The Physics of Falling Raindrops in Diverse Planetary Atmospheres
mmanuscript submitted to
JGR: Planets
The Physics of Falling Raindrops in Diverse PlanetaryAtmospheres
Kaitlyn Loftus , Robin D. Wordsworth , Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA, US School of Engineering and Applied Sciences, Harvard, University, Cambridge, MA, US
Key Points: • We present general methods to calculate raindrop shape, speed, and evaporationrate in diverse planetary atmospheres • We define a dimensionless number that we show can closely capture the behaviorof raindrop evaporation • Maximum stable raindrop size is relatively insensitive to condensible species andatmospheric properties (including air density)
Corresponding author: Kaitlyn Loftus, [email protected] –1– a r X i v : . [ a s t r o - ph . E P ] F e b anuscript submitted to JGR: Planets
Abstract
The evolution of a single raindrop falling below a cloud is governed by fluid dy-namics and thermodynamics fundamentally transferable to planetary atmospheres be-yond modern Earth’s. Here, we show how three properties that characterize fallingraindrops—raindrop shape, terminal velocity, and evaporation rate—can be calculatedas a function of raindrop size in any planetary atmosphere. We demonstrate that thesesimple, interrelated characteristics tightly bound the possible size range of raindrops ina given atmosphere, independently of poorly understood growth mechanisms. Startingfrom the equations governing raindrop falling and evaporation, we demonstrate thatraindrop ability to vertically transport latent heat and condensible mass can be wellcaptured by a new dimensionless number. Our results have implications for precipita-tion efficiency, convective storm dynamics, and rainfall rates, which are properties ofinterest for understanding planetary radiative balance and (in the case of terrestrialplanets) rainfall-driven surface erosion.
Plain Language Summary
The behavior of clouds and precipitation on planets beyond Earth is poorlyunderstood, but understanding clouds and precipitation is important for predictingplanetary climates and interpreting records of past rainfall preserved on the surfacesof Earth, Mars, and Titan. One component of the clouds and precipitation system thatcan be easily understood is the behavior of individual raindrops. Here we show howto calculate three key properties that characterize raindrops: their shape, their fallingspeed, and the speed at which they evaporate. From these properties, we demonstratethat, across a wide range of planetary conditions, only raindrops in a relatively narrowsize range can reach the surface from clouds. We are able to abstract a very simpleexpression to explain the behavior of falling raindrops from more complicated equa-tions, which should facilitate improved representations of rainfall in complex climatemodels in the future.
Within a planetary condensible cycle, precipitation is the transport of the con-densible species in a condensed phase (liquid or solid) through the atmosphere and, forterrestrial planets, to the surface. Extensive vertical displacement relative to the localair mass distinguishes precipitation from clouds. Because precipitating particles canfall far from the air mass where they form, they redistribute both heat and the con-densible species within an atmosphere. Precipitation is a transient state, but thoughits effects are largely indirect, they have immense consequences for planetary climate.The behavior of precipitation is essential to setting planetary radiative balance.Precipitation’s role in transporting condensible mass from the atmosphere to the sur-face (or the deep atmosphere on gaseous planets) exerts a strong influence on therelative humidity distribution (Sun & Lindzen, 1993; Romps, 2014; Lutsko & Cronin,2018; Ming & Held, 2018), cloud lifetimes and occurrence rates (Zhao et al., 2016; See-ley et al., 2019), and condensible surface distributions (Abe et al., 2011; Wordsworthet al., 2013). These properties, in turn, have direct radiative implications via thegreenhouse effect and albedo changes (e.g., Pierrehumbert et al., 2007; Pierrehumbert,2010; Shields et al., 2013; Yang et al., 2014; Pachauri et al., 2014). The role of pre-cipitation in dictating radiative balance is especially important on dry planets (Abeet al., 2011) and planets in or near a runaway greenhouse state (e.g., Pierrehumbert,1995; Leconte et al., 2013).On Earth, global precipitation patterns play a critical role in determining localecology and have significant societal impacts (Margulis, 2017). On terrestrial plan- –2–anuscript submitted to
JGR: Planets ets generally, the intensity, frequency, and spatial distribution of liquid precipitationare essential in governing surface erosion via runoff and physical weathering (e.g.,Margulis, 2017) as well as chemical weathering fundamental to the carbon-silicate cy-cle (Walker et al., 1981; Macdonald et al., 2019; Graham & Pierrehumbert, 2020).Finally, interpreting solar system geological records shaped by fluvial erosion—e.g.,ancient Mars’ large-scale valley networks and crater modifications (e.g., Craddock &Howard, 2002), modern Titan’s lakes and rivers (e.g., Lorenz et al., 2008), and ArcheanEarth’s fossilized raindrops (Som et al., 2012; Kavanagh & Goldblatt, 2015)—requiresan understanding of changes in precipitation events as planetary conditions vary.Despite the importance of precipitation, little progress has been made on howprecipitation physically behaves in different planetary environments (e.g., Vallis, 2020).Previous studies have tended to view precipitation behavior primarily as a functionof cloud formation and evolution. Cloud physics is complicated by extreme nonlin-earities, gaps in theoretical understanding of fundamental processes bridged only byempiricisms, and dependencies on spatial and temporal scales that span many ordersof magnitude. In turn, planetary clouds are studied via a hierarchy of models withvarying tradeoffs among complexity, robustness of included processes, and ease of in-terpretation (e.g., Rossow, 1978; Carlson et al., 1988; Ackerman & Marley, 2001; Leeet al., 2016; Powell et al., 2018; Gao et al., 2018). Still, climate models of terres-trial planets commonly represent clouds and precipitation using modern-Earth-tunedparameterizations with essentially ad hoc parameter sweeps (e.g., Wordsworth et al.,2013; Urata & Toon, 2013; Yang et al., 2014; Komacek & Abbot, 2019).Given the transition from cloud to precipitation is complex and poorly modeledeven on the well-observed modern Earth (e.g., Rogers & Yau, 1996; Pruppacher &Klett, 2010; Flato et al., 2014), any insights on precipitation behavior become almostimpossible to extract when precipitation is considered only as an afterthought of cloudbehavior. An alternative approach is to consider the behavior of individual precip-itating particles independently of their formation conditions. This strategy is muchmore tractable in different planetary environments because individual precipitatingparticles are governed by thermodynamics and fluid dynamics that are both relativelywell understood and fundamentally transferable to planetary regimes beyond mod-ern Earth’s. Lorenz (1993) took this approach and used key properties of individualmethane-nitrogen raindrops on Titan to highlight fundamental differences betweenrainfall on Titan and Earth and hypothesize consequences for storm intensities.Previous planetary science studies have also attempted to use the simplicity ofraindrop physics to place constraints on paleo-air pressures on Archean Earth (Somet al., 2012; Kavanagh & Goldblatt, 2015) and early Mars (Craddock & Lorenz, 2017;Palumbo et al., 2020) via maximum raindrop sizes before breakup. This use of raindropphysics hints at the possible productivity of this approach. However, even maximumraindrop size has not been considered systematically in general planetary atmospheresbefore, and the conclusions reached by these studies have in some cases been incon-sistent. Specifically, a recent study by Craddock and Lorenz (2017) reached oppositeconclusions versus other studies (e.g., Komabayasi et al., 1964; Lorenz, 1993; Clift etal., 2005; Pruppacher & Klett, 2010; Som et al., 2012; Kavanagh & Goldblatt, 2015;Palumbo et al., 2020) on the dependence of maximum stable raindrop size on planetaryair density, leading to entirely different conclusions about paleo-air densities.In this paper, we build on Lorenz (1993) to establish a comprehensive and gener-alized picture of the “life and death” of a single raindrop over a wide range of planetaryconditions. Like Lorenz (1993), we neglect the “birth” of the raindrop (i.e., the growthof a cloud particle of negligible vertical velocity to a precipitating particle below acloud, essentially done growing). Our approach here is distinct from previous workas we present fully generalized methods, tie each component of our methodology backto fundamental physics, and focus on how the well-understood behavior of an indi- –3–anuscript submitted to
JGR: Planets vidual raindrop can provide insight into the rest of the condensible cycle in differentplanetary environments. This work lays a foundation for building physically driven mi-crophysics parameterizations for generalized mesoscale models and global circulationmodels (GCMs).We limit our focus to liquid precipitating particles (“raindrops”) because theyhave a unique shape for a given mass of condensible. The shape degeneracy of solidprecipitating particles is a major challenge (e.g., Pruppacher & Klett, 2010, chapter2.2) that we do not treat here. However, once the non-uniqueness of solid precipitatingparticle shapes is addressed, our methodology is applicable to solid particles as well.Water is the most familiar condensible species, but all methodology we present hereis generalized for any liquid condensible, e.g., methane-nitrogen raindrops on Titan oriron raindrops on WASP-76b (Ehrenreich et al., 2020). Except when we assume theexistence of a planetary surface, our methodology is also general to both terrestrialand gaseous planets.In section 2 we present methods to calculate falling raindrop shape, terminalvelocity, and evaporation rate in a generic atmosphere. We show how these charac-teristics can place upper and lower bounds on raindrop size in section 3. Section 4uses the methodology developed to probe raindrop characteristics and size bounds indifferent atmospheres. In section 5 we discuss the implications of our results for dif-ferent planetary atmospheres and microphysics parameterizations as well as possibleextensions of this work to solid precipitating particles. We summarize our results insection 6.
In isolation, a precipitating particle does two things: (1) it falls and (2) it evapo-rates. To calculate the rate at which a particle falls and evaporates requires knowledgeof the relationship between particle mass and shape. Unlike solid precipitating parti-cles, whose forms vary widely, raindrops have equilibrium shapes that can be uniquelycalculated for a given mass of liquid condensible, air density, and surface gravity. Aunique shape allows us, in a known external environment, to describe a raindrop withonly a single size variable. Here, we use equivalent radius r eq , which is the radius araindrop of mass m would have if it were spherical, i.e., m = 43 πρ c ,‘ r eq (1)where ρ c ,‘ is the density of the liquid condensible. Falling raindrops adopt a range of shapes depending on their size—though neverthe teardrop shape inscribed in the public imagination (Blanchard, 2004). As raindropsgrow in mass, they evolve from spheres to oblate spheroids to shapes resembling thetop of a hamburger bun (e.g., Pruppacher & Pitter, 1971; Beard & Chuang, 1987). (Anoblate spheroid is generated from an ellipse rotated about its minor axis.) Spheres aremerely a specific subset of oblate spheroid, and the more complex shapes have virtuallyindistinguishable dynamical properties from oblate spheroids (Green, 1975; Beard &Chuang, 1987; Szakáll et al., 2010); therefore, we simplify our shape calculations byprescribing that raindrops take the shape of an oblate spheroid with semi-major axis a (oriented perpendicular to the fall direction) and semi-minor axis b (oriented parallelto the fall direction). Given this assumption, we can describe raindrop shape withonly an axis ratio b/a and make use of many existing analytic expressions. Geometricproperties of oblate spheroids used in this paper are given in Appendix A. –4–anuscript submitted to JGR: Planets
In equilibrium, the deviation of a raindrop surface from a minimum-energy-statesphere can be calculated considering the first law of thermodynamics: the change inenergy from the surface’s increased surface tension must be balanced by the workdone to expand the surface’s enclosed volume into a region of different pressure (e.g.,Pruppacher & Klett, 2010, chapter 10.3.2). We follow Green (1975) in accountingfor the key pressures at raindrop equator and assuming an oblate spheroid shape (seeAppendix B for more detail), which gives r eq = r σ c-air g ( ρ c, ‘ − ρ air ) (cid:18) ba (cid:19) − s(cid:18) ba (cid:19) − − (cid:18) ba (cid:19) − + 1 . (2)Here σ c-air is the surface tension between the liquid condensible and air, g is the localgravitational acceleration, and ρ air is the local air density. For a given r eq , equation(2) can be solved numerically for b/a to characterize the raindrop’s shape. The two key forces acting on a falling raindrop in a planetary atmosphere are thegravitational force F g and the aerodynamic drag force F drag . The effective gravitationalforce on a raindrop is F g = 43 πr eq ( ρ c, ‘ − ρ air ) g (3)after accounting for the raindrop’s buoyancy within the air fluid. Drag force on araindrop is F drag = 12 C D Aρ air v (4)where C D is the drag coefficient of the raindrop, A is cross sectional area of theraindrop, ρ air is the local air density, and v is the raindrop’s fall speed relative toair. Raindrop cross sectional area is a function of raindrop shape and size. C D is afunction of raindrop shape and flow regime. The latter can be characterized by thedimensionless Reynolds number Re Re ≡ v‘ρ air η air (5)where ‘ is a characteristic raindrop length scale that we take to be r eq and η air isthe dynamic viscosity of local air. Viscosity varies with both air composition andtemperature; we calculate η air in a generic air mixture following Reid et al. (1977)chapter 9.Calculating C D theoretically for a general-shaped object in a general flow is nottractable (e.g., Stringham et al., 1969), and thus C D is typically evaluated via exper-imentally based parameterizations. As long as the flow regime is correctly capturedby a scale analysis, defaulting to experimentally based parameterizations for genericatmospheric conditions in and of itself is not problematic—though limitations in thecoverage of the parameter space used to fit expressions must be considered.In this work, we use the drag parameterization C D = (cid:18) Re (cid:16) . Re . (cid:17) + 0 . (cid:0) . × Re − . (cid:1) − (cid:19) C shape (6)(Clift & Gauvin, 1970; Loth, 2008). C shape is a correction term to account for raindropdeviations from spherical shape fit by Loth (2008) for b/a ≤ across a variety of fallingobject shapes: C shape = 1 + 1 . f SA − . + 6 . f SA − (7)where f SA is the ratio of the surface area of the oblate spheroid raindrop to the surfacearea of a sphere of radius r eq . The formulation of C D in equation (6) is primarily –5–anuscript submitted to JGR: Planets based on the parameterization of drag for a sphere with Re < . × by Clift andGauvin (1970). C shape is most accurate when significant raindrop deviation from asphere occurs within the Newtonian flow regime of Re between about 750 to 3.5 × (Clift et al., 2005; Loth, 2008).Compared to the other C D parameterizations we considered (Salman & Verba,1988; Lorenz, 1993; Ganser, 1993; Hölzer & Sommerfeld, 2008), we found equation(6) with (7) best reproduced the velocity dependence on raindrop radius for Earthvalues with experimental validation over wide ranges of Re and oblate spheroid axisratios. We note that we neglect corrections to C D for non-continuum regime effects(commonly referred to as the Cunningham or slip-flow correction factor) as we arenot concerned with the behavior of very small ( . µm) particles (e.g., Seinfeld &Pandis, 2006, chapter 9.2). (Such corrections only become important when particlesize becomes comparable to the mean free path of local air molecules.)Terminal velocity v T occurs when the raindrop is no longer accelerating, and thegravitational force F g is balanced by the aerodynamic drag force F drag . Under modernEarth atmospheric conditions, the timescale for raindrops of a fixed size to reachterminal velocity is very small compared to their lifetimes (Pruppacher & Klett, 2010,chapter 10.3.5). To test the generality of this rapid raindrop acceleration assumption,we numerically integrated water raindrop motion accounting for variable accelerationacross a large range of plausible planetary conditions (air composition, surface gravity,air pressure, air temperature).We find that the Earth-based empiricism is generally true: even the largest possi-ble stable raindrops (described in section 3.4) with the longest acceleration timescalesreach 99% of their terminal velocity after starting from rest within the first 1% of theirtotal fall distance and 5% of their total fall time (Figure S1). Further, when consider-ing the effect of raindrop size changes due to evaporation on reaching terminal velocity,we find that the differences in raindrop fall speed between self-consistent treatmentof raindrop acceleration and assuming terminal velocity is instantly reached are, atmaximum, on the order of 10% and typically much smaller (Figure S2). Thus, forsimplicity, we henceforth make the standard assumption that raindrop falling speedrelative to air v is the raindrop’s terminal velocity, which can be uniquely determinedfor a given raindrop size and shape.Equating F drag and F g and substituting the appropriate oblate spheroid geometryyields a terminal velocity of v T = − s
83 ( ρ c, ‘ − ρ air ) ρ air gC D (cid:18) ba (cid:19) r eq . (8) ρ air and g are known from planetary atmospheric properties. b/a is uniquely deter-mined from r eq with equation (2). The nonlinear dependence of C D on v (through Re)requires we solve equation (8) numerically.A raindrop’s vertical speed relative to a planet’s surface—d z /d t , its change inaltitude z per unit time t —is the sum of the its velocity relative to air, here assumedto be v T , and the vertical velocity of the raindrop’s local air w :d z d t = v T + w. (9)We are focused on raindrop physics here and so treat w as a free parameter in theanalysis that follows. Raindrop evaporation occurs when the atmosphere surrounding the drop is sub-saturated in condensible gas. The preferred phase of the condensible molecules at the –6–anuscript submitted to
JGR: Planets drop’s surface becomes gas rather than liquid. As the condensible is transferred fromthe liquid phase in the raindrop to the gas phase in the air, the air closest to the rain-drop surface deviates in temperature and relative humidity from the local atmosphericstate; the relative humidity adjacent to the drop surface increases while the tempera-ture drops due to the latent heat required for the liquid-to-gas phase transition. Boththese effects serve to lower the thermodynamic impetus to evaporate. Thus, in addi-tion to the environmental level of sub-saturation, the rate at which evaporation occursis dictated by the rate at which heat and condensible gas can be transported awayfrom the raindrop surface.Quantitatively, the change in raindrop equivalent radius with time t can then beformulated from geometry and appropriate boundary conditions asd r eq d t = f V,mol D c-air µ c r eq ρ c, ‘ R (cid:18) RH p c,sat ( T air ) T air − p c,sat ( T drop ) T drop (cid:19) (10)(see Rogers & Yau, 1996, chapter 7 for a derivation). RH is the relative humidity of thelocal air; R is the ideal gas constant; µ c is the molar mass of the condensible in its gasphase; ρ c, ‘ is the density of the liquid condensible; D c-air is the diffusion coefficient forthe condensible gas in air; p c,sat is condensible gas saturation pressure; T air is the localair temperature far from the drop’s surface; T drop is the temperature at the raindrop’ssurface; and f V,mol is a ventilation factor that accounts for how much raindrop motionenhances condensible molecule transport relative to a stagnant drop.Conservation of heat at the raindrop surface yields the following differential equa-tion governing T drop :d T drop d t = 3 r eq c p ,c, ‘ (cid:18) L c d r eq d t − f V,heat K air ρ c, ‘ r eq ( T drop − T air ) (cid:19) (11)(Rogers & Yau, 1996, chapter 7). c p ,c, ‘ is the specific heat at constant pressure of theliquid condensible; L c is the condensible’s latent heat of vaporization at T drop ; K air isthermal conductivity of air; f V,heat is a ventilation factor that accounts for how muchraindrop motion enhances heat transport relative to a stagnant drop. Without any fur-ther simplifications, equations (10) and (11) must be solved together numerically frominitial conditions given their mutual dependencies. (We will discuss simplifications tocalculating T drop in detail in section 3.3.)Relative humidity RH (also known as saturation) is defined as the ratio of thelocal condensible gas partial pressure p c to p c,sat at T air , i.e, RH ≡ p c /p c,sat ( T air ) . D c-air is a function of temperature, pressure, and air composition that we calculate followingReid et al. (1977) chapter 11 and Fairbanks and Wilke (1950). K air is a function oftemperature as well as air composition that we calculate following the Eucken method(Reid et al., 1977, chapter 10). Note that the formulation of raindrop temperaturein equation (11) only considers heat transport via conduction. For high temperaturecondensibles, heat transport via radiation will also need to be considered.The ventilation factors arise from fluid dynamical effects not analytically calcu-lable, so, as for C D , we must evaluate f V,mol and f V,heat from parameterizations basedon experiments. Here, we choose to use the f V,mol parameterization of Beard andPruppacher (1971) and Pruppacher and Rasmussen (1979): f V,mol = . (cid:16) Re . Sc . ¯3 (cid:17) , Re . Sc . ¯3 < . , .
78 + 0 . (cid:16) Re . Sc . ¯3 (cid:17) , Re . Sc . ¯3 ≥ . (12)where Sc is the dimensionless Schmidt number defined asSc ≡ η air D c-air ρ air . (13) –7–anuscript submitted to JGR: Planets
This parameterization is only experimentally validated for Re < but is hypoth-esized to be valid for spheres with Re < × based on theory (Pruppacher &Rasmussen, 1979). Following Pruppacher and Klett (2010, chapter 13.2.3), we cal-culate f V,heat from equation (12) for f V,mol with Sc replaced by the mathematicallyanalogous dimensionless Prandtl number PrPr ≡ η air c p ,air K air (14)where c p ,air is the specific heat at constant pressure for air. We neglect the effects ofturbulence, which can act to increase ventilation.Raindrop shape impacts multiple aspects of ventilation, but in raindrop experi-mental data considered by Pruppacher and Rasmussen (1979), these shape effects can-cel each other such that f V is independent of raindrop shape deformations at largerRe. (Note that this shape independence is only true for liquid raindrops, not solidcondensibles (e.g., Pruppacher & Klett, 2010, chapter 13.3.2).)The above expressions for f V,mol and f V,heat are not definitive and could be im-proved by further experiments over a broader parameter space. The ventilation factorsincrease d r eq /d t by roughly an order of magnitude at large raindrop sizes, so they needto be accounted for despite the somewhat limited coverage of these parameterizations;as we will show in section 4, the qualitative impact of these uncertainties in ventilationfor r eq ( t ) is often limited because large raindrops evaporate mass very gradually.In our formulation of d r eq /d t in equation (10), we have made a number of simpli-fying assumptions that are valid here because we are concerned only with evaporation.(d r eq /d t can describe both evaporation and condensation depending on whether RHis less than or greater than 1.) Our focus on evaporation, rather than condensation,means we are not concerned with the behavior of very small drops, which, as we willshow in Section 4, always evaporate rapidly compared to larger drops. We neglectcorrections to relative humidity at the raindrop’s surface due to surface tension andcondensation nuclei solute effects—commonly known as Kelvin and Raoult effects, re-spectively (e.g., Lohmann et al., 2016, chapter 6)—that are important only at verysmall radii ( r . µm). Note that in atmospheres where a gas component is solublein the liquid condensible (e.g., N in CH on Titan; Thompson et al., 1992), solutecorrections to RH cannot be neglected. (See, for example, Graves et al. (2008) forhow to extend what is presented here to such cases.) Consistent with our treatmentof drag, we also neglect corrections to condensible vapor diffusivity and air thermalconductivity for non-continuum regime effects very close to the drop’s surface, which iseffectively equivalent to assuming that the mean free paths of air and condensible gasmolecules are small compared to the size of the raindrop (e.g., see Lamb & Verlinde,2011, chapter 8.2.2 for further discussion).Beyond corrections only necessary for small drops, we also neglect corrections tothe boundary conditions used to solve for d r eq /d t due to raindrop deformation froma sphere (Lamb & Verlinde, 2011, chapter 8.3). These shape corrections considerablycomplicate manipulating d r eq /d t and cause variations in d r eq /d t of less than 5%. From the raindrop characteristics we outlined in section 2, we can calculate thechange in raindrop equivalent radius with altitude z asd r eq d z = d r eq d t (cid:18) d z d t (cid:19) − (15) –8–anuscript submitted to JGR: Planets where d r eq /d t is given by equation (10) and d z /d t by equation (9). Solving equation(10) also requires coupled evaluation of d T drop /d t from equation (11). Note that thisformulation neglects the time to accelerate to a new terminal velocity as raindrop sizechanges (which is rapid compared to the timescale on which r eq evolves, as discussedin section 2.2).To evaluate equation (15), we first must describe the atmospheric state variablesthat affect parameters required for calculating evaporation rate and terminal velocity— p , T , RH—as functions of z . In this work, we prescribe planetary conditions of p , T ,and RH, at a single z —either at the surface or at the cloud base depending on thecalculation of interest. We also require planetary inputs of g and dry air composition,which are assumed to be fixed.We then follow the standard assumptions for a 1D atmosphere in radiative-convective equilibrium below saturated regions to relate atmospheric properties (e.g.,Pierrehumbert, 2010; Romps, 2017): our pressure-temperature profile follows a dryadiabat, z is related to p (and thus RH and T ) assuming hydrostatic equilibrium, andRH is prescribed assuming the condensible gas is well-mixed (i.e., a constant molarconcentration). Note that assuming constant T and RH from average values below thecloud base does not lead to significantly different d r eq / d z values, but such a simpleprofile does not allow for an internally consistent calculation of cloud base height.We define cloud base as the “lifting condensation level” (LCL), the height atwhich a condensible gas reaches saturation in a parcel of air rising adiabatically: z such that p c ( z LCL ) = p c, sat ( T ( z LCL )) . The LCL errs in predicting cloud base whenlimited cloud condensation nuclei require supersaturation to initiate cloud particleformation. However, as we are concerned with where the raindrop starts evaporating(which requires RH < 100%), this caveat does not concern us.Equation (15) is stiff, so we integrate it using an implicit Runge-Kutta methodof order 5. We define the raindrop’s initial r eq at cloud base as r . We calculate r eq ( z )from the cloud base ( z = z LCL ) to a desired z or until the raindrop fully evaporates.Here we define a “fully evaporated” raindrop as a drop of equivalent radius less than athreshold drop size ∆ r .We set ∆ r = 1 µm; our results are not sensitive to this choice of ∆ r as long as ∆ r (cid:28) r . ∆ r must be non-zero for numerical stability, but ∆ r > also reflects thephysical reality that the cloud drops that form raindrops very strongly thermodynam-ically favor condensing onto preexisting nuclei rather than forming homogeneously. To understand the potential of a raindrop to transport condensible mass and heatwithin an atmosphere, we calculate the cloud-edge (where RH transitions to less than 1and evaporation begins) size threshold where a raindrop can survive to a given height z without totally evaporating, r min ( z ) . We define r min ( z ) as the r such that r min ( z ) − ∆ r evaporates before reaching z , but r min ( z ) reaches the z , i.e., r eq ( z, r = r min ) ≥ ∆ r . r min ( z ) is solved for via bisection by integrating r ( z ) as described in section 3.1 forinitial radii between ∆ r and the maximum raindrop radius described in section 3.4.On terrestrial planets (with a surface at z surf ), clouds that can grow raindrops of r ≥ r min ( z surf ) can move condensible mass from the atmosphere to the surface con-densible reservoir. We can place an lower bound on raindrop size from the cloud-edgesize threshold where a raindrop can survive to the surface without totally evaporating: r min ( z surf ) , which we will henceforth abbreviate to simply r min . On gaseous planets,there is no surface, and raindrops can only evaporate, but their ability to transportmass and heat as a function of height is still important dynamically. –9–anuscript submitted to JGR: Planets
To better understand raindrop evaporation, we simplify equation (15) into adimensionless number that can be more clearly interpreted—and evaluated—than asystem of differential equations requiring numerical integration to solve. First, we needto simplify calculating T drop for an evaporating raindrop from the differential equation(11). We assume T drop changes only as a function of altitude. This is justified by com-paring the timescale on which atmospheric temperature changes ( τ air ) to the timescaleon which raindrop temperature changes ( τ drop ). Assuming a dry adiabatic tempera-ture profile, τ air ≈ ( c p ,air ∆ T air ) / ( g d z/ d t ) where we conservatively set the characteristicchange in air temperature ∆ T air to 1 K. Assuming the atmosphere transfers heat to theraindrop via conduction, τ drop ≈ ( r eq ρ c, ‘ c p ,c, ‘ ) / (3 K air ) . Except for the largest possibleraindrops, under broad planetary conditions τ drop (cid:29) τ air , and hence d T drop /d t =0 is agood approximation at a given altitude.We define ∆ T drop as the equilibrium temperature difference between the air andraindrop, i.e., ∆ T drop ≡ T air − T drop | d T drop / d t =0 . (16)From this definition of ∆ T drop and equation (11), ∆ T drop = D c-air f V ,mol Lµ c K air f V ,heat R (cid:18) p sat ( T air − ∆ T drop ) T air − ∆ T drop − RH p sat ( T air ) T air (cid:19) . (17)This transcendental equation can be solved numerically via a root-finding algorithm.It is commonly simplified to an analytic expression using Clausius Clapeyron, Taylorexpansions, and series of assumptions regarding ∆ T drop being small compared to T air (e.g., Rogers & Yau, 1996, chapter 7).However, we find an analytic approximation that holds better across a broadrange of planetary conditions is to evaluate equation (17) with ∆ T drop values on theright hand side approximated as ∆ T drop ≈ . T air − T LCL ) . (18) T drop must fall between T LCL and T air (i.e, ∆ T drop ∈ [0 , T air − T LCL ] ) because there isno heat source for the drop once T drop = T air , and there is no heat sink for the droponce T drop = T LCL because RH=1 and evaporation ceases. Equation (18) can also beemployed for a back-of-the-envelope calculation of ∆ T drop .Now we can define a dimensionless number Λ to evaluate the tendency of araindrop of radius r eq (and mass m ) toward evaporation within a given vertical lengthscale ‘ . Λ is the ratio of evaporative mass loss during transit through ‘ to raindropmass: Λ ≡ ‘m d m d t (cid:18) d z d t (cid:19) − = 3 ‘r eq d r eq d t (cid:18) d z d t (cid:19) − . (19)Here we have made use of the chain rule, the definition of r eq , and the relationd m/ d r eq = 4 πρ c, ‘ r eq . Expanding the terms in Λ gives Λ = 3 ‘r eq f V,mol D c-air µ c ( w + v T ) ρ c, ‘ R (cid:18) RH p c,sat ( T air ) T air − p c,sat ( T air − ∆ T drop ) T air − ∆ T drop (cid:19) . (20)Altitude-dependent values needed to calculate Λ are evaluated at the midpoint of ‘ . ∆ T drop can be evaluated from equation (17) numerically (most accurate), fromequation (17) and (18) algebraically, or from equation (18) (back-of-envelope). v T canbe evaluated from equation (8) numerically or from a parameterized relationship for acommonly studied planet. Alternatively, v T can be estimated via Stokes law for very –10–anuscript submitted to JGR: Planets small drops or via v T,max ≈ σ c-air ( ρ ‘ ,c − ρ air ) g ) . ( ρ air ) − . for very large drops(Clift et al., 2005, chapter 7.C). Λ values give the expected change in raindrop mass from evaporation relativeto initial mass after falling a given distance. Λ( r eq , ‘ ) ≥ indicates raindrops ofsize r eq will fully evaporate over distance ‘ . Therefore, the fraction of raindrop massevaporated over ‘ can be estimated from min{ Λ ,1}. For a given ‘ , the r eq such that Λ = 1 approximates the minimum radius to reach that distance below the starting z without fully evaporating, r min ( z start − ‘ ) . For simplicity, here we only consider Λ defined when d z /d t <
0, i.e., when a raindrop is falling downward. Though we donot treat raindrop formation here, we note that with some slight modifications thisdimensionless number can also be employed to consider the effectiveness of cloud dropgrowth via condensation.
The final physical process we need to consider is raindrop breakup. Raindropscannot grow to infinitely large sizes because the resistance provided by surface tensionas surface area increases is limited. When surface tension ceases to be the dominantforce experienced by a raindrop, the raindrop rapidly breaks apart.A variety of approaches to estimating this maximum stable raindrop radius r max have been proposed previously; but none are expected to yield quantitatively exactvalues. The physics is additionally complicated in many situations (such as on present-day Earth) by the fact that the practical upper bound on raindrop size is not set fromindividual raindrop breakup but rather from hydrometeor collisions (e.g., Barros et al.,2010). Given the uncertainties, we are therefore primarily concerned here with how r max scales with external planetary properties. In particular, we focus on the effectof air density, which has inconsistently been claimed within the planetary literatureto have no effect on r max (Som et al., 2012; Palumbo et al., 2020) and an extremelysignificant one (Craddock & Lorenz, 2017).Variants of two methods have commonly been used to describe raindrop breakup;we review them here and describe their origins in more detail in Appendix C. First,we can estimate r max by considering when the base of a raindrop becomes unstableto small perturbations from a more dense fluid (liquid condensible) being on top of aless dense fluid (air)—generally referred to as Rayleigh-Taylor instability (Komabayasiet al., 1964; Grace et al., 1978; Lehrer, 1975; Clift et al., 2005; Pruppacher & Klett,2010, chapter 10.3.4). This analysis yields a maximum length scale ‘ RT,max that canbe related back to a maximum equivalent radius r max : ‘ RT,max = π r σ c-air g ( ρ c ,‘ − ρ air ) . (21)There is not a definitive ‘ RT,max , with different authors choosing related, but oftendistinct, length scales.Another common approach for estimating r max in the Earth literature is to cal-culate when the force of surface tension F σ is balanced by the aerodynamic drag force(e.g., Pruppacher & Klett, 2010, chapter 10.3.4). We henceforth refer to this approachas “force balance.” Again, we get a relationship to be solved for r max that depends ona somewhat arbitrary length scale, here pertaining to surface tension ‘ σ ,max : r max ‘ σ ,max = 34 π σ c-air ( ρ c, ‘ − ρ air ) g . (22)We evaluate equations (21) and (22) for r max under different length scales pro-posed in the literature. Length scales are related to r max via the geometry of spheres –11–anuscript submitted to JGR: Planets r eq ( z ) [mm]0100200300400500600 z [ m ] r r end r r end r r end r r end r r end r r end Figure 1: Raindrop altitude z versus equivalent radius r eq for equally log-spaced ini-tial raindrop radii ( r ) near the minimum radius threshold for survival to surface ( r min ).Gray-shaded lines are raindrops that evaporate before reaching the surface while purple-shaded lines are raindrops that successfully reach the surface. Planetary conditions are setto Earth-like as given in Table 1.or oblate spheroids. Both approaches yield similar expressions for r max with some vari-ation in dependence on raindrop shape and constant factors depending on the choiceof length scale. Having described the key physical processes that affect isolated falling raindropsin detail, we now present numerical results for a wide range of planetary conditionsand circumstances pertaining to falling raindrops. We validated our shape and ter-minal velocity calculations against modern Earth observations, experimental results,and empirically based calculations (Figures S3-S4; Gunn & Kinzer, 1949; Best, 1950;Pruppacher & Beard, 1970; Pruppacher & Pitter, 1971; Beard, 1976; Beard & Chuang,1987; Thurai et al., 2009). We also compared, with reasonable agreement, our resultsto previous planetary theoretical results on Titan’s methane-nitrogen raindrops forshape, terminal velocity, and raindrop properties with altitude (Figures S5-S8; Ta-ble S1; Lorenz, 1993; Graves et al., 2008), using Cassini Huygens’ probe data whereappropriate (Fulchignoni et al., 2005; Niemann et al., 2005).
Integrating equation (15), we investigated the behavior of raindrop evaporationfor water raindrops falling from the cloud base to planetary surface under Earth-likeconditions (Table 1). Figure 1 shows the evolution of raindrop radius as function of –12–anuscript submitted to
JGR: Planets T a b l e : P l a n e t a r y p r op e r t i e s u s e d i n c a l c u l a t i o n s p l a n e t z r e f T ( z r e f ) p d r y ( z r e f ) R H ( z r e f ) g f H , d r y f H e , d r y f N , d r y f O , d r y f C O , d r y H L C L n a m e [ K ][ P a ][][ m s − ][ m o l m o l − ][ m o l m o l − ][ m o l m o l − ][ m o l m o l − ][ m o l m o l − ][ k m ] E a r t h - li k e a s u r f a ce . . . . E a r t h s u r f a ce . . . . . . e a r l y M a r s b s u r f a ce . . . J up i t e r L C L c . c . . d . d . S a t u r n L C L c . c . . d . d . K - b e L C L . . f , g . . . c o m p o s i t i o n L C L . . , , , , , . - b r oa d L C L - . - - , , , . - N o t e . I npu t p r o p e r t i e s ( c o l u m n s b e t w ee n t h e d o ub l e v e r t i c a lli n e s ) a r e s p ec i fi e d f o rr e f e r e n ce a l t i t ud e z r e f a ndu s e d t o d e t e r m i n e a t m o s ph e r i c p r o p e r - t i e s und e r t h ec l o ud l a y e r f o ll o w i n g t h e a ss u m p t i o n s o u t li n e d i n s ec t i o n . . p d r y i s t h e p r e ss u r e o f a ll n o n - c o nd e n s i b l e ga ss p ec i e s . f i s t h e d r y m o l a r c o n ce n t r a t i o n . A t m o s ph e r i c s c a l e h e i g h t H ≡ R T ( g µ a v g ) − i s e v a l u a t e d a t z L C L t h r o u g h o u tt h i s p a p e r . V e r t i c a l w i nd s p ee d i ss e tt o0 m s − un l e ss o t h e r w i s e s p ec i fi e d i n t h e t e x t . a T s u r f i s h i g h e r t h a n a v e r ag e E a r t h T s u r f i n o r d e r t o h i g h li g h t a l a r g e rr a n g e o f p o ss i b l e s u r f a ce R H v a l u e s w h il e k ee p i n g T L C L a b o v e f r eez i n g . P u r e N b a c k g r o und a t m o s ph e r e i s a ss u m e d f o r s i m p li c i t y . b Sp ec u l a t i v e v a l u e s f o r h y p o t h e s i ze d w a r m , w e t p e r i o d i n M a r s ’ a n - c i e n t p a s t . c C a r l s o n e t a l. ( ) . d L ec o n t ee t a l. ( ) . e Sp ec u l a t i v e v a l u e s c o m p a t i b l e w i t h c o n s t r a i n t s f r o m B e nn e k ee t a l. ( ) . f C l o u t i e r e t a l. ( ) . g B e nn e k ee t a l. ( ) . –13–anuscript submitted to JGR: Planets altitude z for a number of an initial radii at cloud base until the raindrops eithercompletely evaporate ( r < r min ) or reach the surface ( r ≥ r min ). The results implya strong positive feedback on raindrop evaporation as raindrops grow smaller.Figure 2(a) demonstrates this positive feedback more explicitly. For the sameplanetary conditions as in Figure 1, it shows the fraction of raindrop mass evaporatedat the surface for a range of r values. This curve approaches a step function about r min . Figure 2(b) extends 2(a) by showing fraction of raindrop mass evaporated viacolormap versus surface relative humidity and r . Qualitatively, Figure 2(b) shows thesame sharp cut-off behavior as 2(a). Surface RH affects the quantitative value of r min because it varies RH( z ), which impacts the magnitude of evaporation rate as well asthe height of z LCL —a higher surface RH gives a lower z LCL . Both these effects act tomake r min decrease as surface RH increases.There is not yet an analytical method for estimating average surface RH in ageneric planetary atmosphere—in part because of the poorly understood feedbacks ofprecipitation evaporation on average RH (Romps, 2014; Lutsko & Cronin, 2018)—so,for now, we consider surface RH a prescribed planetary parameter. In this plot, wevary surface RH from 99.9% to 25%—the former arbitrarily close to the threshold forevaporation to begin (RH < 100%) and the latter about the minimum surface RHfor which the temperature at z LCL is above the freezing point of H O given our otherchosen planetary parameter values.Figure 3 is the same as Figure 2(b) except it probes the effect of vertical windspeed w rather than surface RH. Downdrafts ( w < ) increase the falling speed ofraindrops while updrafts ( w > ) decrease the falling speed of raindrops (as long as w + v T < ). Updrafts can transport raindrops upward once the updraft speed exceedsthe magnitude of a raindrop’s terminal velocity. As downdraft speed increases, there isa smoother transition through fraction mass evaporated across r values. As updraftspeed increases, fraction mass evaporated approaches a step function about r min until w + v T,max = 0 . For updrafts speeds greater than this threshold, no r min exists asraindrops are no longer falling.As with surface relative humidity, there is no analytic approach for estimatingaverage w ranges in generic planetary atmospheres (though values can be probed bymesoscale models of sufficient resolution). In this plot, we bound updraft speed fromwhere w exceeds maximum raindrop terminal velocity and then choose a symmetricdowndraft speed bound. (This choice of a lower bound is arbitrary and does notrepresent an end-member case for downdraft speeds.) For simplicity, here we fix w as constant throughout raindrop falling and evaporation. In reality, vertical velocitiesvary both spatially and temporally within a given storm event (e.g., Lohmann et al.,2016), often as a result of the interaction between precipitation particles and ambientair (e.g., Rogers & Yau, 1996). Figure 4 compares using the dimensionless number Λ defined in equation (19)to predict raindrop evaporation behavior to using numerical integration for Earth-likeatmospheric conditions (a)-(b) and across broad planetary conditions (c)-(d). Figure4(a) shows calculations of the fraction of raindrop mass evaporated at the surfacerelative to initial mass at cloud base versus initial radius r using both numericalintegration and the minimum of Λ( r , ‘ = z LCL ) and 1. Figure 4(c) is similar to (a)except it shows the difference in fraction mass evaporated at 500 m below the cloudbase between these two methods at 4 r for 90 different planetary conditions. The“broad” conditions in Table 1 give the ranges over which we vary T , p , and g at cloudbase for background gas atmospheres of pure H , N , and CO . Only one value among –14–anuscript submitted to JGR: Planets f r a c t i o n m a ss e v a p o r a t e d [ ] surface RH = 0.75 r min r [mm]0.30.40.50.60.70.80.9 s u r f a c e R H [ ] TOTAL EVAPORATION r min % m a ss e v a p o r a t e d f r a c t i o n m a ss e v a p o r a t e d [ ] ab Figure 2: (a) Fraction of raindrop mass evaporated at the surface versus initial radius r for an Earth-like planet (Table 1). The purple dot marks r min , the r -threshold for araindrop to reach the surface without totally evaporating. (b) Fraction of raindrop massevaporated (black-white color scale) versus surface relative humidity and r . The same asthe top panel except with varying surface RH; the horizontal light-purple line highlightsthe surface RH slice that the top panel displays. The purple line marks the calculated r min as a function of surface RH. For r < r min , raindrops totally evaporate before reach-ing the surface (hatched region). The dashed dark-purple line highlights the 10% massevaporated contour within the more continuous shading. –15–anuscript submitted to JGR: Planets r [mm]10.07.55.02.50.02.55.07.510.0 w [ m s ] TOTAL EVAPORATION r min % m a ss e v a p o r a t e d f r a c t i o n m a ss e v a p o r a t e d [ ] Figure 3: Fraction of raindrop mass evaporated (black-white color scale) versus verti-cal wind speed w and initial radius r for an Earth-like planet (Table 1). The horizontallight-purple line ( w = 0 ) divides updrafts ( w > ) from downdrafts ( w < ). The purpleline marks the calculated r min as a function of w . For r < r min , raindrops totally evapo-rate before reaching the surface (hatched region). The dashed dark-purple line highlightsthe 10% mass evaporated contour within the more continuous shading. –16–anuscript submitted to JGR: Planets r [mm]0.00.20.40.60.81.0 f r a c t i o n m a ss e v a p o r a t e d [ ] integrationmin{ ,1}10 r min ( z ) [mm]0200400600 z [ m ] integration= 110 r [mm]0.20.10.00.10.2 d i ff e r e n c e i n f r a c t i o n m a ss e v a p o r a t e d [ ] r e l a t i v e e rr o r r m i n ( z L C L ) [ ] abc d Figure 4: (a) Fraction mass evaporated at the surface versus initial raindrop radius r forEarth-like atmospheric conditions (Table 1) evaluated from numerical integration (dashedgray line) and using dimensionless number Λ (purple line). (b) Altitude z versus thresholdinitial raindrop radius for total evaporation at z ( r min ( z ) ) for Earth-like atmospheric con-ditions evaluated from numerical integration (dashed gray line) and using dimensionlessnumber Λ (purple line). (c) Difference in fraction mass evaporated between calculationsusing numerical integration and Λ versus four r values evaluated across broad planetaryconditions (Table 1) at 500 m below cloud base. (d) Relative error in r min ( z ) calculatedusing Λ relative to numerical integration versus three ‘ values across broad planetaryconditions. For (c) and (d), scatter points are semi-transparent to highlight where pointscluster. –17–anuscript submitted to JGR: Planets T , p , and g is changed at a time relative to the “composition” conditions, which areused as a baseline.Figure 4(b) shows calculations of the threshold minimum radius to reach altitude z without fully evaporating ( r min ( z ) ) using both numerical integration and Λ =1. Fig-ure 4(d) is similar to (b) except it shows the relative error in r min ( z ) calculated using Λ instead of numerical integration at three ‘ values for the same 90 different planetaryconditions as described for panel (c).Unsurprisingly, we find that the accuracy of using Λ to calculate the fraction ofraindrop mass evaporated at z decreases for r near r min ( z ) and that the accuracy ofusing Λ to calculate r min ( z ) decreases as the distance between the cloud base and z increases. Nonetheless, Figure 4 demonstrates that Λ can capture the essential behav-ior of fraction mass evaporated and r min ( z ) with a small fraction of the computationalcost of numerical integration ( (cid:28) ). Comparing the use of Λ and a full numericalintegration for calculations at ‘ = 500 m, we found percent errors for r min ( z ) andpercent differences in fraction raindrop mass evaporated were usually less than in magnitude across a broad planetary parameter space, and even the largest errorswere less than 20%. This agreement indicates that Λ is a viable way to predict andinterpret raindrop evaporation regimes. In Appendix D, we use the definition of Λ to show the numerical results of Figures 1 and 2 can be understood from an analyticmathematical perspective. Before we combine the concepts of minimum and maximum raindrop size thresh-olds, we return to the debate in the planetary literature over whether maximum stableraindrop size r max depends on air density. In Figure 5, we compare r max as a func-tion of air density (or air pressure) using break up criteria as presented in section 3.4and previous planetary literature. The two approaches we reviewed—Rayleigh-Taylorinstability and force balance—are dependent on somewhat arbitrary length scales, sowe plot the spread in values from different length scales. For Rayleigh-Taylor insta-bility, we consider ‘ RT,max = 0 . πr eq , . πa, r eq , a ; for force balance we consider ‘ σ ,max = 2 πr eq , πa .From the planetary literature on maximum raindrop size (Rossow, 1978; Lorenz,1993; Som et al., 2012; Craddock & Lorenz, 2017; Palumbo et al., 2020), we plot inFigure 5 the approaches of Lorenz (1993), Craddock and Lorenz (2017), and Palumboet al. (2020), which we will describe in more detail shortly. We do not plot Som etal. (2012)’s quantitative approach as they make use of empirical fits and Earth-basedobservations, but in practice they suggest a similar criterion to Lorenz (1993). Rossow(1978) and the Clift et al. (2005) expression cited in Lorenz (1993) use Rayleigh-Taylorinstability criteria with length scales included in our range.To calculate r max , Lorenz (1993), Craddock and Lorenz (2017), and Palumbo etal. (2020) all begin from the same criterion: the raindrop radius where the dimension-less Weber number equals 4. The Weber number We, which characterizes the ratio ofdrag force to the force of surface tension, is defined asWe ≡ r eq v ρ air σ c-air . (23)This approach is equivalent to the force balance approach under the assumption that,within F drag , cross sectional area times C D is equal to πr eq —an assumption that holdsreasonably well under Earth surface conditions (Matthews & Mason, 1964).To calculate r max , Lorenz (1993) solves for the r eq satisfying We = 4 numeri-cally (to account for the dependence of v on r eq ). As seen in Figure 5, this setup –18–anuscript submitted to JGR: Planets p air [Pa]0.02.55.07.510.012.515.017.520.0 r m a x [ mm ] Craddock & Lorenz (2017)Lorenz (1993)Palumbo et al. (2020)Rayleigh Taylorforce balanceEarth observations10 [kg m ] Figure 5: Maximum stable water raindrop size r max versus air pressure p air or air density ρ air , assuming fixed temperature T = 275 K and RH = 0 (for a linear relationship between p air and ρ air ), with Earth surface gravity and N background gas. Different line colors cor-respond to different methods for calculating r max as labeled in the legend and describedin the text. The Rayleigh-Taylor and force balance methods can use (arbitrarily) differentlength scales, so we plot the span of values from common length scale choices rather thansingle lines. –19–anuscript submitted to JGR: Planets yields an r max that varies by about a factor of 1.5 over the range of ρ air we consider.We view this variation with density as a non-physical dependence introduced by theneglect of the dimensionless C D in representing the drag force in the formulation ofWe (Kolev, 2007, chapter 8). ( C D nonlinearly depends on ρ air through Re.) Varyingthe C D parameterization within this calculation causes comparable changes in r max tovarying ρ air ; thus, we consider the Lorenz (1993) method consistent with no significantdependence of r max on ρ air . To calculate r max , Palumbo et al. (2020) solves for the r eq satisfying We = 4 algebraically after assuming spherical raindrops and C D =1: r max = (cid:18) σ gρ c, ‘ (cid:19) . . (24)Craddock and Lorenz (2017) does not present a simplified expression for calculat-ing r max , only evaluations under different atmospheric conditions and a statement that“larger diameter raindrops are not possible at higher atmospheric pressures.” When wesimplify their presented equations involved in describing r max (their equations (1) and(3)) following their stated assumptions, we arrive at the same r max result as Equation(24). We are only able to reproduce the results of Craddock and Lorenz (2017)’s eval-uations of r max (their Table 1) using an expression for r max inconsistent with lengthunits: r max = (cid:18) σ gρ air ρ c, ‘ (cid:19) . . (25)In addition to theoretical methods of estimating r max , we also plot in Figure 5 arange of claimed maximum raindrop sizes on present-day Earth, both from experimentsand natural observations. This range is consistent in magnitude with all the estimates.We give a range of maximum values as the measurement is an attempt to estimatethe end of the extreme tail of a stochastic process (e.g., Komabayasi et al., 1964;Grace et al., 1978; Clift et al., 2005). Single-value maxima fall between r eq of 4-5mm (Merrington & Richardson, 1947; Beard & Pruppacher, 1969; Ryan, 1976; Hobbs& Rangno, 2004; Gatlin et al., 2015). Gatlin et al. (2015) compiled observations of . × raindrops and found 0.4% of these raindrops had r eq ≥ . mm and only . × − % had r eq ≥ mm—statistics that suggest in practice r max is about 2.5-4mm. Returning to theory, in Figure 5 we see that while different assumptions aboutraindrop shapes in the Rayleigh-Taylor and force balance methods lead to factor of afew quantitative differences in r max , these differences are not sensitive to air density.We conclude that the effects of raindrop shape are ultimately of limited importance inestimating r max due to the ambiguity of raindrop length scales in the calculation setups.Thus the scalings of r max with planetary variables can reliably be seen analytically byassuming spherical raindrops (i.e., setting b / a = 1): r max ∝ r σ c-air g ( ρ c ,‘ − ρ air ) ≈ r σ c-air gρ c ,‘ . (26)The force balance, Rayleigh-Taylor instability, and Weber number methods all yieldthe same approximate scalings, which are effectively independent of air density. (Thesescalings are not novel (e.g., Clift et al., 2005); we simply present them in the contextof this planetary debate on r max .)Therefore, we agree with Palumbo et al. (2020) that Craddock and Lorenz(2017)’s finding that larger raindrops become possible as time advances and Marsexperiences atmospheric escape due the dependence of raindrop stability on ρ air is notjustified. This conclusion is also consistent with extensive modern-Earth based litera-ture considering r max , which does not explicitly highlight any dependence of r max onair density, which non-trivially varies from cloud to surface. Finally, we note that we –20–anuscript submitted to JGR: Planets p surf,dry [Pa] r e q [ mm ] g [m/s ]
300 325 350 375 400 425 450 T surf [K] CLOUD CONDENSATION NUCLEI SIZES r max r min , RH=0.25 r min , RH=0.5 r min , RH=0.75 a bb c Figure 6: Threshold r eq values versus (a) dry planetary surface pressure p surf,dry , (b)surface gravity g , and (c) surface temperature T surf . Planetary conditions not explicitlyvaried follow the Earth-like conditions in Table 1. Dark-gray lines give r eq at the onset ofRayleigh-Taylor instability ( r max ); purple lines give the minimum cloud-edge raindrop ra-dius required for the raindrop to not evaporate before reaching the surface ( r min ) for threesurface RH values. Purple shaded regions show cloud-edge size bounds of raindrops thatcan reach the surface. The gray shaded region sketches cloud condensation nuclei sizes.are not claiming that average raindrop size is insensitive to air density, only that theinstability of individual large raindrops does not have a significant dependence on airdensity. In Figure 6, we calculated r max and r min for water raindrops across a range ofplanetary conditions. For clarity, we plot only one of the r max values from the methodswe discussed previously (Rayleigh-Taylor with ‘ RT,max = 0 . πa ) and r min values foronly three representative surface relatives humidities (RH = 0.25, 0.5, 0.75) and novertical wind ( w = 0 ). We used as default values the Earth-like conditions of Table 1when the x-axis planetary parameter is not varying.The purple shading highlights how our estimates of r max and r min constrain pos-sible raindrop sizes that can transport condensible mass from a cloud to the surfacereservoir. Varying colors of purple shading correspond with size bounds set from vary-ing surface relative humidity values. As the purple shading lightens, the bound includessmaller r min values associated with higher surface relative humidity values. The darkpurple shading (for RH surf =0.25) is the strictest bound. For perspective, cloud dropsgrow from cloud condensation nuclei with many orders of magnitude smaller sizes aswe have schematically indicated with gray shading. On Earth, typical cloud conden-sation nuclei are around 0.05 µm ( × − mm) (Lohmann et al., 2016) while, forconditions considered here, viable raindrop sizes vary by about an order of magnitudewith typical values of tenths of millimeters, or about 10,000 times larger than typicalcloud condensation nuclei.In Figure 6(a), we plot raindrop size bounds as described for variable dry surfacepressures. Neither of the size bounds has a strong dependence on pressure (see alsosection 3.4). r min depends on pressure in multiple ways that largely cancel each otherout. Figure 6(b) shows the impact of surface gravity g on raindrop size bounds. As g increases, r max and r min all systematically decrease like g − . . Larger raindrops arepossible at lower surface gravities, and raindrops must also be larger to survive to thesurface without evaporating. Figure 6(c) highlights the effect of increased evaporation –21–anuscript submitted to JGR: Planets rate in higher air temperatures. r min rises with T surf because evaporation rate andfalling time to surface increase as long as the molar mass of the condensible gas is lessthan the average dry air molar mass—as considered here. Table 2 considers the effect of atmospheric composition on the time and distancefrom cloud base until evaporation ( t evap and z evap , respectively) for H O raindrops withpure H , He, N , O , and CO atmospheres. Other planetary conditions used are givenunder “composition” in Table 1. Atmospheric composition impacts raindrop evapora-tion from three main effects: (1) molar mass and heat capacity impact atmosphericstructure, which governs how vertical distance maps to temperature, pressure, andrelative humidity—all key parameters in calculating d r /d z ; (2) molar mass impactshow a given pressure maps to a density, which impacts raindrop terminal velocity; and(3) molar mass and molecular structure impact the rate at which air can transportlatent heat and condensible gas away from the raindrop.In Table 2, we calculated t evap and z evap considering each of these effects ofcomposition in isolation as well as all together. When only a single effect is considered,all other compositional effects are calculated with pure N . Total number of moleculesat cloud base is held fixed (i.e., pressure is fixed as an ideal gas assumed).As shown Table 2, air composition acts on the integral of d r eq /d z in competingdirections, so all effects must be considered in unison to understand how precipita-tion evaporation will vary with atmospheric composition. We find the time taken toevaporate is comparable across atmospheric conditions. The distance to evaporationis comparable for the higher molecular mass gases with the He atmosphere about 1.75times larger and the H atmosphere about 3.5 times larger. Excluding the noble gasHe, the ability to transport condensible mass in units of scale heights increases asatmospheric molar mass increases.We further probe the variations in raindrop evaporation due to atmospheric com-position in Figure 7 by calculating latent heat absorbed per second (power, P evap ) asa function of (a) atmospheric pressure, (b) vertical distance from cloud base, and (c)falling time for a fixed initial condensible mass sorted into raindrops of three differentradii. P evap is linearly related to the rate of condensible mass evaporation through thelatent heat of vaporization, which is normalized per unit mass. Thus we choose a con-stant initial mass (the mass of the largest raindrop considered) to compare magnitudesof P evap for different initial raindrop sizes. We use the same planetary conditions asTable 2 (under “composition” in Table 1) with background H , N , and CO atmo-spheres.In all three vertical coordinates shown in Figure 7, the peak P evap reached isabout the same for a given r across all three atmospheric compositions. Maximum P evap decreases as r increases (as expected in order to conserve mass with longer falltimes). Figure 7 shows P evap is roughly a quadratic function of log p (a), z LCL − z (b), and t (c) between cloud base and reaching total evaporation for all atmosphericcompositions and initial size values considered.For a given composition atmosphere, increasing r increases p evap , z evap , and t evap . The spread in p evap and z evap at total evaporation between different composi-tions increases as r increases. As seen in Table 2 (given log p is essentially proportionalto altitude over scale height), p evap increases with increasing dry molar mass; z evap de-creases with increasing dry molar mass; and t evap is about constant across the differentcompositions. –22–anuscript submitted to JGR: Planets
Table 2:
Effects of Atmospheric Composition on Raindrop Evaporation composition effect(s) t evap [s] z evap [m] z evap [ H ] H all 769 6970 0.0648 H H v T
435 3960 0.0368H transport 333 1000 0.00933 He all 638 3560 0.0632 He H v T
623 3480 0.0618He transport 329 991 0.0176 N all 707 2090 0.251O all 701 1900 0.260 O H
675 1920 0.263O v T
771 2000 0.274O transport 705 2090 0.285 CO all 769 1960 0.369 CO H
655 1870 0.351CO v T
726 1750 0.330CO transport 855 2510 0.473 Note.
Time to evaporate t evap and falling distance from cloud base before evaporation z evap in meters and relative to atmospheric scale height H for a water raindrop of initialsize r = 0 . mm in different composition atmospheres. Planetary conditions besides drycomposition are given in Table 1 under “composition.” With the effect column, we con-sider the three main impacts of composition on t evap and z evap —atmospheric scale height H , raindrop terminal velocity v T , and transport rate of condensible gas molecules andheat away from the raindrop surface “transport”—together (“all”) as well as in isolation. –23–anuscript submitted to JGR: Planets p [ P a ] H N CO r =0.2 mm r =1.0 mm r =2.0 mm051015202530 z L C L z [ k m ] P evap [W]020040060080010001200 t [ s ] abc Figure 7: (a) Pressure p versus the rate of heat absorbed from local air P evap to evapo-rate the condensed water mass in a raindrop of size 2 mm distributed into raindrops ofinitial radii r (varying line styles) in different atmospheric compositions (varying colors)falling from T LCL = 275
K and p LCL = 7 . × Pa under Earth surface gravity. (b) Sameas (a) except falling distance from cloud base ( z LCL − z ) versus P evap . (c) Same as (a)except falling time t versus P evap . –24–anuscript submitted to JGR: Planets z L C L z [ m ] Earthearly MarsJupiterSaturnK2-18b r max r min ( z ) [mm]10 z L C L z [ H ] a b Figure 8: Distance from cloud base z LCL − z in units of (a) meters and (b) atmosphericscale heights versus the minimum initial radius raindrop to reach altitude z without to-tal evaporation ( r min ( z ) ). Different colored solid lines represent ostensible atmosphericconditions (given by Table 1) for different planets as labeled. All raindrops are composedof water. Thin dashed vertical lines indicate maximum stable raindrop radius r max asestimated via Rayleigh-Taylor instability with ‘ RT,max = 0 . πa , distinguished by planetwith labeled colors. r min ( z ) values are plotted until r min ( z ) = r max or z intersects with theplanet’s surface. –25–anuscript submitted to JGR: Planets
Next we move from considering raindrops in abstract conditions to studyingraindrops in known (or speculated) planetary conditions. In Figure 8, we comparewater raindrops on Earth; warm, wet ancient Mars; Jupiter; Saturn; and exoplanetK2-18b. Warm, wet ancient Mars is a hypothesized climate state 3-4 billion yearsago where Mars was warm compared to the melting point of water and rainfall wasfrequent (e.g., Wordsworth, 2016).K2-18b is an exoplanet without analog in the solar system, falling between thesizes of Earth and Neptune and receiving an Earth-like amount of stellar insolation(Foreman-Mackey et al., 2015; Montet et al., 2015; Cloutier et al., 2019; Bennekeet al., 2019). Though many of K2-18b’s characteristics are imprecisely known, weselected this exoplanet to demonstrate the flexibility of our model as multiple teamshave claimed observational detections of water vapor (Tsiaras et al., 2019; Benneke etal., 2019; Madhusudhan et al., 2020) and one team has hypothesized that observationssuggest the presence of liquid water clouds (Benneke et al., 2019).We set surface conditions on the terrestrial Earth and Mars and cloud baseconditions on the gaseous Jupiter, Saturn, and K2-18b as given in Table 1. We plotaltitude z from cloud base in units of (a) meters and (b) atmospheric scale heightsversus r min ( z ) , the minimum radius raindrop to reach z without totally evaporating.How r min ( z ) values cluster among planets varies depending on whether the distancefrom cloud base is measured in scale heights or meters, but for all z values testedin both unit systems we find r min ( z ) varies among the diverse planetary conditionsconsidered by only a factor of few. This agreement has interesting implications forfuture work aimed at rigorous generalization of raindrop microphysics schemes fromEarth to other planets. O While water is the most familiar liquid condensible, outside of the familiar Earthtemperature range, many other species condense as liquids and can form raindrops.Table 3 compiles a number of liquid condensible species that are cosmochemicallyabundant. Detailed analysis of raindrop behavior requires specifying atmospheric con-ditions in addition to condensible species, so here we consider how basic condensibleproperties vary raindrop behavior relative to water. Again in the interest of simplicity,we do not consider mixtures of condensibles predicted by thermodynamic equilibriumin many atmospheric gas combinations, e.g., N -CH or NH -H O (Thompson et al.,1992; Guillot, Stevenson, et al., 2020).We give the melting temperature T melt for each condensible to give an idea ofthe atmospheric temperatures where each species will be liquid. Lower temperaturecondensibles like CH and NH dominate the observable clouds of the outer solarsystem. The CH cycle of Titan is the only active “terrestrial” condensible cyclebesides Earth’s we can observe in detail.Metal and rock species like Fe and SiO become condensible species at very hightemperatures. Such species are predicted to be condensibles on highly irradiated exo-planets that are favored observational targets for the foreseeable future; Ehrenreich etal. (2020) have already claimed observational evidence of Fe condensing on WASP-76b.On Earth, such high temperatures can be reached during asteroid/meteoroid impacts.Geologically preserved impact spherules (e.g., Johnson & Melosh, 2012a, 2012b) andmicrometeorites (e.g., Tomkins et al., 2016; Payne et al., 2020) both undergo phases intheir life through the atmosphere where their behaviors are described by the raindropphysics we have presented. –26–anuscript submitted to JGR: Planets
Table 3:
Properties of Liquid Condensibles and Their Raindrops condensible T melt ρ ‘ σ r max L E evap ( r ) [K] [kg m − ] [N m − ] [ r max (H O)] [MJ kg − ] [ E evap ( r, H O) ]CH a a b a a a b c O 273 a a d c e e f g h i j h Note.
Temperature-dependent values use T = T melt . a Linstrom and Mallard (2014). b Somayajulu (1988). c Rumble et al. (2017). d Vargaftik et al. (1983). e Assael et al. (2006). f Brillo and Egry (2005). g Desai (1986). h Melosh (2007). i Bacon et al. (1960). j Kingery(1959).Despite the very wide range of condensible species, r max values in Table 3 onlyvary relative to water by a factor of 0.5 to 2. As we have examined in section 3.4, theonly planetary parameter beyond condensible type that affects r max is surface gravity g , which for planetary bodies varies about an order of magnitude (between about 1and 25 m s − ). We therefore find that maximum stable raindrop sizes are remarkablysimilar across a very wide range of planetary conditions and raindrop compositions. This work is merely a first step toward a generalized theory of how precipitationand condensible cycles operate in planetary conditions different from modern Earth.We have considered only single raindrops, independent of their formations. Futureprogress will require development of theory for general planetary atmospheres on thegrowth of raindrops from cloud drops and extensions that include solid precipitatingparticles and their growth. In this context, below we discuss some future applicationsand extensions of this work.
Precipitation efficiency measures how efficiently an atmosphere transports con-densed mass from a cloud downward. Qualitatively, its distribution over time is animportant metric for planetary climate as it shapes cloud coverage (both temporallyand spatially), cloud radiative properties, and relative humidity profiles, which all havelarge consequences for radiative balance (Romps, 2014; Zhao et al., 2016; Lutsko &Cronin, 2018). On a terrestrial planet, precipitation efficiency evaluated at the surfaceis a particularly important quantity as it helps to set the amount of condensible massin the atmosphere.On short timescales, liquid precipitation efficiency is governed by the basic rain-drop physics of falling and evaporation we have presented here. We have demonstratedthat the dimensionless number Λ captures the essential behavior of raindrop evapora-tion and descent. For Λ evaluated with the length-scale from cloud base to the surface,raindrops of initial sizes with Λ > (i.e., r < r min ) will totally evaporate and have –27–anuscript submitted to JGR: Planets no condensible mass reach the surface; raindrops of initial sizes with Λ < . willexperience little evaporation and have the majority of their mass reach the surface;and raindrops of initial sizes with . < Λ ≤ will have both condensible mass evap-orate and reach the surface. Short-scale precipitation efficiency is then fundamentallycontrolled by the cloud-edge condensed mass distribution among these three differentsize categories.How precipitation efficiency on short timescales—governed directly by microphysics—maps to the climatically important temporal distribution of precipitation efficiency isfundamentally influenced by both large-scale atmospheric dynamics and local-scaleconvection (Romps, 2014). Predicting precipitation efficiency distributions in three-dimensional models requires capturing key microphysical behaviors in a parameterizedrepresentation. Our analysis suggests the key microphysical behavior we must captureto physically represent precipitation efficiency on short timescales is the sorting ofcondensed cloud mass into the three different size categories determined by Λ . Thisinterpretation suggests a physical grounding for microphysics’ role in controlling an im-portant yet poorly understood climate parameter (Zhao et al., 2016; Lutsko & Cronin,2018). Tying precipitation efficiency to the raindrop size distribution may thereforeprovide a framework for future improvements in generalized microphysics parameteri-zations.Given this analysis, the bounds on water raindrop sizes in Figure 6 show onecomponent of how precipitation efficiency will be shaped by different planetary condi-tions. A complete picture of precipitation efficiency obviously will also require betterunderstanding of how planetary parameters influence the formation and subsequentmass distribution of raindrops. Nevertheless, the shrinking viable surface-reachingraindrop size range with rising T surf in Figure 6(c) is a striking predicted feature ofwarmer water cycles, which may have implications for a CO -rich early Earth, for earlyVenus, or for exoplanets close to the runaway greenhouse threshold. Evaporating raindrops also influence convective storm dynamics. The verticaltransport of heat and condensible mass from evaporating raindrops causes local varia-tions in air density through changes in both average molar mass and temperature. Theimplications of evaporating raindrops for convection depend on the ratio between anatmosphere’s dry mean molecular mass µ dry and the molecular mass of its condensingspecies µ c as well as the removal rate of latent heat relative to the local T − p profile(Guillot, 1995; Leconte et al., 2017).On modern-Earth, µ c /µ dry ≈ . , so molar-mass-contrast effects are present butmuted. Nonetheless, the interplay of rising air supplying condensible mass and sinkingprecipitating air plays a key role in the evolution and lifetime of a given storm (e.g.,Rogers & Yau, 1996). Variations in µ c /µ dry have been hypothesized to drive stormsystems completely unlike those on Earth—e.g., Saturn’s giant white storms (Li &Ingersoll, 2015; Leconte et al., 2017).As suggested by Table 2 and Figure 7, an atmosphere’s background dry gas alsoinfluences the ability of raindrops to vertically re-distribute latent heat and condensiblemass with respect to z and log p . This effect of background gas composition will alsolikely influence air density changes during storms. Future dynamical studies mightexplore the implications of these variations in raindrop evaporation with backgroundgas for storm evolution and subsequent condensible gas distributions. One examplesolar system application of interest is better constraining how deep and how effectivelyammonia raindrops (originating from melted NH snowflakes or hail-like NH – H O“mushballs”) can transport ammonia on Jupiter (Ingersoll et al., 2017; Li & Chen,2019; Li et al., 2020; Guillot, Stevenson, et al., 2020; Guillot, Li, et al., 2020). –28–anuscript submitted to
JGR: Planets
On terrestrial planets, rainfall rate—the mass flux of liquid condensible hittingthe surface—is a key characteristic for predicting surface erosion and flooding events(e.g., Kavanagh & Goldblatt, 2015; Craddock & Lorenz, 2017; Margulis, 2017). Rain-fall rate depends on liquid condensible mass per unit air, the raindrop size distribution,and raindrop velocities as a function of size (Rogers & Yau, 1996). While we cannotyet make robust predictions for how rainfall rates should vary in different planetaryconditions, we expect such a relationship will be sensitive to g and air density at cloudlevel because of the role of collisional kinetic energy in shaping raindrop size distribu-tions (Low & List, 1982; Rogers & Yau, 1996; Pinsky et al., 2001; List et al., 2009). r min and Λ as a function of r eq evaluated at the surface are useful for contextualizingthe size range of the in-cloud raindrop size distribution that is important for predictingrainfall rates. Future work should investigate how narrow bounds of viable surface-reaching raindrops might constrain rainfall rates across different planetary conditionsin more detail. In this work, we have focused on falling liquid condensible particles because solidparticle shapes are much more complicated and variable than liquid oblate spheroids.The morphology of solid condensed particles exhibits extreme variability because ofthe high sensitivity of crystal orientation to temperature and condensible vapor super-saturations (e.g., Libbrecht, 2017). Further, crystal structure is fixed at deposition, sobecause a solid condensed particle experiences variations in environmental conditionsduring its growth, its final shape is highly sensitive to its growth path. These consid-erations mean generalizing modern-Earth modeling approaches for handling ice shapedegeneracies (e.g., Krueger et al., 1995) is a non-trivial exercise.Understanding of shape is the main limiting factor for applying the methodologypresented here for raindrops to solid condensible particles. Shape is a key parameterin calculating C D and f V , which impact terminal velocity and evaporation rate, re-spectively. Shape also needs to be accounted for more fundamentally in the derivationof evaporation rate because it controls boundary conditions, but analogous mathe-matics to well-investigated electrostatics means such boundary condition accountinghas already been compiled (e.g., McDonald, 1963; Lamb & Verlinde, 2011, chapter 8).Extension to solid particles would also need to account for differences in saturationpressure with respect to solid and liquid condensibles and a larger assortment of la-tent heats because of more available phase changes, but these issues are a question ofcompiling thermodynamic data rather than an inherent lack of understanding. We have compiled and generalized methods for calculating raindrop shape, ter-minal velocity, and evaporation rate in any planetary atmosphere. These propertiesgovern raindrop behavior below a cloud, a simple but necessary component of under-standing how condensible cycles operate across a wide range of planetary parametersbeyond modern-Earth conditions. For terrestrial planets, raindrops sizes capable oftransporting condensed mass to the surface only span about an order of magnitude,a narrow bound when compared to origins in cloud condensation nuclei many ordersof magnitude smaller. We show across a wide range of condensibles and planetaryparameters that maximum raindrop sizes do not significantly vary.With more in depth calculations, we confirm the conclusion of Palumbo et al.(2020) that maximum raindrop size is only weakly dependent on air density, in contrastto the results of an earlier study (Craddock & Lorenz, 2017). By returning to the –29–anuscript submitted to
JGR: Planets physics equations governing raindrop falling and evaporation, we demonstrate raindropability to vertically transport latent heat and condensible mass can be well capturedby a new dimensionless number. Our analysis suggests cloud-edge, mass-weightedraindrop size distribution is a key microphysics-based control on the important climateparameter of precipitation efficiency.
Acknowledgments
Our model and all code used to generate results for this paper are available atan archived Github repository (Loftus, 2021, https://github.com/kaitlyn-loftus/rainprops ). This work was supported by NSF grant AST-1847120. K.L. thanksMiklós Szakáll for providing data on raindrop shape experiments as well as MarkBaum, Ralph Lorenz, and Jacob Seeley for helpful discussions pertaining to numericalmethods, Martian raindrops, and climatic impacts of microphysics parameterizations,respectively. We thank Tristan Guillot and Ralph Lorenz for insightful reviews.
Notation A [m ] cross sectional area a [m] semi-major axis of oblate spheroid b [m] semi-minor axis of oblate spheroid C D [ ] drag coefficient C shape [ ] oblate spheroid shape correction in drag coefficient c p [J K − kg − ] specific heat capacity at constant pressure D c-air [m s − ] diffusion coefficient of condensible gas in air E [J] energy F x [N] force specified by subscript f SA [ ] ratio of the surface area of an oblate spheroid to a sphere of equivalent radius f V [ ] ventilation coefficient, subscript can further specify for molecular or heat trans-port g [m s − ] planetary gravity H [m] atmospheric scale height K air [W m − K − ] thermal conductivity of air L [J kg − ] latent heat of vaporization ‘ [m] length scale m [kg] mass of raindrop P [W] power p [Pa] air pressure p sat [Pa] saturation pressure of condensible gas Pr [ ] Prandtl number R [J mol − K − ] ideal gas constant Re [ ] Reynolds number RH [ ] relative humidity r [m] initial raindrop radius at cloud base r eq [m] equivalent radius r max [m] maximum raindrop radius before breakup r min [m] minimum threshold raindrop radius to reach without total evaporation agiven distance from cloud base; if no length scale is specified, assumed to bedistance to surface Sc [ ] Schmidt number T [K] temperature t [s] time –30–anuscript submitted to JGR: Planets v T [m s − ] terminal velocity w [m s − ] vertical wind speed z [m] vertical space coordinate β [ ] exponential dependence of raindrop velocity on raindrop radius ∆ r [m] threshold radius size for drop to be considered “fully evaporated” ∆ T drop [K] steady-state temperature difference between evaporating raindrop andlocal air η [Pa s] dynamic viscosity Λ [ ] dimensionless number describing raindrop evaporative mass loss λ [m] wavelength µ [kg mol − ] molar mass σ [N m − ] surface tension ρ [kg m − ] density X air describes air property at local altitude X c describes a condensible property X drop describes a raindrop property X ‘ describes a liquid condensible property where ambiguous if “c” references liquidor gas condensible X evap describes property when raindrop finishes evaporating X LCL describes air property at lifting condensation level X surf describes air property at surface Appendix A Oblate Spheroid Geometrical Relationships
Here, we give geometric properties of our assumed oblate-spheroid raindrop withsemi-major axis a , semi-minor axis b , and equivalent radius r eq . Note that a sphere isan oblate spheroid of b/a = 1 .Raindrop semi-major axis a can be determined numerically for a given r eq and b/a from the relationship a = r eq (cid:18) ba (cid:19) − / (A1)(Green, 1975). An oblate spheroid has cross sectional area A of A = πa = πr eq (cid:18) ba (cid:19) − (A2)(Green, 1975) and a volume V of V = 43 πa b = 43 πr eq , (A3)by definition of r eq . The ratio of the surface area of an oblate spheroid to that of asphere f SA is f SA = ( . (cid:0) ba (cid:1) − / + (cid:0) ba (cid:1) / (4 (cid:15) ) − ln h (cid:15) − (cid:15) i , b/a < , b/a = 1 (A4)where (cid:15) = p − ( b/a ) (Loth, 2008). Appendix B Raindrop Shape
The shape of a falling raindrop in air (i.e., the surface of the condensible-airboundary) can be described by the Young-Laplace equation, which governs the surface –31–anuscript submitted to
JGR: Planets boundary between two immiscible (non-mixing) fluids. For a given point on a raindropsurface, the Young-Laplace equation can be written as σ c-air (cid:0) R − + R − (cid:1) = ∆ p (B1)(see Pruppacher & Klett, 2010, chapter 10.3.2 for derivation). σ c-air is surface tensionbetween liquid condensible and air; R and R are the principle radii of curvature,which together describe the shape of the local surface via its curvature (De Gennes etal., 2013); and ∆ p is the difference between internal and external pressures on eitherside of the raindrop’s surface boundary.Because we have pre-assumed an oblate spheroid geometry, we only have toevaluate equation (B1) at one point on the raindrop’s surface rather than integratingover the entire surface, and the principle radii are analytic expressions rather thanvalues which must be iteratively corrected for such integration to be self consistent(see Beard & Chuang, 1987, for shape calculation without oblate spheroid assumption).We follow Green (1975) and solve equation (B1) at a point on the raindrop’s equator.From geometry, for all points on an oblate spheroid’s equator, R = b a − , (B2)and R = a (B3)(see Green, 1975, Appendix B for derivation).In a full accounting of pressures, internal pressure p int should include hydrostaticpressure, pressure from spherical surface tension, and pressure from internal circula-tion within the raindrop. External pressure p ext should include hydrostatic pressure,pressure from aerodynamic drag, and pressure from air turbulence. Following Green(1975), we only consider internal and external hydrostatic pressures and pressure fromspherical surface tension, giving ∆ p = p int − p ext ≈ ( ρ c − ρ air ) gb + 2 σ c-air r − eq (B4)Pressure from drag is fundamental to shaping the raindrop’s shape at largerraindrop sizes (i.e., where the magnitude of F drag begins to approach the magnitudeof F σ ); drag is responsible for the evolution in raindrop shape from oblate spheroid toupper hamburger bun. Drag acts to deform the raindrop so that it is no longer axiallysymmetric about the major axis because drag is not uniformly distributed over theraindrop’s surface.Our assumption of oblate spheroid shape is incompatible with a detailed con-sideration of the effects of drag on raindrop shape. We fully neglect drag becausewe consider the pressure balance at raindrop equator where drag pressure is minimaleven when the total magnitude of the drag force on the raindrop is significant (Green,1975). The consequences of this neglect can be seen to be minimal from comparisonof Green (1975)’s oblate spheroid method to computed or observed raindrops shapeswhere drag is included (Beard & Chuang, 1987; Thurai et al., 2009).We neglect the effects of atmospheric turbulence on raindrop shape as we arecalculating an equilibrium shape; in practice, turbulence acts to induce oscillationsin raindrop shape (which are then viscously damped) rather than changing the equi-librium shape (Beard et al., 2010). We also neglect the effects of internal circulationwithin the raindrop as empirically its neglect has no impact on predicting Earth rain-drop shapes (Thurai et al., 2009) and theoretically internal circulation is expected tobe small for a higher dynamic viscosity liquid raindrop falling through a lower dynamicviscosity air (Clift et al., 2005). –32–anuscript submitted to JGR: Planets
Appendix C Maximum Raindrop Size before Breakup
C1 Rayleigh-Taylor instability
Raindrop breakup can be studied by a linear instability analysis that incorporatescapillary and gravity waves. For wavelengths above a critical wavelength λ ∗ , total wavephase velocity on the base of the drop becomes imaginary; waves rapidly amplify inmagnitude; and the raindrop becomes unstable. Assuming planar surface waves (the3D corrections for drops near the size of r max are small (Dhir & Lienhard, 1973; Graceet al., 1978)), the critical wavelength can be written as λ ∗ = 2 π r σ c-air g ( ρ c ,‘ − ρ air ) (C1)(Komabayasi et al., 1964; Grace et al., 1978).Physically, for a raindrop to not be disrupted by a wave, the zero mode of wave-length λ ∗ / must be less than the characteristic size of the drop: ‘ RT,max = λ ∗ π r σ c-air g ( ρ c ,‘ − ρ air ) . (C2)Different metrics exist for the maximum raindrop length scale relative to the zero modewave ‘ RT,max —e.g., Grace et al. (1978) uses half the equivalent radius’ circumference, . πr eq , while Pruppacher and Klett (2010) uses the maximum physical raindrop diam-eter, a . Different choices of length scale will result in quantitatively different resultsfor the onset of instability by a factor of a few; this discrepancy emphasizes the role ofthis analysis as an estimate of when drops tend to become unstable. Regardless, anychoice of length scale results in the same dependencies on physical parameters—ourprimary concern here. Length scales using semi-major axis a can be related to r eq viaoblate spheroid geometry.While no experimental pressure chamber data exists to explicitly test air density’seffect on maximum raindrop size, numerous chemical engineering experiments havebeen done on the maximum-sized drops of different media falling through various othermedia. Such experiments consistently agree (within about 20%) with the predictionsof equation (21) where “c” is replaced with the drop medium, “air” is replaced with thefall medium, and the dynamic viscosity of the drop medium is much larger than thefall medium’s (like raindrops and air) (Lehrer, 1975; Grace et al., 1978; Clift et al.,2005). More complex wave analysis does not consistently produce better agreementwith experiments (Grace et al., 1978). C2 Surface Tension-Drag Force Balance
The force on a raindrop due to surface tension is F σ = ‘ σ σ c-air (C3)where ‘ σ is a characteristic length scale of the raindrop’s surface, which is convention-ally taken to be πa or πr eq . Again, there is ambiguity in length scales because thissetup is an estimate rather than a rigorous calculation. Drag force is given by equation(4). We note the maximum velocity relative to air of a raindrop falling in isolationis its terminal velocity (where F drag = F g ), and raindrop velocity is a monotonicallyincreasing function with raindrop size. Thus, at the smallest r eq where F drag = F σ , thesetup is analogous to solving for the raindrop size where the force of surface tensionequals the gravitational force, given by equation (3). –33–anuscript submitted to JGR: Planets
Appendix D Analysis with Λ of the Dependence of Raindrop Evap-oration on Size We can understand the behavior of raindrop evaporation with changing rain-drop size by considering the dependence of Λ on r eq . From equation (20), we canapproximately represent Λ explicitly in terms of r eq as Λ ≈ Cr − (2+ β ) eq . (D1) C is a proportionality constant dependent on the environment across ‘ and the speciesof condensible; β is defined such that d z /d t ∝ r β eq . (We have neglected the dependenceof ventilation factors on r eq to get a tractable expression here.)For simplicity, we consider the limiting case where w = 0 , so d z /d t = v T . Forvery small raindrops (Re (cid:28) β = 2 . Forvery large raindrops (as r eq approaches r max ), raindrop terminal velocity approaches aconstant value (Clift et al., 2005, chapter 7.C), and β approaches 0. From the behaviorof C D , β smoothly varies so that β ∈ [0 , (e.g., Lohmann et al., 2016, chapter 7.2.3).Therefore, as r eq decreases, Λ always exponentially increases, with the dependence on r eq moving from Λ ∝ r − eq to Λ ∝ r − eq . The exponential dependence of Λ on r eq meansthat the transition from minimal evaporation to full evaporation occurs over a narrowraindrop size range.One perspective on the width of this transition regime—applicable for raindropsizes varying by orders of magnitude—is to consider the ratio of the r eq such that Λ = 0 . to the r eq such that Λ = 1 (i.e., the ratio of sizes between the drop thatevaporates 10% of its initial mass and the drop that fully evaporates just as it reachesthe end of the prescribed length scale). From equation (D1) and assuming β is constantbetween Λ = 1 and
Λ = 0 . , this ratio is equal to / (2+ β ) , which evaluates to 1.78and 3.16 for β = 2 and β = 0 , respectively. Because β decreases with increasing r eq ,as the r eq that evaluates to Λ = 1 increases (e.g., by increasing ‘ ), the transition widthincreases. Still, regardless of the exact value of β within [0,2], a change in raindropsize of a factor of 2-3 is small compared raindrop growth processes that require manyorder of magnitude changes in drop size.Similar analysis featuring Λ can be used to introduce the additional complexityof non-zero vertical wind speed or to isolate other variables of interest beyond r eq andconsider their effects on raindrop evaporation. References
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Supporting Information for “The Physics of Falling Raindropsin Diverse Planetary Atmospheres”
DOI: 10.1002/TBD =K. Loftus and R.D. Wordsworth , = Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA, US School of Engineering and Applied Sciences, Harvard, University, Cambridge, MA, US
Contents of this file
1. Text S12. Figures S1 to S83. Table S1
Introduction
Figures S1-S2 support that raindrop velocityrelative to air can be well approximated by raindrop termi-nal velocity across broad planetary conditions. Figures S3-S4 show validation tests for our model against observationsand empirical relationships for Earth raindrops. Figures S5-S8 and Table S1 compare our model to previous theoreticalwork focused on Titan raindrops with further descriptiongiven in Text S1.
Text S1.
To compare our methods to previous theoreti-cal work on Titan raindrops by Lorenz (1993) and Graves,McKay, Griffith, Ferri, and Fulchignoni (2008), we presentresults (1) as reported by the previous authors, (2) fromequivalent calculations done by attempting to reproducetheir methods within the basic framework of our model (la-beled as “reproduction”), and (3) from equivalent calcula-tions done using the methodology we describe in this work.We note that the scatter points labeled Graves et al. (2008)in Figures S7 and S8 were digitized by us from their figuresand are not directly reported values. (We know there is atleast one inconsistency in what is plotted in Figures S7 asour digitized starting radius is r = 4 .
80 mm rather thanthe reported r = 4 .
75 mm.)Where possible, we follow condensible and atmosphereproperties as described or referenced in each text for themost direct comparisons; we review the most notable oneshere. Under Titan’s atmospheric conditions, N gas is ex-pected to dissolve into liquid CH at significant molar con-centrations (e.g., Thompson et al., 1992). Lorenz (1993)accounts for this mixture by prescribing a liquid raindropdensity between that expected for pure CH and pure N .Graves et al. (2008) accounts for this mixture more rigor-ously: accounting for the CH -N mixture thermodynamicsoutlined by Thompson et al. (1992) and evaporation of bothCH and N using Raoult’s law. Graves et al. (2008) usespressure, altitude, temperature, and composition data fromCassini’s Huygens probe (Fulchignoni et al., 2005; Niemannet al., 2005). We linearly interpolate this data to be in termsof altitude to use in our numerical integration. References
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Comparison to Table 1 of Graves et al. (2008) a Graves et al. (2008) Graves et al. (2008) this workreproduction r ( z = 0) [mm] 3.34 3.42 2.93 t fall [min] 78 77 80 v ( z = 0) [m s − ] 1.5 1.5 1.5 f CH ( z = 0) [mol mol − ] 0.77 0.77 0.77 f N ( z = 0) [mol mol − ] 0.23 0.23 0.23 T drop ( z = 0) [K] 90.0 90.0 89.9 a Properties of a CH -N raindrop of initial size r = 4 .
75 mm at 8 km after falling to Titan’s surface ( z = 0 km). t fall is the totalfall time, and f X is the liquid molar concentration of species X in the raindrop. In the first column, we give the values reported byGraves et al. (2008). In the second column, we show the results our own attempt to reproduce the methods of Graves et al. (2008). Inthe third column, we show the results of our methods for the same calculation. Figures S7 and S8 show altitude-dependent propertiesof the same raindrop.Copyright 2021 by the American Geophysical Union.0148-0227/21/$9.00 OFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS
X - 3 r [mm]10 m e d i a n r e l a t i v e e rr o r i n v [ ] H N CO p LCL T LCL g r [mm] m a x i m u m r e l a t i v e e rr o r i n v [ ] a b Figure S2. (a) Median relative error in raindrop veloc-ity v from assuming terminal velocity is instantly reachedduring raindrop evaporation (relative to self-consistenttreatment of raindrop acceleration) versus equivalent ra-dius at cloud base r across a broad planetary parame-ter space. (b) Same as (a) except for maximum relativeerror in v . For H , N , and CO background composi-tion atmospheres (marker shape as labeled), we vary inturn T LCL , p LCL , and g (color as labeled) over rangesgiven in Table 1 under “broad” from baseline conditionsof T LCL = 280 K, p LCL = 10 Pa, and Earth surfacegravity. We test 90 different atmospheric conditions withH O raindrops in all cases. Raindrop acceleration is inte-grated from drag and gravitational forces following New-ton’s second law. - 4
LOFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS r eq [mm]0.00.20.40.60.81.0 b / a [ ] our modelPruppacher & Beard (1970)Pruppacher & Pitt (1971)Beard & Chuang (1987)Beard et al. (1991)Thurai et al. (2009) Figure S3.
Raindrop axis ratio b/a versus equivalentradius r eq for H O raindrops and Earth surface gravity.The dark purple line shows our theoretical calculationfrom equation (2) (following Green, 1975). Dark graypoints show results from more comprehensive theoreticalmodels (Pruppacher & Pitter, 1971; Beard & Chuang,1987). Light purple points show experimental results(Beard et al., 1991; Thurai et al., 2009). The dashedlight-gray line shows an empirical fit to additional exper-imental results (Pruppacher & Beard, 1970).
OFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS
X - 5 r eq [mm]02468 v T [ m s ] this workBeard (1976)Gunn & Kinzer (1949)Best (1950) Figure S4.
Raindrop terminal velocity v T versus equiv-alent radius r eq for H O raindrops. Atmospheric param-eters are set to mimic Earth experimental conditions: T = 293 .
15 K, p = 1 . × Pa, RH=0.5, Earthsurface gravity, and dry air composition of 20% O and80% N . The dark purple line shows our theoreticalcalculation from equation (8). The light-purple scat-ter points represent the experimental data of Gunn andKinzer (1949) and Best (1950). The dashed light-purpleline shows our evaluation of the empirical relationship ofBeard (1976). - 6 LOFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS r eq [mm]0.00.20.40.60.81.0 b / a [ ] this workLorenz (1993) reproduction Figure S5.
Raindrop axis ratio b/a versus equivalentradius r eq for CH -N raindrops and Titan surface grav-ity. The dark-purple line shows our theoretical calcu-lation from equation (2) (following Green, 1975). Thelight-purple line shows our reproduction of the theoreticalmethod of Lorenz (1993). (We do not give scatter pointsof values directly from Lorenz (1993) as b/a is calculatedfrom a simple two-component piecewise function.) OFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS
X - 7 r eq [mm]02468 v T [ m s ] this workLorenz (1993) reproductionEarth (H O), 0 kmTitan (CH -N ), 10 kmTitan (CH -N ), 0 kmLorenz (1993) Figure S6.
Raindrop terminal velocity v T versus equiv-alent radius r eq for Earth and Titan conditions (linestyles as labeled). The dark purple line shows our the-oretical calculation from equation (8). The light-purpleline shows our reproduction of the theoretical method ofLorenz (1993). The values of light-purple scatter pointsare taken from Table 1 of Lorenz (1993). - 8 LOFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS r eq [mm]010002000300040005000600070008000 z [ m ] this workGraves et al. (2008)reproductionGraves et al. (2008) Figure S7.
Altitude z versus equivalent radius r eq forTitan conditions for a CH -N raindrop of initial radius r =4.75 mm at 8 km. The dark-purple line is calculatedfollowing the methods outlined in sections 2-3. The light-purple line is calculated via our reproduction of the meth-ods of Graves et al. (2008). The values of light-purplescatter points are taken from Figure 2(b) of Graves et al.(2008). From sensitivity tests, the majority of the differ-ence in the calculated r ( z ) between our method and thatof Graves et al. (2008) is due to our choice to not assumethe ventilation factor for heat transport is equal to theventilation factor for molecular transport. Thus the dif-ference is not a fundamental disagreement of models butrather the result of uncertain parameters quantifiable byfluid dynamics experiments. OFTUS & WORDSWORTH: PHYSICS OF FALLING RAINDROPS
X - 9
86 87 88 89 90 91 92 93 T [K]010002000300040005000600070008000 z [ m ] T air T drop , this work T drop , Graves et al. (2008) reproduction T drop , Graves et al. (2008) Figure S8.
Altitude z versus raindrop temperature T drop (purple lines) and air temperature T air (gray line)for Titan conditions for a CH -N raindrop of initial ra-dius r =4.75 mm at 8 km. The dark-purple line is calcu-lated following the methods outlined in sections 2-3. Thedashed light-purple line is calculated via our reproduc-tion of the methods of Graves et al. (2008). The valuesof light-purple scatter points are taken from Figure 4 ofGraves et al. (2008). The gray line shows T airair