Effect of particle inertia on the alignment of small ice crystals in turbulent clouds

Small non-spherical particles settling in a quiescent fluid tend to orient so that their broad side faces down, because this is a stable fixed point of their angular dynamics at small particle Reynolds number. Turbulence randomises the orientations to some extent, and this affects the reflection patterns of polarised light from turbulent clouds containing ice crystals. An overdamped theory predicts that turbulence-induced fluctuations of the orientation are very small when the settling number Sv (a dimensionless measure of the settling speed) is large. At small Sv, by contrast, the overdamped theory predicts that turbulence randomises the orientations. This overdamped theory neglects the effect of particle inertia. Therefore we consider here how particle inertia affects the orientation of small crystals settling in turbulent air. We find that it can significantly increase the orientation variance, even when the Stokes number St (a dimensionless measure of particle inertia) is quite small. We identify different asymptotic parameter regimes where the tilt-angle variance is proportional to different inverse powers of Sv. We estimate parameter values for ice crystals in turbulent clouds and show that they cover several of the identified regimes. The theory predicts how the degree of alignment depends on particle size, shape and turbulence intensity, and that the strong horizontal alignment of small crystals is only possible when the turbulent energy dissipation is weak, of the order of 1cm2/s3 or less.

Synchronization And State Estimation Of Nonlinear Systems With Unknown Time-Delays: Adaptive Identification Method

The paper studies a novel adaptive identifier proposed in IFAC World Congress 2020 for nonlinear time-delay systems composed of linear, Lipschitz and non-Lipschitz components. To begin with, an identifier is designed for uncertain systems with a priori known delay values, and then it is generalized for systems with unknown delay values. The algorithm ensures the asymptotic parameter estimation and state observation by using gradient algorithms. The unknown delays and plant parameters are estimated by using a special equivalent extension of the plant equation. The algorithms stability is presented by solvability of linear matrix inequalities. Simulation results are invoked to support the developed identifier design and to illustrate the efficiency of the proposed synthesis procedure.

Study of an asymptotic preserving scheme for the quasi neutral Euler–Boltzmann model in the drift regime

We deal with the numerical approximation of a simplified quasi neutral plasma model in the drift regime. Specifically, we analyze a finite volume scheme for the quasi neutral Euler–Boltzmann equations. We prove the unconditional stability of the scheme and give some bounds on the numerical approximation that are uniform in the asymptotic parameter. The proof relies on the control of the positivity and the decay of a discrete energy. The severe non linearity of the scheme being the price to pay to get the unconditional stability, to solve it, we propose an iterative linear implicit scheme that reduces to an elliptic system. The elliptic system enjoys a maximum principle that enables to prove the conservation of the positivity under a CFL condition that does not involve the asymptotic parameter. The linear L2 stability analysis of the iterative scheme shows that it does not request the mesh size and time step to be smaller than the asymptotic parameter. Numerical illustrations are given to illustrate the stability and consistency of the scheme in the drift regime as well as its ability to compute correct shock speeds.

Initial Layer Analysis for a Linkage Density in Cell Adhesion Mechanisms

In this paper we present a non local age structured equation involved in cell motility modeling [5, 9, 11]. It describes the evolution of a density of linkages of a point submitted to adhesion. It depends on an asymptotic parameter ɛ representing the characteristic age of linkages. Here we introduce a new initial layer term in the asymptotic expansion with respect to ɛ. This improves error estimates obtained in [5]. Moreover, we study the convergence of the time derivative of this density and show how a singular term appears when ɛ goes to zero. We show convergence, in the tight topology of measures, to the time derivative of the limit solution and a Dirac mass supported on the initial half-axis. In order to illustrate these results, numerical simulations are performed and compared to the asymptotic expansion for various values of ɛ.

Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules

We prove asymptotic normality of the distributions defined by $q$-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of $\frak{sl}_2$. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to $\frak{sl}_2$. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type $A$ standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, $N$ say.

Asymptotic Parameter Estimation for a Class of Linear Stochastic Systems Using Kalman-Bucy Filtering

The asymptotic parameter estimation is investigated for a class of linear stochastic systems with unknown parameterθ:dXt=(θα(t)+β(t)Xt)dt+σ(t)dWt. Continuous-time Kalman-Bucy linear filtering theory is first used to estimate the unknown parameterθbased on Bayesian analysis. Then, some sufficient conditions on coefficients are given to analyze the asymptotic convergence of the estimator. Finally, the strong consistent property of the estimator is discussed by comparison theorem.

An Optimal Analytic Approximate Solution for the Limit Cycle of Duffing–van der Pol Equation

The present paper is concerned with the accurate analytic solution of the limit cycle of the Duffing–van der Pol equation. Instead of the traditional Taylor series or asymptotic methods, the homotopy analysis technique is employed, which does not require a small perturbation parameter or a large asymptotic parameter. It is known that such a method is extremely powerful in gaining the exact solution of the physical problem in terms of purely trigonometric functions, yet the computational cost of the method is considerably high. We propose here an approach that not only greatly reduces the computational efforts but also presents an easy to implement task of application of the homotopy analysis method to the Duffing–van der Pol equation. The explicit analytical expressions obtained using the proposed approach generates the displacement, amplitude, and frequency of the limit cycle that compare excellently with the numerically computed ones.