240 research outputs found
Enstrophy dissipation in freely evolving two-dimensional turbulence
Freely decaying two-dimensional Navier--Stokes turbulence is studied. The
conservation of vorticity by advective nonlinearities renders a class of
Casimirs that decays under viscous effects. A rigorous constraint on the
palinstrophy production by nonlinear transfer is derived, and an upper bound
for the enstrophy dissipation is obtained. This bound depends only on the
decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from
above in terms of initial data of the flows. An upper bound for the enstrophy
dissipation wavenumber is derived and the new result is compared with the
classical dissipation wavenumber.Comment: No figures, Letter to appear in Phys. Fluid
Local transfer and spectra of a diffusive field advected by large-scale incompressible flows
This study revisits the problem of advective transfer and spectra of a
diffusive scalar field in large-scale incompressible flows in the presence of a
(large-scale) source. By ``large-scale'' it is meant that the spectral support
of the flows is confined to the wave-number region , where is
relatively small compared with the diffusion wave number . Such flows
mediate couplings between neighbouring wave numbers within of each other
only. It is found that the spectral rate of transport (flux) of scalar variance
across a high wave number is bounded from above by ,
where denotes the maximum fluid velocity and is the spectrum
of the scalar variance, defined as its average over the shell .
For a given flux, say , across , this bound requires
This is consistent with recent
numerical studies and with Batchelor's theory that predicts a spectrum
(with a slightly different proportionality constant) for the viscous-convective
range, which could be identified with . Thus, Batchelor's
formula for the variance spectrum is recovered by the present method in the
form of a critical lower bound. The present result applies to a broad range of
large-scale advection problems in space dimensions , including some
filter models of turbulence, for which the turbulent velocity field is advected
by a smoothed version of itself. For this case, and
are the kinetic energy spectrum and flux, respectively.Comment: 6 journal pages, 1 "cartoon" figure, to appear in PR
Impeded inverse energy transfer in the Charney--Hasegawa--Mima model of quasi-geostrophic flows
The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney-Hasegawa-Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber lambda (or free-surface elasticity). Herein, we derive a bound oil the inverse energy transfer, specifically on the growth rate dl/dt of the characteristic length scale e representing the energy centroid. It is found that dl/dt = l(s) >> lambda(-1)) the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.Publisher PDFPeer reviewe
Impeded inverse energy transfer in the Charney--Hasegawa--Mima model of quasi-geostrophic flows
The behaviour of turbulent flows within the single-layer quasi-geostrophic
(Charney--Hasegawa--Mima) model is shown to be strongly dependent on the Rossby
deformation wavenumber (or free-surface elasticity). Herein, we
derive a bound on the inverse energy transfer, specifically on the growth rate
\d\ell/\dt of the characteristic length scale representing the energy
centroid. It is found that \d\ell/\dt\le2\norm q_\infty/(\ell_s\lambda^2),
where \norm q_\infty is the supremum of the potential vorticity and
represents the potential enstrophy centroid of the reservoir, both invariant.
This result implies that in the potential energy dominated regime
(), the inverse energy transfer is strongly
impeded, in the sense that under the usual time scale no significant transfer
of energy to larger scales occurs. The physical implication is that the
elasticity of the free surface impedes turbulent energy transfer in wavenumber
space, effectively rendering large-scale vortices long-lived and inactive.
Results from numerical simulations of forced-dissipative turbulence confirm
this prediction.Comment: 8 pages, 2 figures, accepted for publication in JF
Large-scale energy spectra in surface quasi-geostrophic turbulence
The large-scale energy spectrum in two-dimensional turbulence governed by the
surface quasi-geostrophic (SQG) equation
is studied. The nonlinear transfer of this system conserves the two quadratic
quantities and
(kinetic energy), where denotes
a spatial average. The energy density is bounded and its spectrum
is shallower than in the inverse-transfer range. For
bounded turbulence, in the low-wavenumber region can be bounded by
where is a constant independent of but dependent on the domain
size. Results from numerical simulations confirming the theoretical predictions
are presented.Comment: 11 pages, 4 figures, to appear in JF
Revisiting Batchelor's theory of two-dimensional turbulence
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).Publisher PDFPeer reviewe
Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence
We study the small-scale behavior of generalized two-dimensional turbulence
governed by a family of model equations, in which the active scalar
is advected by the incompressible flow
. The dynamics of this family are characterized by the
material conservation of , whose variance is
preferentially transferred to high wave numbers. As this transfer proceeds to
ever-smaller scales, the gradient $\nabla\theta$ grows without bound. This
growth is due to the stretching term $(\nabla\theta\cdot\nabla)\u$ whose
``effective degree of nonlinearity'' differs from one member of the family to
another. This degree depends on the relation between the advecting flow $\u$
and the active scalar $\theta$ and is wide ranging, from approximately linear
to highly superlinear. Linear dynamics are realized when $\nabla\u$ is a
quantity of no smaller scales than $\theta$, so that it is insensitive to the
direct transfer of the variance of $\theta$, which is nearly passively
advected. This case corresponds to $\alpha\ge2$, for which the growth of
$\nabla\theta$ is approximately exponential in time and non-accelerated. For
$\alpha<2$, superlinear dynamics are realized as the direct transfer of
entails a growth in \nabla\u, thereby enhancing the production
of . This superlinearity reaches the familiar quadratic
nonlinearity of three-dimensional turbulence at and surpasses that
for . The usual vorticity equation () is the border line,
where \nabla\u and are of the same scale, separating the linear and
nonlinear regimes of the small-scale dynamics. We discuss these regimes in
detail, with an emphasis on the locality of the direct transfer.Comment: 6 journal pages, to appear in Physical Review
- …