We show quantitatively how the collision rate of droplets of visible moisture in turbulent air increases very abruptly as the intensity of the turbulence passes a threshold, due to the formation of fold caustics in their velocity field. The formation of caustics is an activated process, in which a measure of the intensity of the turbulence, termed the Stokes number St, is analogous to temperature in a chemical reaction: the rate of collision contains a factor exp(−C/St). Our results are relevant to the long-standing problem of explaining the rapid onset of rainfall from convecting clouds. Our theory does not involve spatial clustering of particles.PACS numbers: 45.50 Tn, 47.55 D, 92.60 Mt, 92.60 Nv It is common experience that rainfall can commence very abruptly from cumulus clouds (which form when the atmosphere is convecting) but has a much slower onset from stratiform clouds in a stable atmosphere. This can happen even when no part of the cloud is below freezing point. It is believed that the difference between convecting and stable clouds arises because the convection gives rise to small-scale turbulent motion which facilitates the coalescence of microscopic water droplets ('visible moisture') into raindrops. This idea has a long history ([1] is a significant contribution containing references to other early papers), but a satisfying theory has been elusive and the topic remains a subject of intensive research, reviewed recently in [2]. Numerical experiments have shown a dramatic increase in the rate of collision of suspended particles when the intensity of turbulence exceeds a certain threshold. This effect was first described in [3], see also [4]. Here we present a simple quantitative theory of this phenomenon, illustrated in figure 1, which shows the collision rate R of an aerosol as a function of a dimensionless parameter termed the Stokes number, St, which contains information about the turbulence intensity, ǫ, and the radius of the water droplets, a (other symbols in the caption are defined later). There is a precipitous increase in the collision rate at a threshold value of St, which was also observed in [3,4]. The current consensus, represented in [2,3,4,5], is that the increased rate of collision involves spatial clustering of particles. One exception is [6], which presents a theory having elements (described later) in common with our own, but which is more complex in its formulation and less precise in its conclusions. Spatial clustering plays no role in our theory.The properties of clouds are very variable and the sizes of visible moisture droplets have a large dispersion, but typically the average of the radius is approximately 10µm and the density is n ≈ 10 8 m −3 [2]. The motion of the droplets is dominated by viscous forces, so that the equation for the position r of a droplet is well approximated byruntil the particles come into contact [1] (here u(r, t) is the velocity field of the air and dots denote derivatives with respect to time). According to Stokes's formula for the viscous drag on...
We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [G. E. Ornstein and L. S. Uhlenbeck, Phys. Rev. 36, 823, (1930)]. Our generalised Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.
We consider the patterns formed by small rod-like objects advected by a random flow in two dimensions. An exact solution indicates that their direction field is non-singular. However, we find from simulations that the direction field of the rods does appear to exhibit singularities. First, 'scar lines' emerge where the rods abruptly change direction by π. Later, these scar lines become so narrow that they 'heal over' and disappear, but their ends remain as point singularities, which are of the same type as those seen in fingerprints. We give a theoretical explanation for these observations.
We discuss the reflection of light by a rheoscopic fluid (a suspension of microscopic rod-like crystals) in a steady two-dimensional flow. This is determined by an order parameter which is a non-oriented vector, obtained by averaging solutions of a nonlinear equation containing the strain rate of the fluid flow. Exact solutions of this equation are obtained from solutions of a linear equation which are analogous to Bloch bands for a one-dimensional Schrodinger equation with a periodic potential. On some contours of the stream function, the order parameter approaches a limit, and on others it depends increasingly sensitively upon position. However, in the long-time limit a local average of the order parameter is a smooth function of position in both cases. We analyse the topology of the order parameter and the structure of the generic zeros of the order parameter field.Comment: 28 pages, 13 figure
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