Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena

Majda, Andrew J.; Kramer, Peter R.

doi:10.1016/s0370-1573(98)00083-0

Cited by 477 publications

(587 citation statements)

References 279 publications

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“…Exactly as in the proof of Lemma 4.1, we obtain 18) where c > 0 is independent of L, A and h. We remark that the extra 1/ √ h factor arises because derivatives of η 0 are of the order 1/ √ h and they appear as multiplicative factors in the expressions for η 1 and η 2 .…”

Section: The Upper Boundsupporting

confidence: 69%

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From homogenization to averaging in cellular flows

Iyer

Komorowski

Novikov

et al. 2014

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a twodimensional domain with L 2 cells. For fixed A, and L → ∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution equilibrates along stream lines.In this paper, we show that if both A → ∞ and L → ∞, then a transition between the homogenization and averaging regimes occurs at A ≈ L 4 . When A ≫ L 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A ≪ L 4 , the principal eigenvalue behaves likeσ(A)/L 2 , whereσ(A) ≈ √ AI is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent L p → L ∞ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.

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Section: The Upper Boundsupporting

confidence: 69%

“…First, for any fixed A, we letσ(A) = (σ ij (A)) denote the effective diffusion matrix obtained in the limit ε → 0 (see [18] for a comprehensive review). If χ A j is the mean zero, 2-periodic solution to − ∆χ A j + Av(x) · ∇χ j = −Av j (x), for j ∈ {1, 2}, (1.7)…”

Section: Introductionmentioning

confidence: 99%

From homogenization to averaging in cellular flows

Iyer

Komorowski

Novikov

et al. 2014

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…All these methods lead to finite diffusion coefficient of Bohm type in the limit K → ∞, which shows that they are not adequate for the two-dimensional incompressible velocity fields. This process was studied especially by means of direct numerical simulations ( [32] and the reference there in) or by developing simplified models [33]. There is a theoretical estimation [34] based on the analogy with the fractal structure of the landscapes.…”

Section: The Decorrelation Trajectory Methods (Dtm)mentioning

confidence: 99%

Trajectory statistics and turbulence evolution

Vlad

Spineanu

2015

Chaos, Solitons & Fractals

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The aim of this paper is to understand the tendency to organization of the turbulence in two-dimensional ideal fluids. We show that nonlinear processes as inverse cascade of the energy and vorticity concentration are essentially determined by trajectory trapping or eddying. The statistics of the trajectories of the vorticity elements is studied using a semianalytic method. The separation of the positive and negative vorticity is due to the attraction produced by a large scale vortex on the small scale vortices of the same sign. More precisely, a large scale velocity is shown to determine average transverse drifts, which have opposite orientations for positive and negative vorticity. They appear in the presence of trapping and lead to energy flow to large scales due to the increase of the circulation of the large vortex. Recent results on drift turbulence evolution in magnetically confined plasmas are discussed in order to underline the idea that there is a link between the inverse cascade and trajectory trapping. The physical mechanisms are different in fluids and plasmas due to the different types of nonlinearities of the two systems, but trajectory trapping has the main role in both cases.

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“…The challenge is thus to reduce the typically huge number of degrees of freedom by modeling the fastest and smallest scales. In this way one may construct a computationally tractable effective equation, which involves only the scales one is interest in [5,6].…”

Section: Introductionmentioning

confidence: 99%

“…In this limit the goal is to derive the expression of the asymptotic diffusion coefficient renormalized by the presence of the small scale velocity field. This can be accomplished exploiting asymptotic methods (see, e.g., [6,8,9,10,11,12,13,14] among the others). However, in many physical circumstances one has that the velocity field may be though as a smallscale advecting velocity field (at scale ℓ) superimposed to a large-scale, slowly varying component (at scale L ≫ ℓ).…”

Section: Introductionmentioning

confidence: 99%

Large-scale effects on meso-scale modeling for scalar transport

Cencini

Mazzino

Musacchio³

et al. 2006

Physica D: Nonlinear Phenomena

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The transport of scalar quantities passively advected by velocity fields with a smallscale component can be modeled at meso-scale level by means of an effective drift and an effective diffusivity, which can be determined by means of multiple-scale techniques. We show that the presence of a weak large-scale flow induces interesting effects on the meso-scale scalar transport. In particular, it gives rise to non-isotropic and non-homogeneous corrections to the meso-scale drift and diffusivity. We discuss an approximation that allows us to retain the second-order effects caused by the large-scale flow. This provides a rather accurate meso-scale modeling for both asymptotic and pre-asymptotic scalar transport properties. Numerical simulations in model flows are used to illustrate the importance of such large-scale effects.

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Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena

Cited by 477 publications

References 279 publications

From homogenization to averaging in cellular flows

From homogenization to averaging in cellular flows

Trajectory statistics and turbulence evolution

Large-scale effects on meso-scale modeling for scalar transport

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