TY - JOUR

T1 - Vanishing enstrophy dissipation in two-dimensional Navier--Stokes turbulence in the inviscid limit

AU - Tran, Chuong Van

AU - Dritschel, David Gerard

PY - 2006/7/25

Y1 - 2006/7/25

N2 - Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re -> infinity, the dissipation approaches zero no slower than (ln Re)(-1/2). The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation - but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

AB - Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re -> infinity, the dissipation approaches zero no slower than (ln Re)(-1/2). The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation - but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

KW - Quasi-geostrophic turbulence

KW - Spectral distribution

KW - Energy

KW - Decay

KW - Equations

UR - http://www.scopus.com/inward/record.url?scp=33746335656&partnerID=8YFLogxK

UR - http://journals.cambridge.org/action/displayIssue?iid=454619

U2 - 10.1017/S0022112006000577

DO - 10.1017/S0022112006000577

M3 - Article

SN - 0022-1120

VL - 559

SP - 107

EP - 116

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -