Kraichnan–Leith–Batchelor similarity theory and two-dimensional inverse cascades
We study the scaling properties and Kraichnan–Leith–Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids ( ${\it\alpha}$ -turbulence models) simulated at resolution $8192^{2}$ . We consider ${\it\alpha}=1$ (surface quasigeostrophic flow), ${\it\alpha}=2$ (2D E...
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Published in | Journal of fluid mechanics Vol. 767; pp. 467 - 496 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Cambridge, UK
Cambridge University Press
25.03.2015
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Subjects | |
Online Access | Get full text |
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Summary: | We study the scaling properties and Kraichnan–Leith–Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids (
${\it\alpha}$
-turbulence models) simulated at resolution
$8192^{2}$
. We consider
${\it\alpha}=1$
(surface quasigeostrophic flow),
${\it\alpha}=2$
(2D Euler flow) and
${\it\alpha}=3$
. The forcing scale is well resolved, a direct cascade is present and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both
${\it\alpha}=1$
and
${\it\alpha}=2$
. The active scalar field for
${\it\alpha}=3$
contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction
$-(7-{\it\alpha})/3$
in all three systems. Since we stop the simulations well before the cascades have reached the domain scale, vortex formation and spectral steepening are not due to condensation effects; nor are they caused by large-scale dissipation, which is absent. One- and two-point p.d.f.s, hyperflatness factors and structure functions indicate that the inverse cascades are intermittent and non-Gaussian over much of the inertial range for
${\it\alpha}=1$
and
${\it\alpha}=2$
, while the
${\it\alpha}=3$
inverse cascade is much closer to Gaussian and non-intermittent. For
${\it\alpha}=3$
the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling
$\mathscr{E}(k)\propto k^{-2}~({\it\alpha}=1)$
and
$\mathscr{E}(k)\propto k^{-5/3}~({\it\alpha}=2)$
in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation (
${\it\alpha}=1$
and
${\it\alpha}=2$
) and non-realizability (
${\it\alpha}=3$
) cause 2D inverse cascades to deviate from the KLB predictions, but that the flow between the vortices exhibits KLB scaling and non-intermittent statistics for
${\it\alpha}=1$
and
${\it\alpha}=2$
. |
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ISSN: | 0022-1120 1469-7645 |
DOI: | 10.1017/jfm.2015.26 |