Research output: Contribution to journal › Article

**Revisiting Batchelor's theory of two-dimensional turbulence.** / Dritschel, David Gerard; Tran, Chuong Van; Scott, Richard Kirkness.

Research output: Contribution to journal › Article

Dritschel, DG, Tran, CV & Scott, RK 2007, 'Revisiting Batchelor's theory of two-dimensional turbulence', *Journal of Fluid Mechanics*, vol. 591, pp. 379-391. https://doi.org/10.1017/S0022112007008427

Dritschel, D. G., Tran, C. V., & Scott, R. K. (2007). Revisiting Batchelor's theory of two-dimensional turbulence. *Journal of Fluid Mechanics*, *591*, 379-391. https://doi.org/10.1017/S0022112007008427

Dritschel DG, Tran CV, Scott RK. Revisiting Batchelor's theory of two-dimensional turbulence. Journal of Fluid Mechanics. 2007 Nov 25;591:379-391. https://doi.org/10.1017/S0022112007008427

@article{d99be448835f4bc09f6f104f290bc772,

title = "Revisiting Batchelor's theory of two-dimensional turbulence",

abstract = "Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X-2/3 k(-1) enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re -> infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(-1) with (omega(2))k(-1)(In Re)(-1).",

keywords = "Dimensional decaying turbulence, Euler equations, Enstrophy dissipation, Contour dynamics, Self-similarity, High-resolution, Energy, Limit",

author = "Dritschel, {David Gerard} and Tran, {Chuong Van} and Scott, {Richard Kirkness}",

year = "2007",

month = "11",

day = "25",

doi = "10.1017/S0022112007008427",

language = "English",

volume = "591",

pages = "379--391",

journal = "Journal of Fluid Mechanics",

issn = "0022-1120",

publisher = "CAMBRIDGE UNIV PRESS",

}

TY - JOUR

T1 - Revisiting Batchelor's theory of two-dimensional turbulence

AU - Dritschel, David Gerard

AU - Tran, Chuong Van

AU - Scott, Richard Kirkness

PY - 2007/11/25

Y1 - 2007/11/25

N2 - Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X-2/3 k(-1) enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re -> infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(-1) with (omega(2))k(-1)(In Re)(-1).

AB - Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X-2/3 k(-1) enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re -> infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(-1) with (omega(2))k(-1)(In Re)(-1).

KW - Dimensional decaying turbulence

KW - Euler equations

KW - Enstrophy dissipation

KW - Contour dynamics

KW - Self-similarity

KW - High-resolution

KW - Energy

KW - Limit

UR - http://www.scopus.com/inward/record.url?scp=40449132762&partnerID=8YFLogxK

U2 - 10.1017/S0022112007008427

DO - 10.1017/S0022112007008427

M3 - Article

VL - 591

SP - 379

EP - 391

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -

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## Journal of Fluid Mechanics (Journal)

David Gerard Dritschel (Editor)2005 → …Activity: Publication peer-review and editorial work types › Editor of research journal

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ID: 397186