Fluid dynamics in clouds: The sum of its parts
S. Ravichandran, Jason R. Picardo, Samriddhi Sankar Ray, Rama Govindarajan
FFluid dynamics in clouds: The sum of its parts
S. Ravichandran, ∗ Jason R. Picardo, † Samriddhi Sankar Ray, ‡ and Rama Govindarajan § Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
This paper is aimed at describing cloud physics with an emphasis on fluid dynamics. As isinevitable for a review of an enormously complicated problem, it is highly selective and reflects ofthe authors’ focus. The range of scales involved, and the relevant physics at each scale is described.Particular attention is given to droplet dynamics and growth, and turbulence with and withoutthermodynamics.
I. GLOSSARY
Aerosol: tiny ( ∼ . − ∼
99% by weight), watervapour (1%), liquid water droplets (0.1%), aerosol parti-cles, trace gases. Clouds are usually in turbulent flow.Caustics: regions of the flow where particles with dif-ferent velocities arrive simultaneously at the same loca-tion.Supersaturation: a system that has more water vapourthan the saturation value prescribed by the Clausius-Clapeyron equation 5.Ventilation effects: the effects of oncoming flow on thegrowth of droplets. Used in the context of water dropletsgrowing by condensation.
II. WHY STUDY CLOUDS
Clouds, since they involve many different phenomenainteracting with each other in complex ways, are of in-terest purely from scientific curiosity. For instance, isit possible to predict what cloud shapes will result forgiven atmospheric conditions? More importantly, per-haps, clouds are also immensely influential in the energyand mass balances in the planet’s atmosphere. In fact,clouds are the last great sources of uncertainty in climatescience.Clouds increase the planet’s albedo, reflecting awaysunlight before it can make it to the surface; they also actto provide a greenhouse effect, trapping energy radiatedaway from the surface. These two opposing effects areboth of much larger magnitudes than any other sourcesin the radiative balance of the planet ([4] chapter 6, [99].The response of clouds to a warming planet—whetherclouds will act to slow down or to accelerate the planet’swarming—is not clear at present (although very recent ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] studies ([91]) suggest that they do indeed act as positivefeedback). This uncertainty is due to the large magni-tudes of the aforementioned effects, and the fact thatclouds are coupled with the global circulation.The selective annual Northward propagation of thecloud-band known as the ITCZ (Inter-Tropical Conver-gence Zone) over longitudes including those of the In-dian landmass, brings the Indian monsoon, among thebiggest weather events, which provides fresh water forclose to two billion people. Understanding the dynam-ics of the ITCZ requires understanding the dynamics ofclouds, which is as yet an open problem ([20]). Theseare only a few of the most compelling reasons to studyclouds.With the advent of machine learning and associatedstatistical and data-driven techniques, and the increasingavailability of dedicated computing power, it is temptingto rely solely on such statistical methods. However,understanding the dynamics is useful not just as ascientific exercise but also pragmatically. Statisticaltechniques—machine learning in particular—are bestused in scenarios for which they have been ‘trained’.Most estimates suggest that the feedback from cloudson the climate is likely to affect the circulation in theatmosphere substantially. The resulting large changesin the dynamics may not be possible to capture withmachine learning techniques. The planet needs us tostudy the dynamics of clouds! ([90].) III. DEFINITION OF THE SUBJECT
The fluid dynamics in clouds covers length scales fromtenths of microns to hundreds of kilometres. Being anonlinear problem, the physics at each scale has an ef-fect on other scales. There are open questions which re-quire an understanding of the basic physics at each scale,and also in the connections between scales. The lowerend of this range concerns the chemistry and chemicalphysics of aerosols. Aerosols are crucial to cloud forma-tion, because they act as nuclei for droplet formation,as will be discussed below. Aerosols are introduced intothe atmosphere in a variety of ways, natural and anthro-pogenic. The production of aerosols, especially sea-salt a r X i v : . [ phy s i c s . f l u - dyn ] F e b aerosol by the mechanics of wave-breaking at the sur-face of the ocean, is an outstanding problem of fluid me-chanics. The upper end of the range of length scalescovers the dynamics of weather and the climate. Signif-icant progress has been made in recent years, aided byadvances in supercomputing, in the ability to make rea-sonable predictions of the dynamics on these scales. Therobustness of these predictions depends, however, on un-derstanding the global dynamics at the ‘sub-grid scales’.The intermediate range, incorporating the interactionsof buoyancy-driven fluid turbulence of (dilute) suspen-sions, is not only exceedingly complex, but also controlsthe dynamics of processes at the largest scales related toweather and climate. We will describe recent progress inunderstanding the dynamics in the intermediate lengthscales.Studies of the cloud dynamics can be based on ob-servations of clouds either in the real world or in thelaboratory, or on analyses or numerical solutions of thefluid dynamical equations of motion. Our work is in thelatter, and we will for the most part restrict ourselvesto discussing theoretical/ numerical studies of clouds, al-though we do make note of some relevant experimentalstudies.An important ingredient in the intermediate scales isthat clouds are usually in turbulent flow. Turbulenceconsists of vortices and regions of shear whose lengthscales span a large range, starting from the biggest scalein a single cloud, of the order of a few kilometres, towhat is known as the Kolmogorov scale η , which is ofthe order of a millimetre in a cloud. A turbulent flow ischaracterised, among other properties, by its Reynoldsnumber, which is a ratio of inertial and viscous forces.Consider a cloud of length scale L ∼ U are of the order of 10m/sec.The kinematic viscosity ν of air is about 10 − m / sec.The Reynolds number is LU/ν ∼ .The range of length scales involved even within theintermediate range in clouds is vast. Accurate direct nu-merical simulations (DNSs), solving the Navier-Stokesequations or their variants, have to resolve the Kol-mogorov scales of turbulence. If these scales have to beresolved in simulations of a cloud of the length scale of100m, each dimension has to be resolved with O (10 )grid points. Such numerical simulations are impossi-ble with today’s computing resources. DNSs of cloudflows are typically conducted only within small boxes,of a few metres in length [57–59]. In other words asmall volume within a cloud is all we can simulate.In effect experiments (on a computer or in the labora-tory) can be performed at Reynolds numbers that aremuch smaller than those found in clouds, in the hopethat this will nevertheless provide useful answers [1, 26–28, 66, 70, 71, 80, 93, 108] at cloud Reynolds numbers(see also section V B 1). Workarounds for this limitationtake the form of large eddy simulations (LESs) which re-solve only the large scales of motion (i.e. they are ‘cloud-resolving’) [39, 51, 77, 84] and use models to account for the smaller scales including the microphysics of phase-change.The radius a of a typical water droplet in a cloudranges from about a micron to a few millimetres. Ob-viously even simulations that resolve the Kolmogorovscales of the flow cannot resolve the scales associatedwith the motion of the water droplets in clouds. Eventhe simplest approach to tracking droplets adds signifi-cantly to the burden of computations, given that thereare O (1000) small droplets per cubic centimetre of cloud.In the simplest approach, the finite-sized water dropletshave to be treated as point particles and tracked in aLagrangian sense. Alternatively, these particles can becoarse-grained into a field. The relative efficacies of thesetwo approaches to particle-dynamics are studied in [64].The effects of finite droplet size and how this changestheir dynamics is discussed at length in section IV. Theseare known as “one-way coupled” approaches, where thefluid equations are solved for without taking into ac-count the fact that fluid is carrying particles and droplets,whereas the dynamics of the particles and droplets aredictated by the fluid motion. For a dilute suspensionof small particles and droplets this is a fair approach.However, larger raindrops can affect the flow and can af-fect the dynamics of each other, and a perfect treatmentwould have to account for the forces of these objects onthe fluid and on each other (two-way or four-way cou-pling). This can make computational costs forbidding.The thermodynamics taking place within a cloud hasan important effect on the dynamics. Phase change re-sults in heat release, which results in buoyancy. Thepotential energy thus gained is converted into the ki-netic energy of turbulence. Thus turbulence in a cloud isfundamentally different from mechanically forced turbu-lence, and turbulence from heat supplied at the bound-aries, which are studied most often. We return to thispoint in section V A. The droplet-growth bottleneck is awell-known open problem. Droplets can grow quickly toabout 10 microns in size in a supersaturated environmenttypical of clouds. Once they are about 50 microns in di-ameter, gravity can aid in the process of droplet growthby enhancing collision probability, with some fraction ofall collisions resulting in coalescence. How droplets growfrom about 10 to about 50 microns is not completely un-derstood yet, and this is known as the droplet-growthbottleneck. Turbulence is widely accepted now to be abig part of the answer, and this is discussed below.Most present-day studies assume that water dropletsare spherical in shape whereas larger drops are sensitiveto gravity and can distort in shape from the spherical,even to the point where they adopt shapes which are lo-cally sheet-like, which then causes breakup into smallerdroplets. Ice crystals are most often not spherical. Thus,the roles of shape, surface tension, gravity and even sur-face chemistry on the dynamics have to be studied. Theseeffects are also important in the thermodynamics of wa-ter droplet growth (discussed in section V A) and in thecollisions-coalescence of water droplets (section IV). FIG. 1. Cloud dynamics as the sum of its components.
Another important attribute is that over the length-scales that clouds occupy in the atmosphere, air is acompressible fluid. Note that the Earth’s atmosphereat a height of ten kilometres is only a tenth as denseas it is on the ground. So a parcel of air, as it rises,will undergo significant expansion, which cannot beneglected by assuming incompressibility. The equationsof compressible fluid motion are significantly morecomplicated than those of an incompressible fluid (whichitself is a “Millennium problem”). The fully compressibleequations of motion for air contain sound waves whichoperate on very short timescales. These sound wavesare unimportant for the dynamics of interest to usandcomputing them would require short timesteps andgreatly increase the computational requirements. For-tunately, since the Mach numbers
M a associated withthe flow are small, the thermodynamic pressure of theambient can be decoupled from the pressure fluctuationsdue to the motion (which are O ( M a ) relative to thethermodynamic means). This allows the use of theanelastic equations for flows over large heights or theincompressible equations for shallow flows, the latterof which is the limit we are concerned with. A sketchof the derivation of the anelastic and incompressibleequations from the compressible equations may be foundin [5, 34]. As the name suggests, the anelastic equations‘filter out’ the sound waves from the dynamics, leavingonly the effects of compressibility on the large scaledynamics. On relatively small scales, the assumption ofincompressibility is reasonable and is typically made instudies of the flows in shallow clouds [70, 71, 93, 108],and even in some idealised studies of deep convection [47].As we see in figure 1, the dynamics of clouds is an in-terplay of particle inertia, thermodynamics, the resultingbuoyancy driven flow. At the largest scales, the effect ofEarth’s rotation, and solar radiation and its modificationby clouds, need to be understood better, and we do notdeal with these topics in the present paper.In summary, studies at different scales have to some-times be carried out in isolation, using approaches andassumptions appropriate for that scale. New physics isrevealed at each scale, and their effects must then beincluded in our studies at other scales. IV. MICROPHYSICS WITHOUTTHERMODYNAMICS
A simple framework, which ignores the effects of ther-modynamics, phase changes and the associated changesin temperature, to understand the physics of a single warm cloud is to model it as a dilute suspension of small,spherical water droplets of radius a which are advectedby a statistically stationary, homogeneous and isotropic,full-developed turbulent flow. Such an approach ignoresthe effect of condensation, arising from a super-saturatedenvironment, by assuming that the starting point of suchstudies are non-precipitating droplets which are alreadycondensed to sizes of about 10 µm ; hence further growththrough condensation over a reasonably short time win-dow, corresponding to the life-time of such a cloud, isunlikely [18, 31, 41, 61, 95].Such a simplification has at least two distinct advan-tages. Firstly, it allows us to formulate and address ques-tions of collisions, coalescences, and gravitational settling(precipitation) in the turbulent setting of a cloud in a pre-cise way. Secondly, given this framework, it lends itselfeasily to the use of tools and ideas developed in the fieldof turbulent transport over the last two decades or so.In typical clouds, given a/η (cid:28)
1, the Reynolds numberassociated with a droplet Re p (cid:28)
1. This allows us todefine the dynamics of a droplet, in the presence of agravitational force g and an (turbulent) advecting fluidvelocity u , in terms of its position x p and velocity v ,through the linearised Stokes drag model with a Stokestime τ p [81]: d x p dt = v ; (1a) d v dt = − v − u ( x p ) τ p + g . (1b)The velocity field of the carrier flow, driven to a statis-tically steady state through a force f , with density ρ f ,a kinematic viscosity ν , and a pressure field P , satis-fies the incompressible, three-dimensional Navier-Stokesequation ∂ u ∂t + ( u · ∇ ) = ν ∇ u − ∇ Pρ f + f ; (1c) ∇ · u = 0 . (1d)Given the assumptions of a small droplet and a dilutesuspension, the underlying flow is assumed to be unaf-fected by the presence of such water droplets.The effect of the finite size and the density contrastof the particle with the carrier flow, which leads to a fi-nite time of relaxation of the particle velocities to thatof the fluid, is captured by the Stokes time τ p = a ρ p νρ f ,where the particle density is given by ρ p ; for clouds (wa-ter droplets in air), the ratio of these two densities is ρ p /ρ f ∼ FIG. 2. Snaphots of tracers (a) and inertial particles (b), with St = 0 .
1, in a three-dimensional turbulent flow. Particles inregions dominated by rotation ( Q >
0) are colored red, while those in regions dominated by straining ( Q <
0) are colored blue.The dissipative dynamics of inertial particles ( St = 0 .
1) causes them to form dynamic clusters, which are seen in panel (b) tomainly reside in straining regions, in accordance with the ejection of inertial particles from rotational zones. number St = τ p /τ η , where τ η = (cid:112) ν/(cid:15) is the character-istic, short-time, Kolmogorov time-scale of the fluid ( (cid:15) is mean energy dissipation rate). Such non-dimensionalnumbers allow an easy comparison between observations,experiments, theory and numerical simulations. The linear Stokes drag model (Eqs. 1a-1b) is, of course,in the heavy-particle limit ρ p (cid:29) ρ f , a simplification ofthe Maxey-Riley equation [62] for the motion of a spher-ical particle (with Re p (cid:28)
1) in a flow: ρ p d v dt = ρ f D u Dt +( ρ p − ρ f ) g − νρ f a (cid:18) v − u − a ∇ u (cid:19) − ρ f (cid:18) d v dt − DDt (cid:20) u + a ∇ u (cid:21)(cid:19) − ρ f a (cid:114) νπ (cid:90) t √ t − ξ ddξ ( v − u − a ∇ u )d ξ (2)where DDt denotes the full convective derivative and itfactors in the effects of the force due to the undis-turbed flow ρ f D u Dt , the buoyancy ( ρ p − ρ f ) g , theStokes drag νρ f a (cid:16) v − u − a ∇ u (cid:17) , the added mass ρ f (cid:16) d v dt − DDt (cid:104) u + a ∇ u (cid:105)(cid:17) , and the Basset history ρ f a (cid:112) νπ (cid:82) t √ t − ξ ddξ ( v − u − a ∇ u )d ξ effects. Withoutgoing into a rigorous demonstration of how Eq. 2 reducesto Eq. 1b, we can immediately see that for heavy par-ticles the force due to the undisturbed flow is negligibleand the only effect of gravity is a net acceleration down-wards. Furthermore, the Faxen corrections ∼ a ∇ u forsmall particles are negligible as is the Basset term in such dilute suspensions of passive particles. (On the last as-pect, we refer the reader to a recent work by Prasath etal. [75] for a detailed analysis of the Basset history term).Indeed recent work by Saw et al. [88] have confirmed bycomparing experimental data with those obtained froma numerical simulation of Eqs. 1, that the approxima-tion discussed here are indeed reasonably valid for thedilute suspensions of small, but heavy, particles that weconsider in this paper.Before we proceed further, it might be useful at thisstage to comment on how Eqs. 1 are solved on thecomputer [22]. (See Ref. [69] for a discussion on thenature of such simulations in two-dimensional flows.)The incompressible Navier-Stokes equations (Eqs. 1c-1d)are typically solved, in three dimensions, on a triple-periodic 2 π cube with N collocation points; in the re-sults being reviewed in this paper, N has ranged from512 up to 2048 yielding Taylor-scale based Reynoldsnumber which range from (approximately) 120 to 450.The flow is driven to a statistically steady, homoge-neous and isotropic, turbulent state with an externalforcing. There is of course considerable freedom in theway we choose to force the fluid; two particularly pop-ular choices are one with a constant energy injection aslow wave-numbers [89] and another where (again at low-wavenumbers) a second-order Ornstein–Uhlenbeck pro-cess [60] is adapted to provide a more random forcing.The equations themselves are solved through a standardpseudo-spectral method where spatial derivates are takenin Fourier space to allow an easy integration in time ofwhat essentially becomes an algebraic equation.Solving for the particle dynamics (Eqs. 1a-1b) are lessinvolved. The only non-trivial element in this is to usesuitable interpolation schemes to obtain the fluid veloc-ity, which is calculated on a regular Eulerian grid, at typ-ically off-grid particle positions. Several such schemes ex-ist and the ones commonly used are the cubic spline, theB-spline, the trilinear, or the cubic interpolation schemes(see e.g. [104]).The consequences of the linear Stokes drag model havebeen studied extensively [8, 9, 23, 45, 48, 65] since thepioneering work of Bec [6, 7]. A detailed discussion ofthis is certainly beyond the scope of this present paper.However, in what follows, it is useful to recall two cen-tral features of the dynamics defined by Eqs. 1, namely preferential concentration and caustics .The dissipative dynamics of the particle motion leadsto a preferential sampling of the flow and hence a pref-erential concentration of particles as opposed to a homo-geneous distribution in the flow (as seen for tracers) forfinite Stokes numbers: Particles with finite sizes evolveto (dynamically changing) attractors with fractal dimen-sions. Traditionally, this preferential concentration orinhomogeneities in the distribution of particles is cap-tured through the correlation dimension D obtainedfrom calculating the probability P < ( r ) of two particlesbeing within a distance r , whence P < ( r ) ∼ r D . Thecorrelation dimension D is of course a function of theStokes number St . In the limit of vanishing St (tracers),particles must distribute homogeneously and hence, in athree-dimensional flow, D = 3. For very large valuesof St , the motion of particles are essentially decorrelatedfrom the flow and thus have a more ballistic behaviour.This results, yet again, in space-filling and consequently D = 3. At intermediate values of St , however, the par-ticles distributions are inhomogeneous D < D ) for St ∼ O (1). We referthe reader to Figs. 1 and 2 in Ref. [9] for an illustrationof these effects.From the perspective of flow structures, the cluster-ing of heavy inertial particles can be tied to their ejec- tion from vortical or rotational regions of the flow. Oneway to quantify this behaviour is via the Q -criterion [33],which uses the local velocity gradient matrix A to definea quantity Q ≡ ( R ij − S ij ) / R = ( A + A T ) / S = ( A − A T ) / Q > < Q measured along the trajectories of tracers and inertialparticles with various values of St . The undersamplingof vortical regions by inertial particles is clearly visible;indeed the effect is significant even for St as small as 0.03(relevant for 10 − µ m cloud droplets). This effect getsstronger upto St ≈ .
5, beyond which the particles beginto decorrelate from the underlying flow structures, un-til eventually, for large St , the PDF approaches that fortracers (cf. St = 8 . η ), conditioned onthe local value of Q , and for various St . As St increasesfrom zero, the density of particles in regions of moderatestraining increases, while the density in rotational regionsreduces. This is illustrated visually by Fig. 2(b), whereinmost of the dense particle clusters are blue ( Q < St in regions of very high straining, indicating that iner-tial particles cannot cluster in such intense regions of theflow, but rather prefer mildly straining zones.A second important ingredient in this story is caus-tics [78]. Caustics are defined here as regions in the flowwhere droplets of different velocities can arrive simulta-neously at the same location. In other words, caustics areregions of the flow where droplet velocity cannot be de-scribed as a field. These are significant for the followingreason. Droplets of moderate Stokes number ( ∼ . ∼ (cid:112) Γ τ p from the vortex centre,even the smallest droplets are centrifuged out [78], andcaustics can form. It was shown [30, 78] that collisionsbetween small droplets can be greatly enhanced in thecaustics region, giving rise to a small number of dropletswhich can cross the bottleneck, and become the seeds forfurther coalescence events.Given this context, let us return once more to ques-tions within the framework of turbulent transport whichare pertinent to the problem of droplet dynamics in awarm cloud. In particular, the questions that we discussin this paper have to do with collisions, coalescences, andprecipitation. Specifically these are best discussed by an- FIG. 3. Panel (a) presents PDFs of Q sampled by tracers and inertial particles with various values of St . Panel (b) shows theaverage coarse-grained particle number density, conditioned on the local value of Q , for various St . swering the following questions: How fast—and where—do droplet collide? How fast do droplets grow (by coa-lescence)? And, how fast do droplet settle under grav-ity? (The issue of the structure of such aggregates whenthey collide, but not coalesce, will not be covered in thisoverview [11, 42].)Turbulence is thought to play a dominant role in en-hancing the droplet-droplet collision rates, and in turnthe droplet-size distributions as well as the initiation timeof rain, in typical warm clouds[36, 95]. The underly-ing mechanism instrumental in this is not only prefer-ential concentration, discussed above, but also, through slings [14, 36] and caustics [35, 109], the extreme ve-locities with which particles can approach each other.Therefore in order to gain insights which can help buildmesoscopic models for collision kernels, it is importantto have reliable estimates of the typical relative veloc-ities between droplets which are about to collide in aturbulent flow.This issue was addressed by Saw et al. [88] who,through experiments, numerical simulations and theory,studied the probability distribution functions of the ve-locity differences between pairs of particles, measuredalong the line-of-sight, when they are quite close to eachother. In the experiment, a turbulent flow was generatedwithin a 1 m -diameter acrylic sphere [14] with Taylormicro-scale Reynolds numbers R λ as high as 190 corre-sponding to values of η as low as 180 µm . In such a flow,a bi-disperse population of droplets were introduced, viaa spinning disc [107], with mean diameters 6 . µm and 19 µm which are much smaller than η . Given that theexperiment was able to achieve three different R λ , andhence η , it was possible to obtain particle trajectoriesfor 6 different Stokes numbers through stereoscopic La-grangian Particle Tracking [67]. In the numerical sim-ulations, of the sort described above, particle trajecto-ries were obtained for the similar Stokes numbers andfor comparable values of R λ which allowed a meaningfulcomparison of theory and experiments.A convenient measure of the statistics of how dropletsimpact on each other, is through the probability distri-bution function of the longitudinal component of the ve-locity differences v (cid:107) between pairs of particles and con-ditioned on their separation r . In Fig. 1 of Ref. [88],these distribution functions for four different values ofthe Stokes number and, in each case, conditioned on threedifferent values of the separation r are shown. The agree-ment between the experimental and numerical data, es-pecially for the larger values of St , is a confirmation ofthe validity of the linearised Stokes drag model of Eqs. 1.However, it is worth pointing out that these distributions,which seem to fit the form of stretched exponentials [53]and at odds with the compressed exponential predictionfor large Stokes numbers [46], show consistently, for rea-sons still not clear, a greater convergence between thenumerical and experimental data for the left tail (ap-proaching pairs) than for the right tail (separating pairs).Furthermore, a closer examination of these distributionsshow that droplet-pairs approach each other with increas-ing relative velocities—and a possible increase in collisionrates—as their Stokes number increases consistent withother evidences of the sling effect [14].These distributions were of course conditioned on small( O ( η )) but still finite separations; in the context of under-standing collisions amongst droplets in a cloud, we shouldexamine the relative velocities at contact or at least whenthey are even closer to each other ( r → r β ; experimental and numerical data suggests that β hasa mild Stokes-dependence and for reasonably large val-ues of St (not entirely valid for droplets in a cloud at itsinfancy), corresponding to extreme velocity differences,its value is consistent to the saturation exponent ξ ∞ ofthe higher-order moments of relative velocities [10]. Thepositive exponent β , which is small for small droplets,nevertheless suggests the possibility of mild impact ve-locities on contact.The particle dynamics in a more realistic cloud is ofcourse non-stationary as droplets grow and change theirsizes and numbers. A step in this direction is to ac-count for the polydispersity in a suspension. James andRay [50] investigated this problem for suspensions in twoand three-dimensional turbulent flows. Interestingly, theauthors were able to derive the typical impact velocitiesbetween particle pairs which, in principle, could have dif-ferent Stokes numbers. This was conveniently done byassuming a reference particle with a Stokes number St and a second one with St ; in the usual mono-disperseproblem, St = St . Assuming the smoothness of theunderlying fluid velocity at small inter-particle separa-tions, the impact velocity ∆ was theoretically calculated(under suitable approximations) for pairs of droplets withdifferent Stokes numbers and validated against numericalsimulations.As we discussed before, because of the collision-coalescence processes, it is safe to assume that particledynamics in a cloud may well be non-stationary. As aresult of this James and Ray have also looked at thecollision rates in a non-stationary phase and found anenhancement of this, when compared to the collisionrates in the statistically stationary phase for all non-zeroStokes numbers. Interestingly, this ratio peaks to 2 (Fig.4 in Ref. [50]) when the Stokes number of the collid-ing pairs are around 0.2. Although the results obtainedsuggests a lack of universality, this observation might beone possible explanation of possible run-away processeswhich explains the rapid growth of droplets from tinynuclei, seed rain drops. We also refer the reader to re-cent studies in Refs. [16, 17, 43, 44, 63] on the issue ofsuch poly-disperse suspensions and the relative velocitiesof colliding inertial particles in turbulent flows.All of this inevitably leads us to the question of how thesize distribution of droplets evolve when we turn on ac-tual coalescences in a suspension advected by a turbulentflow. A time-honoured theoretical framework for study-ing such problems is the Smulochowski’s equation witha stationary coalescence rate or kernel. Starting with FIG. 4. Snaphot of intense rotational (red) and straining(blue) regions in a turbulent flow, visualized using the Q -criterion. Intense vortical/rotational regions take the formof tubes or worms that are enveloped by strongly strainingsheet-like structures, forming vortex-strain wormrolls. Repro-duced from [74]. an initial infinite bath of particles of the same size (andmass), such an approach inevitably leads to a growthin the population of droplets of larges sizes as a simplepower-law in time. Specifically, assuming an initial infi-nite bath of particles of mass 1, the number of particleswith mass i (also, in these units, an integer since coales-cence imposes mass conservation) grows as n i ( t ) (cid:39) n i ( t/t i ) i − . (3)(The time t i appearing in the exponent is taken as an av-erage time-scales set by the different stationary collisionrates between all particle pairs which add up to give the i -th particle).However, when such particles are in a dilute suspen-sion carried by a turbulent flow, how accurate are theseestimates emerging from Smulochowski’s equation? Thisquestion was answered, through a combination of theoryand numerical simulations, by Bec et al. in Ref. [13].This work carefully analysed the contribution to the co-alescence rate coming not from the microphysics of ad-hesion but the fact that particles are in a turbulent flow.This non-trivial contribution was shown to factor in theanomalous part δ in the scaling of the third-order struc-ture function in a turbulence-advected passive-scalar field (cid:104) ( θ ( x + r ) − θ ( x )) (cid:105) ∼ | r | − δ and leads to a population ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ( b ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ( c ) FIG. 5. Panel (a) presents the 0.1 and 0.03 level contours of the joint probability distribution function of the values of R and S , measured at the positions of inertia-less ( St = 0) particles (dashed) and at their collision locations (solid). Panels(b) and (c) present distributions of the collision angles and the collision velocities, respectively, conditioned on whether thecollisions occur in rotational ( Q > +0 .
3, red) or straining ( Q < − .
3, blue) regions. The contribution of these regions to thejoint distribution in panel (a) is shown by the red/blue shading. Taken together, these panels show that collisions in strainingregions tend to occur in a head-on or rear-end fashion, which results in a higher collision approach velocity, and thereby ahigher collision frequency. Reproduced from [74]. growth of the form n i ( t ) (cid:39) n i (cid:0) t/ ˜ t i (cid:1) (1 − δ )( i − . (4)Given that the (universal) value of δ ≈ .
18, this formsuggests a more rapid growth of droplets with massesdifferent from the bath of monomers of mass 1.The accuracy and correctness of the theoretical calcu-lations were benchmarked against state of the art directnumerical simulations with an initial suspension of 1 bil-lion monomers which were then allowed to coalesce andform droplets of other sizes in the same paper [13]. Figure2 in this work shows the accuracy of the prediction (4)at early times; a further confirmation of the importanceof anomalous scaling was obtained via the probabilitydensity function of the inter-coalescence times betweenthe initial monomers (of mass 1) and other droplets ofdifferent sizes which were subsequently formed (Fig. 3in Ref [13]). Remarkably, these results are perhaps theonly ones which show how anomalous scaling in turbu-lence shows up as a leading order effect in a more ap-plied problem such as the one of coalescences in turbulenttransport.This work thus established a plausible argument tosuggest a rapid growth in droplets at short times througha complex (Lagrangian) correlated sequence of events.However the fate of these droplets at long time still re-mains a large unexplored issue.All of these measurements are of course central inbuilding up models for collision and coalescences in awarm cloud. However, they do not help us, in a directway, to uncover the correlation, if any, between colli-sions and the flow structures peculiar to turbulence. In- deed, even small-scale, homogeneous and isotropic turbu-lence, which we may expect to encounter in the core ofa cumulus cloud, is rich in structure: It is perforatedby a hierarchy of rotational and straining flow struc-tures [25, 32, 52, 94, 97, 110], as shown in Fig. 4. Thesestructures are a physical manifestation of the intermit-tency of the velocity gradient field, which distinguishesfully-developed turbulence from a simple random Gaus-sian field [49, 68, 101]. One may ask, therefore, whetherthe collisions between inertial particles or droplets aresensitive to these flow structures, and thereby to thenon-Gaussian nature of turbulence. This question wasaddressed recently by Picardo et al. [74], who measuredthe relative values of rotation ( R ) and straining ( S )at the locations of collisions, and compared them to thevalues sampled by particle trajectories. They found thatcollisions among small St particles are disproportionatelyfrequent in straining regions, much more than what maybe anticipated from preferential concentration alone. Infact, this effect is not fundamentally tied to inertia, butpersists even in the limit of St →
0. for which the par-ticles are homogeneously distributed. (In this limit, theparticles are effectively tracers, but with a small fictionalradius that enables the detection of collisions.)Figure 5(a) presents contours of the joint probabilitydistribution of the values of R and S , measured bothwhere inertia-less particles reside (dashed contours) andwhere they collide (solid contours). Here, straining (ro-tational) regions with Q < − . > +0 .
3) are shadedin blue (red). Clearly, there there is an oversampling(undersampling) of straining (vortical) regions by colli-sions, compared with the particle trajectories. This dis-crepancy is a result of the very different flow geometryin these regions. Colliding particles in straining regionstend to approach each other in a head-on or rear-endfashion, whereas particles in rotational regions approacheach other in a side-on manner and undergo glancing col-lisions. This is shown by the conditioned-PDFs of the col-lision angle θ (the angle between the relative velocity andposition vectors of the two colliding particles), presentedin Figure 5(b). For a given magnitude of the underlyingfluid velocity, head-on collisions are faster than side-oncollisions, as shown in Figure 5(c), because a larger com-ponent of the relative-velocity of the particles is trans-lated into the collision velocity. For the same numberdensity of particles, this results in a higher collision fre-quency in straining regions.Particle inertia acts to enhance this preference of col-lisions for straining regions, upto St ≈ .
3. This isseen in Fig. 6(a), which presents the average value of Q = ( R − S ) / St . At small St , inertia selec-tive enhances the collision velocity in straining regions,as shown in Fig. 6(b), and therefore increases the fre-quency of collisions in straining regions relative to ro-tational regions. At larger values of St , however, theparticles begin to decorrelate from the underlying flowstructures and both particle and collision locations be-gin to distribute uniformly (cf. the inset of Fig. 6(a))and also collide with comparable velocities in strainingand vortical regions (Fig. 6(c)). This mismatch betweenwhere inertial (for small St ) particles reside and collidewas earlier observed by Perrin and Jonker [72], who alsoanalyzed the influence of flow structures on collisions us-ing the eigenvalues of the velocity gradient matrix [73],which enable a finer classification of structures than thesimple Q -criterion.We have seen that collisions between small St particlesare sensitive to the local underlying structure of the flow.As St approaches unity, however, collisions are affectednot just by the structure at the collision location, but byall the flow structures encountered by the particles, uptoa time of about τ p prior to the collision, or upto a dis-tance of | v | τ p around the collision. This raises the possi-bility of intense vortical and straining regions conspiringto generate violent, rapid collisions, due to their peculiarvortex-strain worm-roll geometry (cf. Fig. 4): Particlesin intense vortex tubes will be ejected with large slip ve-locities into the enveloping straining sheets, where theyhave a high chance of colliding with large relative veloc-ities. Picardo et al. [74] found evidence for this scenario,by Lagrangian backtracking of particles that collided instraining regions: the particles which collided in the leasttime after being ejected from a vortex, were indeed theones that originated from the strongest vortices, collidedin the strongest straining regions and with the largestcollision velocities.This effect of vortex-strain worm rolls was found tobe prominent only for St beyond about 0.5. This valuemay seem too large for small cloud droplets, and indeedit is when one considers the particle relaxation time rela- tive to the mean Kolmogorov time-scale of the turbulentflow. However, at the extremely large Re of in-cloudturbulence, there are likely to be a few, very intense,intermittent vortices, with a local flow time scale thatis much smaller than the mean Kolmogorov time-scale.This means that local, effective St of particles in thevicinity of such intense vortices will be closer to unity,making the vortex ejection and collision scenario rele-vant. Even if such intense vortices occupy a very smallvolume fraction of the flow, the rapid collisions gener-ated will be able to act as a seed that initiates the run-away growth of droplets by gravitational driven collision-coalescence [56].Unfortunately, the Re values that can be directly simu-lated on a computer are still orders of magnitude smallerthan what is expected for a cloud. Therefore, it is notpossible to directly investigate the effect of high- Re flowstructures. However, one can gain a basic understand-ing of how such structures may influence the motionand collisions of droplets by using model vortex flows.Such an analysis was carried out in two-dimensions, usingpoint and gaussian vortices, by Ravichandran et al. [78]and Deepu et al. [30], and later extended to a three-dimensional Burgers vortex [21] by Agasthya et al. [2].These studies show that particles near the core of a strongvortex are ejected more rapidly than particles fartheraway. This leads to a large increase in the local parti-cle density around the periphery of the vortex (Fig. 3of [2]), as well as large relative velocities between neigh-bouring particles (caustics). These factor combine to sig-nificantly enhance collisions in the vicinity of the vortex.Figure 7 shows the coarse-grained collision density Θ (ob-tained from a large ensemble of simulations) as a func-tion of the radial distance r from the axis of a Burgersvortex, which serves as a model for the intense vortextubes [24, 37] observed in three-dimensional turbulence(cf. Fig. 4) [32, 97]. Three cases are presented, corre-sponding to a mild, wide vortex ( r core = 0 .
4) and an in-tense thin vortex ( r core = 0 . σ = 0 .
08 and 0 . V s , i.e., the compo-nent of the particle velocity along the direction of grav-ity, of a particle with a Stokes time τ p is simply (1b) V s = τ p g − (cid:104) u z ( x p , t ) (cid:105) . In the absence of a flow or a uni-form sampling of the flow by the particles, (cid:104) u z ( x p , t ) (cid:105) = 0and leading to a predictable settling velocity V s = τ p g .However in the presence of a background turbulent flow,0 FIG. 6. Panel (a) presents the average value of Q measured where inertial particles or droplets collide (black-solid) and reside(dashed-red), as a function of St . The inset presents the same result for a wider range of St using a semi-log scale. Panel (b)presents the average collision velocity, conditioned on whether collisions occur in regions dominated by rotation (red) or strain(blue), as a function of St . Adapted from [74].FIG. 7. Coarse-grained collision density Θ, i.e. the number ofcollision per unit volume, as a function of the radial distance r from the axis of a tubular Burgers vortex. r core is a measureof the size of the core of the vortex, i.e. of how intenselythe vorticity in concentrated about the vortex-axis. σ is ameasure of the straining flow, that is directed inward alongthe radial direction and outward along the vortex-axis. Thisstraining is essential for maintaining a concentrated vortextube in a viscous fluid, where vorticity continuously diffusesoutward [24]. Here, we see that this straining flow also actsto enhance the collisions around intense vortices. Reproducedfrom [2]. it has been known that the settling velocity can be en-hanced through an oversampling of the regions where thefluid velocity is downwards. A systematic and quantita-tive understanding of this phenomenon was carried outby using extensive numerical simulations and theory byBec et al. in Ref. [12].A convenient way to estimate this enhancement ofthe settling velocity is to measure the relative increase∆ V = ( V s − τ p g ) / ( τ p g ) = −(cid:104) u z ( x p , t ) (cid:105) / ( τ p g ) as a functionof the Stokes number. In Fig. 2 of Ref [12], the authorsshowed the ∆ v is indeed positive and a non-monotonicfunction of St with a peak at St ∼
1. This enhancement,in the limit of small values of St was understood by show-ing that the correlation (cid:104) u z ∇ ⊥ · v (cid:105) = τ p g (cid:10) ( ∂ z u z ) (cid:11) > ∇ ⊥ · v < u z <
0. Such an asymptoticsalso suggests that for small Stokes numbers ∆ v ∝ St .The large Stokes asymptotics, dominated by the bal-listic motion of particles resulting in a short-time correla-tion with the fluid velocity being sampled by the particle,is more involved. However, the remarkable thing aboutthis asymptotics is it makes a scaling prediction on ∆ v in terms of the Reynolds, Froude and Stokes numberswhich are shown, through numerical simulations (Fig. 2in Ref. [12], to be exceptionally accurate.From the perspective of warm clouds, the nature of1setting of droplets under gravity has one further im-portant consequence. Bec et al. [12] showed that grav-itational settling, especially when the effect of grav-ity is pronounced, is accompanied by a quasi-two-dimensionalisation of the particle dynamics (Fig. 3 inRef. [12]). A consequence of this is the estimation of thecollision rate κ ∼ r γ which is the average longitudinal ve-locity differences between pairs of same-sized particles ata separation r ∼ a (cid:28) η as they approach each other. Itwas shown that since γ = ξ + D −
1, where ξ is exponentof the order-1 structure function constructed from parti-cle velocities, the approach rates must be influenced bythe nature of clustering D brought about through grav-itational settling. Figure 4 in Ref. [12] summarises theseexponents with their dependence on both the Stokes andFroude numbers and shows how, under the influence ofgravity, inter-particle approach velocities are diminished,through a renormalised effective Stokes number, as theeffect of gravity begins to dominate. Indeed, these re-sults suggest that for 30 µm -sized water droplets, typicalin warm clouds, collision rates are almost doubled whenwe factor in the interplay of both gravitational and tur-bulent effects on their mixing. These ideas of settling arenow being extended to spheroidal particles in turbulentflows which serve as an effective model for ice crystals incolder clouds [3, 86].The findings above are obtained at moderate Reynoldsnumber, while the Reynolds numbers in a cloud are sev-eral orders of magnitude higher. We highlight the impor-tance of understanding how flow structures and other fea-tures change with increasing Reynolds numbers. More-over buoyancy effects can be important, as discussed be-low. V. MICROPHYSICS WITHTHERMODYNAMICS
In the previous section we discussed how turbulenceaffects the dynamics of droplets, in that it clusters theminto portions of the flow, thus encouraging droplet col-lisions and merger. We assumed that droplets do notaffect the turbulence. Mechanically speaking, this is afair assumption through most stages of droplet growth.This is because water droplets are very small, and form avery dilute suspension, in that occupy only a millionth ofthe volume in a cloud. However, droplets can distort theturbulence that drives them through the thermodynam-ics associated with phase change. Some of the physicsbehind these effects was explained in Ravichandran andGovindarajan [79].
A. Thermodynamics of phase-change
The condensation of water vapour into water and theevaporation of liquid water into vapour (hereafter justphase-change) are governed by the Clausius-Clapeyron law, dp s dT = L v p s R v T , (5)where p s is the equilibrium water vapour pressure at thetemperature T , L v is the enthalpy of vaporisation, and R v is the gas constant for water vapour. The Clausius-Clapeyron law can be derived from the condition thatthe vapour-liquid system is at equilibrium at the giventemperature (see, e.g. [19, Chapter 5]). This equationcan be integrated assuming L v and R v are constants (thisis a reasonable assumption) to give p s = p s exp (cid:18) L v R v (cid:20) T − T (cid:21)(cid:19) . (6)Further approximation is possible for small temperaturechanges ( T ≈ T ) to give p s = p s exp (cid:18) L v ( T − T ) R v T (cid:19) . (7)Due to its exponential nature, the amount of watervapour that can exist in equilibrium is a rapidly chang-ing function of the ambient temperature; the equilibriumvapour pressure roughly doubles for every 10K increasein temperature.While the Clausius-Clapeyron law governs the equilib-rium vapour pressure, it says nothing about how thisequilibrium is to be reached. Chemical reactions orchanges of phase that are thermodynamically favouredmay nevertheless not occur because the reactions havehigh energies of activation. As a result, pockets of airwith higher concentrations of water vapour than given byequation 6 are very commonly found in the atmosphere.The system of air and water vapour is then said to be ‘su-persaturated’. In fact a cloud is often on average super-saturated. So excess water vapour is available, which canthen condense. However, thermodynamically for sponta-neous condensation, i.e., condensation without any pre-existing nucleation site, we need about 400% supersatu-ration, whereas such supersaturation is impossible underatmospheric conditions, where it is almost never greaterthan 5%. We therefore need cloud condensation nuclei onwhich condensation can occur, and droplets and aerosolparticles provide such surfaces. These nuclei are typicallysmall particles of salt or dust and a background concen-tration of these nuclei of about 100 − − exists inthe atmosphere. This number concentration is a functionof how polluted the air is, typically being larger over thecontinents than over the ocean. This number concentra-tion, then, also decides how supersaturated the air canbe. Supersaturations for polluted air are typically 1% orlower, while higher supersaturations are seen in marineclouds (see, e.g. [76], chapter 2). Nuclei smaller than acritical size (called the Kohler radius) reach an equilib-rium radius and do not grow beyond this (due to the factthat the saturation vapour pressure is a function of the2radius). Nuclei that are larger than the Kohler radiusgrow to become water droplets in clouds. These waterdroplets continue to grow by absorbing the water vapourin the atmosphere. The resulting release of the latentheat of vaporisation drives the large scale dynamics ofclouds, as we sketch below.A relation describing the rate at which the waterdroplets grow and consume water vapour can be derivedassuming the water droplets are small enough for ven-tilation effects to be negligible (see [19] chapter 7, [76],chapter 13). This gives a dadt = s − Cρ w , where C = O (cid:0) (cid:1) ms/kg is a thermodynamic constantwhich is a function of the ambient temperature. Therate of growth of a droplet is inversely proportional to itsradius. As we have seen in section IV, this is one of thefactors that makes explaining rain-formation challenging.If this relation is applied to a system of n droplets perunit volume, ignoring interactions, the rate at which thewater vapour in the system is consumed is dρ v dt = − ρ v /ρ s − τ s , (8)where τ s is a time-scale and ρ s = p s / ( R v T ) is the satura-tion vapour density. This condensation of vapour resultsin the heating of the flow at a rate dTdt = L v C p (cid:18) ρ v /ρ s − τ s (cid:19) . The latent heat of vaporisation, L v ≈ . × J/kg/K is a large value. As a result of this, despite the smallamounts by weight of water vapour and liquid found inclouds (typical values are O (1 − g/kg ), the amountsof heat released are enormous and can be O ( M W/m (see, e.g. the discussion in [66] on typical heating ratesin clouds). The heating thus provided increases the tem-perature, and the resulting buoyancy drives the upwardflow of the cloud. Differential heating of regions of theflow and the resulting buoyancy differences drive the tur-bulence in the flow. We look next at how particle inertia,phase change and buoyancy interact in clouds. B. Interactions of particle inertia,thermodynamics, and buoyancy-driven flow
We refer again to the box-diagram in fig. 1 showing thedifferent interacting phenomena in clouds. All clouds arecomposed of water vapour, water droplets, and aerosolparticles suspended in turbulent flow. Particle inertia,phase-change, and buoyancy-driven turbulent flow are allactive in clouds. However, depending on the type of cloudand the range of parameters, simplifying approximationsmay be made which ignore one or more of these effects.We discuss some relevant examples below which illustratethis.
FIG. 8. The entrainment coefficient in heated plumes firstincreases and then decreases to zero (i.e. the plume stopsentraining). Reproduced from [66].
1. Cumulus clouds: phase-change+buoyancy+turbulence
Cumulus clouds are tall, heap-like clouds found in theatmosphere, and are crucial to the maintenance of heatand mass balance in the atmosphere. Their importancein the dynamics of the atmosphere has long been recog-nised, with competing attempts to model them as variousfree-shear flows like jets, plumes, or thermals ([98, 100]).These clouds, driven by the release of latent heat in theflow, differ significantly in their dynamics from jets andplumes without such latent heating, and have been anobject of study for 60 years. A fuller review of these ef-forts may be found in [29]. The parameter of interest inthe study of these clouds is the entrainment rate—therate at which the cloud drags in ambient (dry) air fromits environment. Entrainment dilutes the cloud, leadingto its ultimate demise. Entrainment in free-shear flowsis still a topic of active research, and the addition of vol-umetric (i.e. not at the source on the ground) heatingcomplicates the picture.Progress has been made through laboratory experi-ments on cumulus clouds, showing the role of the la-tent heat release, reported by [66], building on work in[15, 105, 106]. The addition of latent heat to the flowseems to not only accelerate the flow but to shut down theentrainment of ambient air into the bulk of the flow. Thisshutdown of entrainment (shown in figure 8) is argued tobe because the heating disrupts the coherent structuresin the shear layer of the flow ([15, 66]). This then leadsto the cloud remaining undiluted for longer and reachinghigher altitudes than if the flow were a pure plume or jet.As we have argued, the heating in the cloud arisesout of the condensation of water vapour onto liquidwater droplets in the cloud, and thus a full descriptionof the dynamics would include all the phenomena listedin figure 1. However, especially for growing cumulusclouds which have not reached the precipitating state,the droplet size may be assumed to be small enough thatthe particle inertia is negligible. This is a reasonable as-3sumption since the size distribution in non-precipitatingcumulus clouds peaks at 10 µ m. This being the case,the droplet inertia is small and the droplets follow thevelocity of the fluid. This in turn allows the dropletsto be coarse-grained into a liquid-water field, a stepthat saves significant computational effort. Despite thechallenges involved, progress has been made becauseof the simplifying assumptions discussed. Large eddysimulations of cumulus clouds showing this behaviourhave been reported by [85]. Laboratory experimentsof cumulus clouds have been reported by [66]. Directnumerical simulations for achievable Reynolds numbersare reported in [80].
2. Mammatus clouds: phase-change + particle settling +buoyancy
A type of cloud that is perhaps not as consequential asthe cumulus clouds, but no less fascinating, is the mam-matus cloud. Typically found underneath cumulonimbusanvil outflows, and therefore acting as harbingers of in-clement weather. The reasons for the formation of theseclouds is a matter of ongoing research, and a comprehen-sive review of the various proposed mechanisms may befound in [92], with follow-up studies in [54, 55]. A promis-ing explanation involves the combination of the settlingof water droplets out of the cumulonimbus anvils, theirsubsequent evaporation forming a layer of air below theanvil that is denser than the ambient air, and the even-tual instability due to this density inversion. Unlike insection V B 1, particle settling cannot be neglected. Astudy by [83] finds that the size of the mammatus lobesis proportional to the settling velocity (which increasesfor small droplet sizes like the square of the droplet size)and to the time-scale for phase-change (which is propor-tional to the inverse of the mean droplet size and thenumber concentration).
3. Droplets in clouds, redux: particleinertia+turbulence+phase-change
The phenomenon of warm-rain initiation and thedroplet-growth bottleneck that has been a long-standingunsolved problem in cloud physics, as discussed insection IV. In studies focussing on the fluid mechanics ofdroplet collisions-coalescence, the effects of phase-changeand thermodynamics are typically neglected. In theregime of interest—when droplets have grown to largeenough sizes that their growth rates are small—thisassumption is justifiable. For most of the lifetimeof a cloud droplet, however, the thermodynamics ofcondensation cannot be neglected. The interactions ofphase-change and particle inertia are thus relevant inthe dynamics of clouds: broad droplet size distributionsare important in the rapid growth of falling droplets through coalescence with smaller droplets.A first step in understanding the interactions ofparticle inertia and thermodynamics was taken by [96]who study how droplets interact with vortices. Clouds,being turbulent flows, are a tangle of strong vortices.Vortices, then, are suitable models for idealised studies.As we have seen in section IV, vortices expel inertialparticles. When these inertial particles are also nucleifor condensation, the cores of vortices are voided ofnuclei for condensation and therefore have higher vapourconcentrations than the outside. [96] argue that thisshould have two consequences: first, that any dropletsthat remain trapped in the vortical region will end upexperiencing larger-than-average supersaturations, andthus be able to grow to sizes not predicted by only con-sidering cloud-average values of supersaturation; second,that the supersaturations produced in the cores of thevortices should lead to the nucleation of condensationnuclei well above cloud-base, which allows a broadeningof the droplet size distributions (on the lower side). Theuse by [96] of this mechanism in explaining warm-raininitiation has been questioned in the literature (see[40, 102, 103]), but the central message that particleinertia and thermodynamics interact in complex waysremains relevant, in our view.Clouds are, as we have seen, turbulent flows where theturbulence is driven by the energy provided by condens-ing water vapour. While large velocities can be gener-ated by large values of buoyancy, turbulence itself, as wehave also argued, is generated by spatial inhomogeneities in the heating. [79]propose a model of how this can beachieved starting from initially homogeneous conditions,thereby providing a route by which turbulence can sus-tain itself by ‘feeding’ on the latent heat of vaporisa-tion. The mechanism builds on the aforementioned effectof inertial particles being centrifuged out of vortical re-gions. This leaves the vortices more supersaturated (as in[96], but keeping track of temperature changes), but alsocolder than their surroundings which have been heated bythe condensation of water vapour. The resulting densityinhomogeneities lead to baroclinic torques which gener-ate vorticity and thus turbulence in the flow. We mentionin passing that the buoyant vortices that result have in-teresting dynamics of their own (see [82]).Growing clouds are also a class of free-shear flows.The edges of clouds, where the shear layers separatesaturated regions from unsaturated regions, are thusregions of large inhomogeneities in vapour and dropletconcentration. This results in the generation of strongsustained turbulent flow. [57–59] study this dynamicsin an idealised setup where the temperature is heldconstant, but water droplets are allowed to grow orshrink in response to local conditions. The authorsargue that, as a consequence of the high flow Reynoldsnumbers in a cloud, the mixing of dry ambient air withthe saturated cloudy air will be highly inhomogeneous:4 k -10 -7 -4 -1 E ( k ) k -10 -7 -4 -1 E ( k ) No inertiaInertia k -3 k -10 -7 -4 -1 E ( k ) k -10 -7 -4 -1 E ( k ) t = 40 t = 50 t = 30t = 10 FIG. 9. The energy density E ( k ) vs the wavenumber k in sim-ulations with and without accounting for the effects of parti-cle inertia. The addition of particle inertia effects provides aroute for the transfer of energy to smaller scales starting fromhomogeneous conditions. Similar to a figure in [79] i.e. that parcels of saturated air and parcels of subsat-urated air will often be found close to each other. As aresult, particles in the shear layer can have complicatedgrowth histories; the size distribution of such dropletswill also be very broad as a result. They quantify thisinhomogeneity in terms of a Damkoehler number whichis a ratio of the time-scale of phase-change to a char-acteristic timescale of the flow. For large Damkoehlernumbers, which would be expected in the high-Reynoldsnumber flows in the shear layers of clouds, they showthat using a large Damkoehler number Da (cid:29) CONCLUSION AND FUTURE DIRECTIONS
The fluid dynamics of clouds is only part–if the mostcomplex part–of climate science. We have argued herethat understanding the dynamics is important in the faceof the uncertainties of climate change. We hope to haveconvinced readers of the immense challenges and oppor-tunities the field presents, both as an exercise in scientificcuriosity and because of the far-reaching implications.Both these facets arise from the range of length- andtime-scales present in the dynamics. The approach wefavour—of understanding the parts in service of under-standing the whole—has led to several semi-independentprogrammes of research which have to be eventually beunited. Under present computational limitations the goalof these studies is to be able to parameterise the dynam-ics and improve global climate models.Some areas of active research where the questions arereasonably well-defined and can be addressed computa-tionally are a) A fuller understanding of entrainment infree-shear flows in general, and cloud-flows in particu-lar. Open problems include the magnitude of the en-trainment and the details of the process by which volu-metric heating alters the entrainment; by extension, en-trainment in merging plumes/cumulus clouds as a way ofunderstanding the dynamics of convective aggregation;and the incorporation of the resultant better models forentrainment into global climate simulations: perhaps byextending super-parameterisation method (see [77]). b)Droplet-resolved simulations for collision-coalescence, inorder to overcome limitations of the geometric collisionsapproach; to include effects of droplet splintering, dropletinteractions, and flow-droplet coupling; and to include ef-fects of differences in liquid/gas properties like density,temperature, conductivity in the phase-change process.
ACKNOWLEDGEMENTS
SR is supported under Swedish Research Council grantno. 638-2013-9243. JRP acknowledges funding from theIITB-IRCC seed grant. SSR acknowledges DST (India)project MTR/2019/001553 for support. RG and SSRacknowledge support of the DAE, Govt. of India, underproject no. 12-R&D-TFR-5.10-1100.5 [1] D. Abma, T. Heus, and J.-P. Mellado. Direct Nu-merical Simulation of Evaporative Cooling at the Lat-eral Boundary of Shallow Cumulus Clouds.
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