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Register a new userLocalization and advectional spreading of convective currents under parametric disorder
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We address a problem which is mathematically reminiscent of the one of Anderson localization, although it is related to a strongly dissipative dynamics. Specifically, we study thermal convection in a horizontal porous layer heated from below in the presence of a parametric disorder; physical parameters of the layer are time-independent and randomly inhomogeneous in one of the horizontal directions. Under such a frozen parametric disorder, spatially localized flow patterns appear. We focus our study on their localization properties and the effect of an imposed advection along the layer on these properties. Our interpretation of the results of the linear theory is underpinned by numerical simulation for the nonlinear problem. Weak advection is found to lead to an upstream delocalization of localized current patterns. Due to this delocalization, the transition from a set of localized patterns to an almost everywhere intense global flow can be observed under conditions where the disorder-free system would be not far below the instability threshold. The results presented are derived for a physical system which is mathematically described by a modified Kuramoto-Sivashinsky equation and therefore they are expected to be relevant for a broad variety of dissipative media where pattern selection occurs.
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We study transport of a weakly diffusive pollutant (a passive scalar) by thermoconvective flow in a fluid-saturated horizontal porous layer heated from below under frozen parametric disorder. In the presence of disorder (random frozen inhomogeneities of the heating or of macroscopic properties of the porous matrix), spatially localized flow patterns appear below the convective instability threshold of the system without disorder. Thermoconvective flows crucially effect the transport of a pollutant along the layer, especially when its molecular diffusion is weak. The effective (or eddy) diffusivity also allows to observe the transition from a set of localized currents to an almost everywhere intense global flow. We present results of numerical calculation of the effective diffusivity and discuss them in the context of localization of fluid currents and the transition to a global flow. Our numerical findings are in a good agreement with the analytical theory we develop for the limit of a small molecular diffusivity and sparse domains of localized currents. Though the results are obtained for a specific physical system, they are relevant for a broad variety of fluid dynamical systems.
We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a new first-passage-time problem for Lagrangian tracers that are advected by a turbulent flow: By direct numerical simulations of the three-dimensional (3D) incompressible, Navier-Stokes equation, we obtain the time $t_R$ at which a tracer, initially at the origin of a sphere of radius $R$, crosses the surface of the sphere textit{for the first time}. We obtain the probability distribution function $mathcal{P}(R,t_R)$ and show that it displays two qualitatively different behaviors: (a) for $R ll L_{rm I}$, $mathcal{P}(R,t_R)$ has a power-law tail $sim t_R^{-alpha}$, with the exponent $alpha = 4$ and $L_{rm I}$ the integral scale of the turbulent flow; (b) for $l_{rm I} lesssim R $, the tail of $mathcal{P}(R,t_R)$ decays exponentially. We develop models that allow us to obtain these asymptotic behaviors analytically. We show how to use $mathcal{P}(R,t_R)$ to develop social-distancing guidelines for the mitigation of the spreading of airborne aerosols with viruses such as SARS-CoV-2.
We study the spreading and leveling of a gravity current in a Hele-Shaw cell with flow-wise width variations as an analog for flow {in fractures and horizontally heterogeneous aquifers}. Using phase-plane analysis, we obtain second-kind self-similar solutions to describe the evolution of the gravity currents shape during both the spreading (pre-closure) and leveling (post-closure) regimes. The self-similar theory is compared to numerical simulations of the partial differential equation governing the evolution of the currents shape (under the lubrication approximation) and to table-top experiments. Specifically, simulations of the governing partial differential equation from lubrication theory allow us to compute a pre-factor, which is textit{a priori} arbitrary in the second-kind self-similar transformation, by estimating the time required for the current to enter the self-similar regime. With this pre-factor calculated, we show that theory, simulations and experiments agree well near the propagating front. In the leveling regime, the currents memory resets, and another self-similar behavior emerges after an adjustment time, which we estimate from simulations. Once again, with the pre-factor calculated, both simulations and experiments are shown to obey the predicted self-similar scalings. For both the pre- and post-closure regimes, we provide detailed asymptotic (analytical) characterization of the universal current profiles that arise as self-similarity of the second kind.
The phenomenon of localization usually happens due to the existence of disorder in a medium. Nevertheless, certain quantum systems allow dynamical localization solely due to the nature of internal interactions. We study a discrete time quantum walker which exhibits disorder free localization. The quantum walker moves on a one-dimensional lattice and interacts with on-site spins by coherently rotating them around a given axis at each step. Since the spins do not have dynamics of their own, the system poses the local spin components along the rotation axis as an extensive number of conserved moments. When the interaction is weak, the spread of the walker shows subdiffusive behaviour having downscaled ballistic tails in the evolving probability distribution at intermediate time scales. However, as the interaction gets stronger the walker gets exponentially localized in the complete absence of disorder in both lattice and initial state. Using a matrix-product-state ansatz, we investigate the relaxation and entanglement dynamics of the on-site spins due to their coupling with the quantum walker. Surprisingly, we find that even in the delocalized regime, entanglement growth and relaxation occur slowly unlike marjority of the other models displaying a localization transition.
We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a Taylor-Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using DNS and find that its emerging butterfly-type structure -- a spatio-temporal variation in the form of upward and downward traveling waves -- is in a very good agreement with the linear stability analysis, which indicates the presence of two dominant absolute AMRI modes in the flow giving rise to this global butterfly pattern.
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