Passive Scalar Diffusion Research Articles

SLE on Doubly-Connected Domains and the Winding of Loop-Erased Random Walks

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLEκ with parameter κ=2. In this note, some properties of an SLEκ trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE2 on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE2 with fixed endpoints. A solution for the endpoint distribution of SLE4 on the cylinder is obtained and a relation to reflected Brownian motion pointed out.

Slowly oscillating modes in the passive scalar diffusion in stratified turbulence

Scalar fluxes in unsteady stratified turbulence are investigated when vertical mean gradients of both the density (active scalar) and the passive scalar exist. Analytical solutions by the rapid distortion theory show that there is a “slowly” oscillating mode in the passive scalar flux, whose oscillation period is twice as large as that of the density flux. For passive scalars with high Schmidt number Sc>2Pr∕(1+Pr) (Pr: Prandtl number of density), this slow mode with small initial decay becomes dominant, so that temporal oscillation of Kc (turbulent diffusion coefficient of the passive scalar) persists with larger amplitude than the density diffusion coefficient Kρ. On the other hand, at low Schmidt number Sc<2Pr∕(1+Pr)<2, slow mode suffers strong initial decay, so that Kc and Kρ decay at the similar rate. These results illustrate the importance of molecular diffusivity and unsteadiness in weak turbulence which would make significant difference between Kc and Kρ. A criterion for which the linear mechanisms become important is also derived. When the buoyancy Reynolds number Reb (=ϵ∕νN2, ϵ: energy dissipation rate; N: Brunt–Väisälä frequency) is smaller than Re(l0)[=u(l0)l0∕ν], i.e., the Reynolds number defined at the integral scale l0, turbulent energies, and fluxes would be affected by linear processes. This suggests that large value of Reb does not necessarily mean strong nonlinearity, and the differential diffusion previously observed in stratified turbulence for Reb≲103 could be due to linear processes.

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Peclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Peclet), to two algebraic singularities at both unit and zero scalar values (large Peclet).

An upper bound for passive scalar diffusion in shear flows

This study is concerned with the diffusion of a passive scalar Θ(r,t) advected by general n-dimensional shear flows u=u(y,z,…,t)x̂ having finite mean-square velocity gradients. The unidirectionality of the incompressible flows conserves the streamwise scalar gradient, ∂xΘ, allowing only the cross-stream components to be amplified by shearing effects. This amplification is relatively weak because an important contributing factor, ∂xΘ, is conserved, effectively rendering a slow diffusion process. It is found that the decay of the scalar variance ⟨Θ2⟩ satisfies d⟨Θ2⟩∕dt≥−Cκ1∕3, where C>0 is a constant, depending on the fluid velocity gradients and initial distribution of Θ, and κ is the molecular diffusivity. This result generalizes to axisymmetric flows on both the plane and sphere having finite mean-square angular velocity gradients.

1006 二重拡散の成層乱流におけるアクティブスカラーとパッシブスカラー拡散の相違(1)(OS10-2 大気海洋の流体現象と環境流体,OS10 大気海洋の流体現象と環境流体,オーガナイズドセッション)

1006 二重拡散の成層乱流におけるアクティブスカラーとパッシブスカラー拡散の相違(1)(OS10-2 大気海洋の流体現象と環境流体,OS10 大気海洋の流体現象と環境流体,オーガナイズドセッション)

Parameterization of meandering phenomenon in a stable atmospheric boundary layer

Accounting for the current knowledge of the stable atmospheric boundary layer (ABL) turbulence structure and characteristics, a new formulation for the meandering parameters to be used in a Lagrangian stochastic particle turbulent diffusion model has been derived. That is, expressions for the parameters controlling the meandering oscillation frequency in low wind speed stable conditions are proposed. The classical expression for the meandering autocorrelation function, the turbulent statistical diffusion theory and ABL similarity theory are employed to estimate these parameters. In addition, this new parameterization was introduced into a particular Lagrangian stochastic particle model, which is called Iterative Langevin solution for low wind, validated with the data of Idaho National Laboratory experiments, and compared with others diffusion models. The results of this new approach are shown to agree with the measurements of Idaho experiments and also with those of the other atmospheric diffusion models. The major advance shown in this study is the formulation of the meandering parameters expressed in terms of the characteristic scales (velocity and length scales) describing the physical structure of a turbulent stable boundary layer. These similarity formulas can be used to simulate meandering enhanced diffusion of passive scalars in a low wind speed stable ABL.

Rapid distortion theory for differential diffusion

Rapid distortion theory (RDT) is used to examine differential diffusion of active and passive scalars in unsheared, initially isotropic turbulence. RDT is well suited to study differential diffusion because it applies to strongly stratified flows with weak turbulence—that is, the conditions under which differential diffusion occurs. The theory reproduces several key features of the evolution of scalar fluxes and scalar flux spectra observed in direct numerical simulations (DNS). Predictions of the diffusivity ratio match laboratory results well when a parameter of the theory is related to a parameter of the experiments. RDT also allows parameters such as molecular diffusivities to be varied over a wider range than DNS can currently reach. RDT may prove to be a useful tool for computing mixing in weakly turbulent parts of the stratified ocean interior and possibly for parameterizing subgrid scale mixing in general circulation models.

A multi-relaxation lattice kinetic method for passive scalar diffusion

A kinetic Lattice-Boltzmann scheme for advection–diffusion is presented. The scheme is based on a matrix formulation of the lattice Boltzmann method which permits to handle non-isotropic diffusion–advection problems. In addition, by adjusting the kinetic eigenvectors defining the collision matrix to the local value of the flow velocity, the scheme is also shown to preserve isotropy under genuinely two-dimensional flow.

On probability density function equations for particle dispersion in a uniform shear flow

The paper examines a fundamental discrepancy between two probability density function (PDF) models, the kinetic model (KM) and generalized Langevin model (GLM), currently used to model the dispersion of particles in turbulent flows. This discrepancy is manifest in particle dispersion in an unbounded simple shear flow where model predictions for the values of the streamwise fluid-particle diffusion coefficients are not only different but are of opposite sign. It is shown that this discrepancy arises through a neglect of the inertial convection term in the GLM equation for the mean carrier flow velocity local to a particle which eventually leads to algebraic forms for the particle-fluid diffusion coefficients. Evaluating this term for a Gaussian process leads to identical results for both PDF formulations. This also resolves a fundamental long-standing discrepancy in previous forms reported for the passive scalar diffusion coefficients in a simple shear flow where similar assumptions were made. Avoiding this assumption, the exact solutions are given for the dispersion of particles in this simple shear flow case derived from the solution of the GLM PDF equation which show explicitly the dependence on the particle response time and the strain rate, both normalized on the integral timescale of the turbulence. The analysis shows that the particle diffusion coefficient in the streamwise direction is negative when the strain rate a certain value. The origin of negative diffusion coefficients is explained and their influence is shown in the way in which the mean concentration and mean velocity flow fields of the particle and carrier flow (seen by the particle) evolve with time for particles released from the centre of the shear.

Thermal Conduction in Magnetized Turbulent Gas

Using numerical methods, we systematically study in the framework of ideal MHD the effect of magnetic fields on heat transfer within a turbulent gas. We measure the rates of passive scalar diffusion within magnetized fluids and make the comparisons (1) between MHD and hydrodynamic simulations, (2) between different MHD runs with different values of the external magnetic field (up to the energy equipartition value), and (3) between thermal conductivities parallel and perpendicular to the magnetic field. We do not find apparent suppression of diffusion rates by the presence of magnetic fields, which implies that magnetic fields do not suppress heat diffusion by turbulent motions.

Turbulent thermal diffusion and barodiffusion of passive scalar and dispersed phase of particles in turbulent flows.

It is shown that direct interaction approximation closure solutions for passive scalar and dispersed inertial particles in compressible turbulent flows result in the phenomena, proposed by Elperin et al. [Phys. Rev. E 58, 3113 (1998)], of turbulent thermal diffusion and turbulent barodiffusion. A more rigorous analysis of Lagrangian history direct interaction approximation for the dispersed phase in the kinetic approach framework is used to accurately quantify the phenomena.

Sensitive enhanced diffusivities for flows with fluctuating mean winds: a two-parameter study

Enhanced diffusion coefficients arising from the theory of periodic homogenized averaging for a passive scalar diffusing in the presence of a large-scale, fluctuating mean wind superimposed upon a small-scale, steady flow with non-trivial topology are studied. The purpose of the study is to assess how the extreme sensitivity of enhanced diffusion coefficients to small variations in large-scale flow parameters previously exhibited for steady flows in two spatial dimensions is modified by either the presence of temporal fluctuation, or the consideration of fully three-dimensional steady flow. We observe the various mixing parameters (Péclet, Strouhal and periodic Péclet numbers) and related non-dimensionalizations. We document non-monotonic Péclet number dependence in the enhanced diffusivities, and address how this behaviour is camouflaged with certain non-dimensional groups. For asymptotically large Strouhal number at fixed, bounded Péclet number, we establish that rapid wind fluctuations do not modify the steady theory, whereas for asymptotically small Strouhal number the enhanced diffusion coefficients are shown to be represented as an average over the steady geometry. The more difficult case of large Péclet number is considered numerically through the use of a conjugate gradient algorithm. We consider Péclet-number-dependent Strouhal numbers, S = QPe−(1+γ), and present numerical evidence documenting critical values of γ which distinguish the enhanced diffusivities as arising simply from steady theory (γ < −1) for which fluctuation provides no averaging, fully unsteady theory (γ ∈ (−1, 0)) with closure coefficients plagued by non-monotonic Péclet number dependence, and averaged steady theory (γ > 0). The transitional case with γ = 0 is examined in detail. Steady averaging is observed to agree well with the full simulations in this case for Q [les ] 1, but fails for larger Q. For non-sheared flow, with Q [les ] 1, weak temporal fluctuation in a large-scale wind is shown to reduce the sensitivity arising from the steady flow geometry; however, the degree of this reduction is itself strongly dependent upon the details of the imposed fluctuation. For more intense temporal fluctuation, strongly aligned orthogonal to the steady wind, time variation averages the sensitive scaling existing in the steady geometry, and the present study observes a Pe1 scaling behaviour in the enhanced diffusion coefficients at moderately large Péclet number. Finally, we conclude with the numerical documentation of sensitive scaling behaviour (similar to the two-dimensional steady case) in fully three dimensional ABC flow.

Probability density function of a passive scalar gradient

Fractal and multi-fractal models are used to derive the probability density function (PDF) for the gradient of a passive scalar diffusing in a random velocity field. Although the PDF is symmetrical, it is non-Gaussian and compares reasonably well with the measured PDF. In the process, we also provide a unified systematic framework to derive the appropriate scalar microscales for the following regimes: inertial-diffusive range; viscous-convective range; viscous-diffusive range.

Direct numerical simulation of differential diffusion with Schmidt numbers up to 4.0

Highly resolved direct numerical simulations are used to study the differential diffusion of passive scalars of above-unity Schmidt number in low Reynolds number turbulence. Both the coherency and the spectrum of the two-scalar difference obey Batchelor scaling based on the parameters of the less diffusive scalar. A reverse spectral transfer effect due to velocity modes in the far dissipation range is identified, which may be important at very high Schmidt numbers.

An anelastic, scale-separated model for mixing, with application to atmospheric transport phenomena

We present and analyze a simplified, scale-separated, anelastic fluid model which is designed to assess the influence of weak compressibility in the diffusive transport of a passive scalar. Our model incorporates a slowly varying, density induced, anisotropy into a fixed, rapidly varying (small scale) fluid flow. This anisotropy is physically motivated through the anelastic mass balance which retains vertical density variations occurring over the atmospheric scale height (∼8 km). Consequently, these steady flows are nonsolenoidal over large scales and approximately incompressible on small scales. We apply homogenization methods to calculate the large scale, effective mixing experienced by a passive scalar diffusing in the presence of this small scale flow. Over large scales, the evolution of the scalar field is governed by an effective, variable coefficient, anelastic mixing equation. The variable coefficients entering this equation are shown to depend nontrivially upon both the large scale anisotropy as well as the structure of the small scale fluid flow. We establish that anelastic effects produce anisotropic mixing properties not shared by the analogous incompressible closure. Specifically, the mixing equation possesses exact nontrivial bounded steady states, whereas the incompressible regime has only constant (i.e., spatially homogeneous) steady states. Furthermore, the anelastic mixing equation is shown numerically to possess local regions of trapped (Fig. 3) and focused (Figs. 6 and 7) contaminants, behavior not possible in the analogous incompressible model. These results imply that anelastic effects, which occur naturally in the atmosphere, provide mechanisms which locally reduce mixing and generate inhomogeneities in large scale concentration fields.

The Anomalous Diffusion of the Self-Interacting Passive Scalar in the Turbulent Environment

Two variants of the statistical model of diffusing self-interacting passive scalar θ(x, t) driven by the incompressible Navier–Stokes turbulence were studied by means of the field-theoretical renormalization group technique and ∊-expansion scheme, where ∊ denotes the parameter of the forcing spectrum. Dual ∫ ddxdt[θ(x, t)]2 and triple ∫ ddxdt[θ(x, t)]3 interaction terms of the action represent two different mechanisms of the self-interaction matching two alternative values of the critical dimension: d c =4 and d c =6. The major part of the calculations was carried out in the one loop order, nevertheless, the inclusion of the specific two loop contributions represents the important step of the analysis of some renormalization group functions. In the basic model variant the effective action is renormalizable for the supercritical dimensions d > d c . This theory exhibits the presence of the asymptotical regime, which is stable for the inertial-conductive range of wave numbers. It was also shown that stability of this regime remains preserved for a variety of the parametric paths connecting domain ∊>0, d>d c with ∊<2, d=3. In the second model variant, the effective action is constructed to be renormalizable at dimensions d≥d c and to justify the realizability of the continuation from ∊>0, d>d c to ∊< 2, d=3. This variant of the model was analyzed using "double expansion" method with the expansion parameters (d-d c )/2 and ∊. The negative correction ζ(ζ≃0.039 for d=3) to the universal Richardson exponent 4/3 is the physical consequence stemming from the calculations.

The Meso-NH Atmospheric Simulation System. Part I: adiabatic formulation and control simulations

Abstract. The Meso-NH Atmospheric Simulation System is a joint effort of the Centre National de Recherches Météorologiques and Laboratoire d'Aérologie. It comprises several elements; a numerical model able to simulate the atmospheric motions, ranging from the large meso-alpha scale down to the micro-scale, with a comprehensive physical package, a flexible file manager, an ensemble of facilities to prepare initial states, either idealized or interpolated from meteorological analyses or forecasts, a flexible post-processing and graphical facility to visualize the results, and an ensemble of interactive procedures to control these functions. Some of the distinctive features of this ensemble are the following: the model is currently based on the Lipps and Hemler form of the anelastic system, but may evolve towards a more accurate form of the equations system. In the future, it will allow for simultaneous simulation of several scales of motion, by the so-called "interactive grid-nesting technique". It allows for the in-line computation and accumulation of various terms of the budget of several quantities. It allows for the transport and diffusion of passive scalars, to be coupled with a chemical module. It uses the relatively new Fortran 90 compiler. It is tailored to be easily implemented on any UNIX machine. Meso-NH is designed as a research tool for small and meso-scale atmospheric processes. It is freely accessible to the research community, and we have tried to make it as "user-friendly" as possible, and as general as possible, although these two goals sometimes appear contradictory. The present paper presents a general description of the adiabatic formulation and some of the basic validation simulations. A list of the currently available physical parametrizations and initialization methods is also given. A more precise description of these aspects will be provided in a further paper.

Scalar intermittency and the ground state of periodic Schrödinger equations

Recent studies of a passive scalar diffusing in a rapidly fluctuating Gaussian distributed linear shear layer have demonstrated intermittency in the form of broad tails and non-symmetric limiting probability distribution functions. In this paper the authors explore similar issues within the context of a large class of rapidly fluctuating bounded periodic shear layers. We compute the evolution of the moments by analogy to an N dimensional quantum mechanics problem. By direct comparison of an appropriate system of interacting and non-interacting quantum particles, we illustrate that the role of interaction is to induce a lowering of the ground state energy, which implies that the scalar PDF will have broader than Gaussian tails for all large, but finite times. We demonstrate for the case of Gaussian random wave initial data involving a zero spatial mean, that the effect of this energy shift is to induce diverging normalized flatness factors indicative of very broad tails. For the more general case with Gaussian random initial data involving a non-zero spatial mean, the distribution must approach that of a Gaussian at infinite times, as required by homogenization theory, but we show that the approach is highly non-uniform. In particular our calculation shows that the time required for the system to approach Gaussian statistics grows like the square of the moment number.

Probability distribution function for the gradient of a passive scalar diffusing in isotropic turbulence: Mapping-closure model.

A mapping-closure model is used to derive the probability distribution function (PDF) for the gradient of a passive scalar diffusing in a random velocity field. This PDF is non-Gaussian and shows exponential tails observed in laboratory experiments.

Intermediate dissipation range for passive scalars

The power-law scaling behavior via a multifractal model for a passive scalar diffusing in a random velocity field breaks down in an intermediate dissipation range. However, upon using a suitable rescaling, the multifractal in question is shown still to exhibit a multi-scaling behavior with certain universal features.