Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension $D_{0}$ of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology.
Deep Dive into Signatures of fractal clustering of aerosols advected under gravity.
Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension $D_{0}$ of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology.
The transport of finite-size particles plays important role in several fields, from cloud physics [1] to plankton dynamics [2]. The recent interest [3,4] in this problem comes in part from the fact that the dynamics of these particles is dissipative due to the drag force. This makes dynamical systems tools and concepts, such as attractors and dimensions, applicable. In the presence of gravity, a net vertical motion occurs due to the density difference between fluid and particle. For heavy particles (aerosols), this leads to raindrop falling in the atmosphere [5] and to the sedimentation of plankton [2] and marine snow [6] in the ocean.
In such situations, knowledge of the advection dynamics of the aerosols is of fundamental importance, whereas usually only data obtained at a given level (height) are available. This occurs often, for instance, in meteorology, in the case where the aerosols are raindrops. In fact, it is much easier to obtain direct measurements of the raindrops when they reach the ground level than before, i.e., when they are being advected in the air flow. The derivation of approaches to obtain information on the advection dynamics of aerosols solely from data observed at a given level is, therefore, an instrumental and relevant task. In particular, here we are interested in approaches to obtain the fractal dimension of the set where the aerosols cluster while they are advected.
We report the uncovering of two independent fractal signatures of chaotic advection under gravity. Both make the computation of the fractal dimension D 0 of the strange attractor in the N -dimensional configuration space possible without prior knowledge of the advection dynamics. First, we show that the time series of the instants of arrival of advected aerosols in a small detector placed at a given level has a fractal dimension which is equal to
We assume that D 0 < N , which implies that the attractor in the full 2N -dimensional phase space is D 0dimensional since a set of dimension D 0 , when projected into a space of dimension N , typically remains
. Second, we show that the spatial distribution of the aerosols reaching a line at a given level contains discontinuities (jumps) at points that form a fractal set whose dimension is again equal to d 0 [8]. We illustrate our findings with a numerical experiment.
The dimensionless form of the governing equation for the path r(t) of aerosols much denser than the fluid, subjected to Stokes drag and gravity, reads as [9]:
where ṙ is the velocity of the aerosol, u = u(r(t), t) is the fluid velocity field evaluated at the position r(t) of the aerosol, and n is a unit vector pointing upwards in the vertical direction. Throughout this paper we consider the vertical direction along the axis y. The inertia parameter A (larger values for smaller inertia) can be written in terms of the densities ρ p and ρ f of the aerosol and of the fluid, respectively, the radius a of the aerosols, the fluid’s kinematic viscosity ν, and the characteristic length L and velocity U of the flow. It is A = R/St, where R = ρ f /ρ p ≪ 1 and St = (2a 2 U )/(9νL) is the Stokes’s number of the aerosol. As seen from ( 2), the gravitational parameter W provides the dimensionless settling velocity in a medium at rest. The actual settling velocity is the result of two effects: the gravitational attraction (buoyancy) and an updraft, if present. We consider the settling velocity to be comparable with U , implying a W of the order of unity.
For convenience, we treat the case where the fluid flow is two-dimensional, N = 2. In this situation, the phase space of the advection dynamics of the aerosols is 4-dimensional, since the aerosols are not constrained to move with the same velocity as their corresponding fluid elements. For the sake of concreteness, let us consider the time-smoothened version of the alternating sinusoidal shear flow of Ref. [10]. In dimensionless form it is given as u x (r, t) = 0.5 (1 + tanh(γ sin(2πt))) sin(2πy), u y (r, t) = 0.5(1 -tanh(γ sin(2πt))) sin(2πx).
(3) This flow is defined on the unit square with periodic boundary conditions, and is periodic in time with a unit period. The vertical direction corresponds to the y-axis.
Apart from the spatial (sinusoidal) factor, each velocity component consists of two plateaus in time with a rapid but smooth crossover if γ = 20/π. This is a simple analytically given model which, nevertheless, possesses a paradigmatic property of both chaotic and turbulent flows [11]: the intense stretching of material elements. Substituting Eq. ( 3) into Eq. ( 2) and fixing R = 10 -3 and W = 0.8, there are regions of the parameter St for which the dynamics of the aerosols is ruled by a strange attractor. We note that the time-independence of the attractor’s dimension D 0 , which will be essential in what follows, is valid for a broad class of randomly timedependent flows as well (see conclusion). A bifurcation diagram and our analysis suggest the existence
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