### Abstract

We analyze from first principles the advection of particles in a velocity field with Hamiltonian H(x, y)=- V_{1}y-- V_{2}x+W_{1}(y)-W_{2}(x), where W_{i}, i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratio ρ=max(|-V_{1}|/Ū; |-V_{2}|/Ū), with Ū=〈|W_{1}′|^{2}〉^{1/2}=rms fluctuations. Remarkably, the model is exactly solvable for ρ ≥1 and well suited for Monte Carlo simulations for all ρ, providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines for ρ=0, deriving power laws for p_{nc}(L) and 〈λ(L)〉, which are, respectively, the escape probability and the length of escaping trajectories for a box of size L, L » 1. We also obtain a characterization of the "statistical topography" of the Hamiltonian H. Second, we study the large-scale transport of advected particles with ρ > 0. For 0 <ρ < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for ρ≥ 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variables t^{-}v/2[x(t)-〈x(t)〉] and t^{-v/2}[y(t)-〈(t)〉]. The large-scale motions are shown to be either Fickian (ν=1), or superdiffusive (ν=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for ρ ≥ 1 and extended to the regime 0<ρ<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of {Mathematical expression}) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.

Original language | English (US) |
---|---|

Pages (from-to) | 1227-1304 |

Number of pages | 78 |

Journal | Journal of Statistical Physics |

Volume | 72 |

Issue number | 5-6 |

DOIs | |

State | Published - Sep 1993 |

### Fingerprint

### Keywords

- percolation
- superdiffusion
- Trapping

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*72*(5-6), 1227-1304. https://doi.org/10.1007/BF01048187

**Trapping, percolation, and anomalous diffusion of particles in a two-dimensional random field.** / Avellaneda, Marco; Elliott, Frank; Apelian, Christopher.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 72, no. 5-6, pp. 1227-1304. https://doi.org/10.1007/BF01048187

}

TY - JOUR

T1 - Trapping, percolation, and anomalous diffusion of particles in a two-dimensional random field

AU - Avellaneda, Marco

AU - Elliott, Frank

AU - Apelian, Christopher

PY - 1993/9

Y1 - 1993/9

N2 - We analyze from first principles the advection of particles in a velocity field with Hamiltonian H(x, y)=- V1y-- V2x+W1(y)-W2(x), where Wi, i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratio ρ=max(|-V1|/Ū; |-V2|/Ū), with Ū=〈|W1′|2〉1/2=rms fluctuations. Remarkably, the model is exactly solvable for ρ ≥1 and well suited for Monte Carlo simulations for all ρ, providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines for ρ=0, deriving power laws for pnc(L) and 〈λ(L)〉, which are, respectively, the escape probability and the length of escaping trajectories for a box of size L, L » 1. We also obtain a characterization of the "statistical topography" of the Hamiltonian H. Second, we study the large-scale transport of advected particles with ρ > 0. For 0 <ρ < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for ρ≥ 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variables t-v/2[x(t)-〈x(t)〉] and t-v/2[y(t)-〈(t)〉]. The large-scale motions are shown to be either Fickian (ν=1), or superdiffusive (ν=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for ρ ≥ 1 and extended to the regime 0<ρ<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of {Mathematical expression}) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.

AB - We analyze from first principles the advection of particles in a velocity field with Hamiltonian H(x, y)=- V1y-- V2x+W1(y)-W2(x), where Wi, i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratio ρ=max(|-V1|/Ū; |-V2|/Ū), with Ū=〈|W1′|2〉1/2=rms fluctuations. Remarkably, the model is exactly solvable for ρ ≥1 and well suited for Monte Carlo simulations for all ρ, providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines for ρ=0, deriving power laws for pnc(L) and 〈λ(L)〉, which are, respectively, the escape probability and the length of escaping trajectories for a box of size L, L » 1. We also obtain a characterization of the "statistical topography" of the Hamiltonian H. Second, we study the large-scale transport of advected particles with ρ > 0. For 0 <ρ < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for ρ≥ 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variables t-v/2[x(t)-〈x(t)〉] and t-v/2[y(t)-〈(t)〉]. The large-scale motions are shown to be either Fickian (ν=1), or superdiffusive (ν=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for ρ ≥ 1 and extended to the regime 0<ρ<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of {Mathematical expression}) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.

KW - percolation

KW - superdiffusion

KW - Trapping

UR - http://www.scopus.com/inward/record.url?scp=21144480095&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144480095&partnerID=8YFLogxK

U2 - 10.1007/BF01048187

DO - 10.1007/BF01048187

M3 - Article

VL - 72

SP - 1227

EP - 1304

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -