g = 981 cm/s 2 , ρ a = 1.2×10 -3 g/cm 3 and c P,a = 1.0×10 7 erg/(g K), we obtain ∼ 0.04 cm and ∆T ∼ drag × 850 K. Everyday experience does tell us that the heating should be less than a few Kelvin (the same could be applied to e.g. driving a car on the highway). Therefore drag < 10 -2 , yielding a temperature increase which is negligible compared to other uncertainties. Another way to see this is as follows: The temperature increase in the boundary layer across the mushball is proportional to gravity (2.6 times higher in Jupiter), but is inversely proportional to the product of gas density and heat capacity. At 1 bar in Jupiter, this product is similar to that at sea level on the Earth, but deeper in Jupiter, where mushballs evaporate, it is an order of magnitude higher. Therefore, the increase in temperature in the boundary layer is expected to be smaller than for a similar situation on Earth. Since everyday experience tells us that drag heating of cars on the highway or of human skydivers is small (limited to a few Kelvins at most), it must be even smaller (and therefore negligible) for mushballs in Jupiter.

A3 Parameter sensitivity

We have focused on a set of fiducial model parameters and a simple model to show that mushballs in Jupiter can form and potentially transport ammonia downward efficiently. In Table A1, we study how varying these parameters affect the mushballs characteristics and how far they penetrate into Jupiter's atmosphere. The pressure at which mushballs evaporate, P max , their depth measured from the 1 bar level and the fall duration are provided both for the nominal ventilation factors (Eq. ( 9) and plain lines in Fig. 3) and for 10 times lower values (dashed lines in Fig. 3).

The first four lines of Table A1 correspond to our fiducial case (Fig. 3), for 4 values of the water ice abundance, from xH2O = 100 to 1200 ppmv. These lead to mushballs similar to the largest hailstones on Earth, with a maximum diameter between about 10 to 18 cm and a maximum mass between 0.1 and 0.9 kg. Their maximum freefall velocity v max can reach more than 200 m/s, reaching full melting pressures P max between 6.4 to over 20 bars in as much as 2.7 hours to as little as 30 minutes. The final NH 3 fraction at evaporation is always high, ensuring an efficient transport of ammonia.

When decreasing the upward velocity v up to 10 m/s, the growth of mushballs is suppressed. For all but the highest H 2 O values, they melt soon after reaching the 0 • C level at pressures close to 5 -6 bar. They can reach deeper levels for higher H 2 O crystal concentrations, but the value of f NH3 is then too low for an efficient transport of ammonia. Conversely, increasing v up to 100 m/s leads to a fast mushball growth and a penetration depth that can reach the 27 bar level in the most favorable conditions (including a ventilation factor that is ten times lower than the nominal one).

The value of the range over which a strong updraft is present (nominally between P bottom = 5 bar and 0.4 bar) has only a limited effect on the outcome, and results with P bottom = 2 bar are relatively close to the nominal case. Similarly, a change of the initial seed radius of the ice crystal, d0 , or in the relative humidity above cloud base, H a , lead to very small changes in the final results.

Changing E NH3 , the collection efficiency in the region where NH 3 • H 2 O forms, leads to a limited suppression of growth for high H 2 O ice abundances. However, for xH2O = 100 and 300 ppmv, we notice an increase of the mushball size and penetration instead. This is because for this case, the mushball starts evaporating in a region where the updraft is still present and is transported back upward where it continues to grow. This leads to several cycles of growth and evaporation before the mushball is large enough to fall through the zone and fully evaporate. A full consideration of this case is beyond the scope of this paper, but it is reminiscent of hail on Earth which is known to undergo multiple episodes of growth [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF].

Finally, the last set of cases correspond to an assumed low density of both H 2 O and NH 3 • H 2 O ice. They lead to an extremely fast growth of the mushballs in less than 1.5 hours even for the low xNH3 case. Of course, these cases are extreme because compaction effects would be expected to be very significant, but they show that the 0.14 assumed density of ice crystals is an important parameter. In the Earth atmosphere, graupel particles with mm sizes have densities ranging from 0.05 to 0.9 g cm -3 (see Table 2.8 of [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF]).

Overall, Table A1 shows that for a majority of cases, mushballs grow and deliver below the 5 bar level a mixture with a high concentration of NH 3 (f NH3 > 0.1), in less than an hour. For all these cases, evaporative downdrafts would be expected to form and lead to a further transport of ammonia and water in the deep atmosphere. For highly favorable cases (high updraft velocities, high abundance of H 2 O ice crystals and high collection rates) mushballs can penetrate deeper than the 20 bar level. However, although these may look promising to explain the Juno results directly, the evaporation of water ice takes place at levels below the water cloud where it may not be recycled efficiently. Instead the events for which water ice evaporate mostly above the water cloud base and the mushball core is deposited below appear most favorable.

Appendix B Nomenclature

Table B1 provides the main quantities used in this article and their default values.

Figure 1 .

 1 Figure 1. H2O-NH3 equilibrium phase diagram[START_REF] Weidenschilling | Atmospheric and cloud structures of the Jovian planets[END_REF] as a func-

Figure 2 .

 2 Figure 2. Terminal velocity of ice (or ammonia-ice) particles with diameters from 1 µm to 1m, for three pressure levels, 1, 5 and 20 bar inside Jupiter's atmosphere. The plain lines correspond to the full formulation. The dotted lines are the result from assuming a constant drag C d =0.6, applicable to large Earth hailstones (Rasmussen & Heymsfield, 1987). For comparison the Earth case for a pressure of 400 mbar and a temperature of -20 • C is shown as a dashed line. Two examples for the Earth case are shown: The circle corresponds to 5cm hailstones observed in a particularly powerful storm which occurred in Oklahoma on 1976/05/29, with updrafts of ∼50 m/s[START_REF] Nelson | The Influence of Storm Flow Structure on Hail Growth[END_REF]. The diamond corresponds to a giant hailstone collected on 1970/09/03

  and we choose P top = 0.4 bar, P bottom = 5 bar and δ P = 0.05. (The precise values are not important, as long as the updraft takes place at pressures between say, 0.5 and 1.5 bar).

Figure 3 .

 3 Figure 3. Characteristics of hail/mushballs as a function of pressure in Jupiter, for three values of the abundance of water-ice particles in the upper atmosphere: 100 ppmv (black), 600 ppmv (blue), and 1200 ppmv (red), assuming an updraft velocity of 50 m/s (see text). The first panel shows the diameter of hailstones, the second one the percentage of NH3 molecules that they contain, the third one their terminal velocity and the fourth one the time spent since their formation. The dotted lines correspond to cases in which the ventilation factor has been decreased by a factor 10 compared to the nominal value (see text). The temperatures in Jupiter's atmosphere are indicated on the right. The grey area corresponds to the location of the water cloud base, i.e. between 4.8 bar and 6.7 bar according to the Juno measurements[START_REF] Li | The water abundance in Jupiter's equatorial zone[END_REF].

Figure 4

 4 Figure4examines the evolution of the internal structure of the mushballs. We identify six evolution phases:

Figure 4 .

 4 Figure 4. The phases and internal structure of mushballs.

Figure 5 .

 5 Figure 5. Simple experiment to illustrate the importance of localized downdrafts in fluid mixtures. Here, at t=0, a tea spoon of fat milk from the refrigerator (∼10 • C) is added to a glass of water at room temperature (∼20 • C). Although the milk would be able to dissolve homogeneously

Figure A1 .

 A1 Figure A1. Comparison of the evolution of hailstones obtained from wind tunnel experiments (horizontal error bars), dedicated calculations (Rasmussen & Heymsfield, 1987) (dotted curves) and our simple model (plain lines). The three panels show the evolution with altitude of the hailstone diameter (left), terminal velocity (center) and time (right). The colored lines correspond to different initial diameters: 0.5 cm (black), 1 cm (purple), 2 cm (blue) and 3 cm (orange).

  3 •H 2 O ice, NH 3 •H 2 O ice and H 2 O ice) and liquid NH 3 •H 2 O in white and blue colors. (The solid NH 3 • 2H 2 O phase discovered later -see

  3 •H 2 O surface layer. At room temperatures, Dliq NH3 ∼ 10 -5 cm 2 /s, but we must account that it is a strong function of temperature. Laboratory measurements show that the viscosity of the liquid NH 3 •H 2 O mixture increases by up to 3 orders of magnitude at T = 176.2 K

  3 liquid and H 2 O ice. By assuming full thermodynamic equilibrium and that H 2 O•NH 3 liquid contains 2/3 H 2 O and 1/3 NH 3 , we derive xNH3

Table A1 .

 A1 Effect of model parameters on mushball characteristics and penetration depth.

Acknowledgments

This paper is dedicated to the memory of our friend and colleague Adam Showman, curious mind, great scientist and wonderful man. We thank the referees and particularly Kensuke Nakajima for helpful comments. TG thanks Stéphanie Cazaux, Yohai Kaspi, Ravit Helled, the IWG and AWG Juno teams for discussions that led to this work and Gaël Guillot and Flora Bellone for experimental tests on milk downdrafts and on ice melting in the family oven. T.G. acknowledges support from the Centre National d'Etudes Spatiales and the Japan Society for the Promotion of Science. The codes used to make the figures in this paper are available at http://doi.org/10.5281/zenodo.3749609. manuscript submitted to JGR: Planets

b Assuming a solar N/O ratio [START_REF] Lodders | Solar System Abundances and Condensation Temperatures of the Elements[END_REF].

c Approximate value based on measurement in Earth clouds [START_REF] Pruppacher | Microphysics of Clouds and Precipitation[END_REF]). d Value obtained using Jupiter's mean radius [START_REF] Guillot | THE INTERIORS OF GIANT PLANETS: Models and Outstanding Questions[END_REF]). e https://en.wikipedia.org/wiki/Latent heat. f Dean (1999). g https://www.engineeringtoolbox.com/diffusion-coefficients-d 1404.html.

h [START_REF] Livingston | General Trends for Bulk Diffusion in Ice and Surface Diffusion on Ice[END_REF].

i Galileo probe profile [START_REF] Seiff | Thermal structure of Jupiter's atmosphere near the edge of a 5-µm hot spot in the north equatorial belt[END_REF]


Abib Alimi

Matteo Frigo

Samuel Deslauriers-Gauthier

Rachid Deriche

Quantitative assessment of multi-scale tractography: bridging the resolution gap with 3D-PLI

Introduction

The in vivo validation of diffusion MRI (dMRI)-based tractography has been shown to be a challenging task [Maier-hein et al.]. Therefore, we have been investigating how 3D Polarized Light Imaging (3D-PLI) could be used as a validation tool for dMRI-based fiber orientation estimation and tractography. PLI is an optical imaging technique that provides us with high-resolution fiber orientation measurements at micrometer scale. For this reason, it has been presented as a good candidate for the aforementioned validation tasks [Axer et al, 2011, Alimi et al, 2019submitted]. In some previous works [alimi2017,18isbi, 18ismrm,19,19submitted] we introduced an approach to close the resolution gap between dMRI and 3D-PLI. The study of the brain network from the topological point of view has seen an increasing interest in the last years [START_REF] Sizemore | Cliques and cavities in the human connectome[END_REF][START_REF] Chung | Topological distances between brain networks[END_REF]. In this work, we show how tractograms obtained at different spatial scales using 3D-PLI human brain datasets can be inspected using homology theory to perform a quantitative comparison between them. In particular, we investigate the persistence of the number of connected components in brain networks estimated from data at different resolutions.

Methods

Analytical Fiber ODF in 3D-PLI : From 3D-PLI dataset, FOD can be reconstructed [Axer et al, 2016, Alimi et al, 2018isbi, 2019ismrm, 2019isbi, 2019Me-dIAsubmitted] and computed at different spatial scales by changing the size of the so-called super-voxel (SV) which allows downsampling the micrometer resolutions up to the millimeters (of dMRI). In [Alimi et al, 2018isbi, 2019ismrm, 2019isbi, 2019MedIAsubmitted] our FOD is analytically described on a spherical harmonics basis and elegantly and efficiently computed via the spherical Fourier transform and by means of the 2D Diracs delta δ function. This analytical approach is computationally efficient and allows fast computations of FODs from 3D-PLI data.

Multi-scale tractography : The FODs can be computed by varying the SV sizes, allowing to perform tractography at different spatial resolutions. Seeding from the gray matter -white matter interface we obtained 10 million streamlines at each resolution with the iFOD2 algorithm implemented in Mrtrix3 [Touriner et al, 2019]. A weight was associated to each streamline using SIFT2 [START_REF] Smith | SIFT2: Enabling dense quantitative assessment of brain white matter connectivity using streamlines tractography[END_REF] to re-establish the balance between streamline density and the fiber density computed from the fODFs. The chosen atlas is defined such as every voxel in the gray matter -white matter interface is a different region. In this way, proximity in the atlas is coupled with proximity of the assigned labels. Finally, a weighted connectivity matrix was computed at each resolution using the defined streamline weights and atlas.

Homology analysis: The analysis of the homology of the obtained networks was performed by measuring the persistence of the 0-th Betti β 0 on the natural filtration of the graph. This was achieved by counting the number of connected components of the graph when thresholded at the levels defined by the unique values of the edge weights in the network. The shape of the obtained threshold-β 0 curves give an insight on the robustness of β 0 with respect to the chosen threshold for a specific network.

Human Brain Dataset: The studied dataset consists of 50 unstained histological coronal slices of the right hemisphere. Each slice is 70 microns thick with an in-plane resolution of 64 × 64 µm 2 . Further details about this datasets' preparation and imaging can be found in [Schmitz et al, 2018]. The SV size which closes the gap with dMRI resolution has dimension of SV = 25 × 25 × 25 natines voxels isotropic.

Results and discussion

Figure 1 displays the generated stramlines from the 3D-PLI high-resolution data up to the relatively low dMRI resolution. From (A) to (C), the spatial resolution decreases however the integrity of the brain fiber pathways are preserved.

The first row of figure 2 shows the connectivity matrices obtained with different SV sizes. Notice the pervasive presence of small communities of regions at every resolution. This is due to the existence of many U-fibers in the occipital lobe as demonstrated from histology and dissection studies by [Sachs et al. 1893] and [START_REF] Vergani | Intralobar fibres of the occipital lobe: a post mortem dissection study[END_REF]] Moreover, the second row of figure 2 shows the sparsity pattern of the obtained weighted connectivity matrices, highlighting the presence of weaker connections between further regions. These weaker connections are more present in the connectome obtained with the lowest resolution from a SV size of 25 × 25 × 25. Figure 3 shows the evolution of the Betti 0 number β 0 with respect to the thresholding level applied to the matrix. The value of β 0 for the original connectomes (threshold equal to 0) is 67 for the highest resolution dataset, 64 for the mid resolution dataset and 63 for the lowest resolution dataset. Since each region is composed of only one voxel, these differences correspond to minimal changes in the obtained tractograms and should not be interpreted as structural differences caused by the diverse super-voxel sizes. Up to a threshold level of around 15000 units the three curves have a similar behaviour, showing how thresholding at low levels does not affect the comparability of the β 0 features of the connectomes. On the contrary, as soon as the 15000 units level is reached, the differences between the number of connected components of the three connectomes become higher, losing comparability between the β 0 features. The comparability is restored for high values of the threshold, where the connectome is reduced to a graph made almost uniquely of isolated nodes. This case is not interesting for our comparison and is included only to show the correct termination of the analysis.

Conclusion

In the context of defining a validation technique for dMRI-based tractography by means of 3D-PLI imaging, this work represents an important step bridging the concept of brain connectivity and the work we did in the last years on closing the imaging resolution gap. The employed approach makes use of the concept of 0-th Betti number, allowing a direct comparison of the topology of the brain networks obtained with data at different super-voxel resolutions. Our results demonstrated that analogies in β 0 between these connectomes may be observed only if the corresponding networks are thresholded up to a relatively small value, hence taking into account both the stronger and the weaker connections regardless of the chosen spatial resolution. This paves the way for 3D-PLI to be a potential validation approach for dMRI-based tractography.

Acknowledgment

This work was partly supported by ANR "MOSIFAH" under ANR-13-MONU-0009-01 and the ERC under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665:CoBCoM). We would like to thank professor Marcus Axer and his group for providing the 3D-PLI human brain datasets.

Maier-Hein, Klaus H., et al. "The challenge of mapping the human connectome based on diffusion tractography." Nature communications 8.1 (2017): 1349.

Jouk et al., Three-dimensional cartography of the pattern of the myofibers in the second trimester fetal human heart, Anatomy and embryology, vol. 202, no. 2, pp. 103-118, 2000. Figure 3: Each line represents the β 0 curve associated to a connectome obtained from data at a specific resolution. The value of the three curves at zero threshold is 67 for the 5x5x5 SV resolution, 64 for the 10x10x10 SV resolution and 63 for the 25x25x25 SV resolution. The fact that the three curves terminate at different thresholds happens because the maximal entry in the corresponding connectivity matrix is different for each connectome.

Axer et al., A novel approach to the human connectome: ultra-high resolution mapping of fiber tracts in the brain, Neuroimage, vol. 54, no. 2, pp. 1091Neuroimage, vol. 54, no. 2, pp. -1101Neuroimage, vol. 54, no. 2, pp. , 2011

Figure 1 :

 1 Figure 1: Panels A-B-C show the streamlines passing through the central coronal slice of the studied dataset with the usual orientation-based colouring. It is possible to notice the different white matter structures that are generated with data at different SV resolution. The figure on the top left is the transmittance image obtained from PLI. The figure on the bottom left is the used white matter mask.

Figure 2 :

 2 Figure 2: The first row shows the weighted connectivity matrices corresponding to the tractograms obtained at the three studied resolutions. The second row shows the sparsity pattern of the connectomes in the first row.

  . Alimi et al., Solving the inclination sign ambiguity in three dimensional polarized light imaging with a pde-based method, in Biomedical Imaging (ISBI 2017), 2017 IEEE 14th International Symposium on. IEEE, 2017, pp. 737-740. Axer et al., Estimating fiber orientation distribution functions in 3D-polarized light imaging, Frontiers in neuroanatomy, vol. 10, 2016. Alimi et al., Regularizing the ODF estimate with the Laplace-Beltrami operator in 3D Polarized Light Imaging, CoBCoM 2017 -Computational Brain Connectivity Mapping Winter School Workshop, Nov. 2017. Alimi et al., An analytical fiber odf reconstruction in 3d polarized light imaging, in Biomedical Imaging (ISBI 2018), 2018 IEEE 15th International Symposium on. IEEE, 2018, pp. 1276â-1279.