It was previously shown that the superdroplet algorithm for modeling the collision-coalescence process can faithfully represent mean droplet growth in turbulent clouds. An open question is how accurately the superdroplet algorithm accounts for fluctuations in the collisional aggregation process. Such fluctuations are particularly important in dilute suspensions. Even in the absence of turbulence, Poisson fluctuations of collision times in dilute suspensions may result in substantial variations in the growth process, resulting in a broad distribution of growth times to reach a certain droplet size. We quantify the accuracy of the superdroplet algorithm in describing the fluctuating growth history of a larger droplet that settles under the effect of gravity in a quiescent fluid and collides with a dilute suspension of smaller droplets that were initially randomly distributed in space (lucky droplet model). We assess the effect of fluctuations upon the growth history of the lucky droplet and compute the distribution of cumulative collision times. The latter is shown to be sensitive enough to detect the subtle increase of fluctuations associated with collisions between multiple lucky droplets. The superdroplet algorithm incorporates fluctuations in two distinct ways: through the random spatial distribution of superdroplets and through the Monte Carlo collision algorithm involved. Using specifically designed numerical experiments, we show that both on their own give an accurate representation of fluctuations. We conclude that the superdroplet algorithm can faithfully represent fluctuations in the coagulation of droplets driven by gravity.

Coagulation driven by supersonic turbulence is primarily an astrophysical problem because coagulation processes on Earth are normally associated with incompressible fluid flows at low Mach numbers, while dust aggregation in the interstellar medium for instance is an example of the opposite regime. We study coagulation of inertial particles in compressible turbulence using high-resolution direct and shock-capturing numerical simulations with a wide range of Mach numbers from nearly incompressible to moderately supersonic. The particle dynamics is simulated by representative particles and the effects on the size distribution and coagulation rate due to increasing Mach number is explored. We show that the time evolution of particle size distribution mainly depends on the compressibility (Mach number). We find that the average coagulation kernel C-ij scales linearly with the average Mach number M-rms multiplied by the combined size of the colliding particles, that is, < Cij > similar to <(a(i) + a(j))(3)> M-rms tau(-1)(eta), is proposed and can serve as a benchmark for future studies. We argue that the coagulation rate < R-c > is also enhanced by compressibility-induced compaction of particles.

We investigate the effect of turbulence on the combined condensational and collisional growth of cloud droplets by means of high-resolution direct numerical simulations of turbulence and a superparticle approximation for droplet dynamics and collisions. The droplets are subject to turbulence as well as gravity, and their collision and coalescence efficiencies are taken to be unity. We solve the thermodynamic equations governing temperature, water vapor mixing ratio, and the resulting supersaturation fields together with the Navier-Stokes equation. We find that the droplet size distribution broadens with increasing Reynolds number and/or mean energy dissipation rate. Turbulence affects the condensational growth directly through supersaturation fluctuations, and it influences collisional growth indirectly through condensation. Our simulations show for the first time that, in the absence of the mean updraft cooling, supersaturation-fluctuation-induced broadening of droplet size distributions enhances the collisional growth. This is contrary to classical (nonturbulent) condensational growth, which leads to a growing mean droplet size, but a narrower droplet size distribution. Our findings, instead, show that condensational growth facilitates collisional growth by broadening the size distribution in the tails at an early stage of rain formation. With increasing Reynolds numbers, evaporation becomes stronger. This counteracts the broadening effect due to condensation at late stages of rain formation. Our conclusions are consistent with results of laboratory experiments and field observations, and show that supersaturation fluctuations are important for precipitation.

We show that the growth rate of dust grains in cold molecular clouds is enhanced by the high degree of compressibility of a turbulent, dilute gas. By means of high-resolution (1024(3)) numerical simulations, we confirm the theory that the spatial mean growth rate is proportional to the gas-density variance. This also results in broadening of the grain-size distribution (GSD) due to turbulence-induced variation of the grain-growth rate. We show, for the first time in a detailed numerical simulation of hydrodynamic turbulence, that the GSD evolves toward a shape that is a reflection of the gas-density distribution, regardless of the initial distribution. That is, in case of isothermal, rotationally forced turbulence, the GSD tends to be a lognormal distribution. We also show that in hypersonic turbulence, decoupling of gas and dust becomes important and that this leads to an even further accelerated grain growth.

Condensational growth of cloud droplets due to supersaturation fluctuations is investigated by solving the hydrodynamic and thermodynamic equations using direct numerical simulations (DNS) with droplets being modeled as Lagrangian particles. The supersaturation field is calculated directly by simulating the temperature and water vapor fields instead of being treated as a passive scalar. Thermodynamic feedbacks to the fields due to condensation are also included for completeness. We find that the width of droplet size distributions increases with time, which is contrary to the classical theory without supersaturation fluctuations, where condensational growth leads to progressively narrower size distributions. Nevertheless, in agreement with earlier Lagrangian stochastic models of the condensational growth, the standard deviation of the surface area of droplets increases as t(1/2). Also, for the first time, we explicitly demonstrate that the time evolution of the size distribution is sensitive to the Reynolds number, but insensitive to the mean energy dissipation rate. This is shown to be due to the fact that temperature fluctuations and water vapor mixing ratio fluctuations increase with increasing Reynolds number; therefore the resulting supersaturation fluctuations are enhanced with increasing Reynolds number. Our simulations may explain the broadening of the size distribution in stratiform clouds qualitatively, where the mean updraft velocity is almost zero.

This Ph.D. thesis examines the challenging problem of how turbulence affects the growth of cloud droplets in warm clouds. Droplets grow by either condensation or collision. Without turbulence, the condensation process driven by a uniform supersaturation field is only efficient when droplets are smaller than 15 μm (radius). Gravitational collision becomes effective when the radius of droplets is larger than 50 μm. The size gap of 15–50 μm, in which neither condensation nor collision processes dominate droplet growth, has puzzled the cloud microphysics community for around 70 years. It is the key to explaining the rapid warm rain formation with a timescale of about 20 minutes. Turbulence has been proposed to bridge this size gap by enhancing droplet growth processes, and thereby, to explain rapid warm rain formation. Since cloud–climate interaction is one of the greatest uncertainties for climate models, the fundamental understanding of rapid warm rain formation may help improve climate models.

The condensational and collisional growth of cloud droplets in atmospheric turbulence is essentially the problem of turbulence-droplet interaction. However, turbulence alone is one of the unresolved and most challenging problems in classical physics. The turbulence–droplet interaction is even more difficult due to its strong nonlinearity and multi-scale properties in both time and space. In this thesis, Eulerian and Lagrangian models are developed and compared to tackle turbulence–droplet interactions. It is found that the Lagrangian superparticle model is computationally less demanding than the Eulerian Smoluchowski model.

The condensation process is then investigated by solving the hydrodynamic and thermodynamic equations using direct numerical simulations with droplets modeled as Lagrangian particles. With turbulence included, the droplet size distribution is found to broaden, which is contrary to the classical theory without supersaturation fluctuations, where condensational growth leads to progressively narrower droplet size distributions. Furthermore, the time evolution of droplet size distributions is observed to strongly depend on the Reynolds number and only weakly on the mean energy dissipation rate. Subsequently, the effect of turbulence on the collision process driven by both turbulence and gravity is explored. It is found that the droplet size distribution depends moderately on the mean energy dissipation rate, but is insensitive to the Reynolds number. Finally, the effect of turbulence on the combined condensational and collisional growth is investigated. In this case, the droplet size distribution is found to depend on both the Reynolds number and the mean energy dissipation rate. Considering small values of the mean energy dissipation rate and high Reynolds numbers in warm clouds, it is concluded that turbulence enhances the condensational growth with increasing Reynolds number, while the collision process is indirectly affected by turbulence through the condensation process. Therefore, turbulence facilitates warm rain formation by enhancing the condensation process, which predominantly depends on the Reynolds number.

We investigate the effect of turbulence on the collisional growth of um-sized droplets through high- resolution numerical simulations with well resolved Kolmogorov scales, assuming a collision and coalescence efficiency of unity. The droplet dynamics and collisions are approximated using a superparticle approach. In the absence of gravity, we show that the time evolution of the shape of the droplet-size distribution due to turbulence-induced collisions depends strongly on the turbulent energy-dissipation rate, but only weakly on the Reynolds number. This can be explained through the energy dissipation rate dependence of the mean collision rate described by the Saffman-Turner collision model. Consistent with the Saffman-Turner collision model and its extensions, the collision rate increases as the square root of the energy dissipation rate even when coalescence is invoked. The size distribution exhibits power law behavior with a slope of -3.7 between a maximum at approximately 10 um up to about 40 um. When gravity is invoked, turbulence is found to dominate the time evolution of an initially monodisperse droplet distribution at early times. At later times, however, gravity takes over and dominates the collisional growth. We find that the formation of large droplets is very sensitive to the turbulent energy dissipation rate. This is due to the fact that turbulence enhances the collisional growth between similar sized droplets at the early stage of raindrop formation. The mean collision rate grows exponentially, which is consistent with the theoretical prediction of the continuous collisional growth even when turbulence-generated collisions are invoked. This consistency only reflects the mean effect of turbulence on collisional growth.

Small-scale dynamos are expected to operate in all astrophysical fluids that are turbulent and electrically conducting, for example the interstellar medium, stellar interiors, and accretion discs, where theymay also be affected by or competing with large-scale dynamos. However, the possibility of small-scale dynamos being excited at small and intermediate ratios of viscosity to magnetic diffusivity (the magnetic Prandtl number) has been debated, and the possibility of them depending on the large-scale forcing wavenumber has been raised. Here, we show, using four values of the forcing wavenumber, that the small-scale dynamo does not depend on the scale separation between the size of the simulation domain and the integral scale of the turbulence, i.e. the forcing scale. Moreover, the spectral bottleneck in turbulence, which has been implied as being responsible for raising the excitation conditions of small-scale dynamos, is found to be invariant under changing the forcing wavenumber. However, when forcing at the lowest few wavenumbers, the effective forcing wavenumber that enters in the definition of the magnetic Reynolds number is found to be about twice the minimum wavenumber of the domain. Our work is relevant to future studies of small-scale dynamos, of which several applications are being discussed.

Turbulence is argued to play a crucial role in cloud droplet growth. The combined problem of turbulence and cloud droplet growth is numerically challenging. Here an Eulerian scheme based on the Smoluchowski equation is compared with two Lagrangian superparticle (or superdroplet) schemes in the presence of condensation and collection. The growth processes are studied either separately or in combination using either two-dimensional turbulence, a steady flow or just gravitational acceleration without gas flow. Good agreement between the different schemes for the time evolution of the size spectra is observed in the presence of gravity or turbulence. The Lagrangian superparticle schemes are found to be superior over the Eulerian one in terms of computational performance. However, it is shown that the use of interpolation schemes such as the cloud-in-cell algorithm is detrimental in connection with superparticle or superdroplet approaches. Furthermore, the use of symmetric over asymmetric collection schemes is shown to reduce the amount of scatter in the results. For the Eulerian scheme, gravitational collection is rather sensitive to the mass bin resolution, but not so in the case with turbulence. Plain Language Summary The bottleneck problem of cloud droplet growth is one of the most challenging problems in cloud physics. Cloud droplet growth is neither dominated by condensation nor gravitational collision in the size range of 15 mu m similar to 40 mu m [1]. Turbulence-generated collection has been thought to be the mechanism to bridge the size gap, i.e., the bottleneck problem. This study compares the Lagrangian and Eulerian schemes in detail to tackle with the turbulence-generated collection.

The bottleneck problem of cloud droplet growth is one of the most challenging problems in cloud physics. Cloud droplet growth is neither dominated by con-densation nor gravitational collision in the size range of 15–40 μm in radius. Turbulence-generated collision has been thought to be the mechanism to bridge the size gap, i.e., the bottleneck problem. This study develops the numerical approaches to study droplet growth in atmospheric turbulence and investigates the turbulence effect on cloud droplet growth. The collision process of in-ertial particles in turbulence is strongly nonlinear, which motivates the study of two distinct numerical schemes. An Eulerian-based numerical formulation for the Smoluchowski equation in multi-dimensions and a Monte Carlo-type Lagrangian scheme have been developed to study the combined collision and condensation processes. We first investigate the accuracy and reliability of the two schemes in a purely gravitational field and then in a straining flow. Discrepancies between different schemes are most strongly exposed when con-densation and coagulation are studied separately, while their combined effects tend to result in smaller discrepancies. We find that for pure collision simulated by the Eulerian scheme, the mean particle radius slows down using finer massbins, especially for collisions caused by different terminal velocities. For the case of Lagrangian scheme, it is independent of grid resolution at early times and weakly dependent at later times. Comparing the size spectra simulated by the two schemes, we find that the agreement is excellent at early times. For pure condensation, we find that the numerical solution of condensation by the Lagrangian model is consistent with the analytical solution in early times. The Lagrangian schemes are generally found to be superior over the Eulerian one interms of computational performance. Moreover, the growth of cloud droplets in a turbulent environment is investigated as well. The agreement between the two schemes is excellent for both mean radius and size spectra, which gives us further insights into the accuracy of solving this strongly coupled nonlinear system. Turbulence broadens the size spectra of cloud droplets with increasing Reynolds number.