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 $k<k_d$, where $k_d$ is relatively small compared with the diffusion wave number $k_\kappa$. Such flows mediate couplings between neighbouring wave numbers within $k_d$ of each other only. It is found that the spectral rate of transport (flux) of scalar variance across a high wave number $k>k_d$ is bounded from above by $Uk_dk\Theta(k,t)$, where $U$ denotes the maximum fluid velocity and $\Theta(k,t)$ is the spectrum of the scalar variance, defined as its average over the shell $(k-k_d,k+k_d)$. For a given flux, say $\vartheta>0$, across $k>k_d$, this bound requires $$\Theta(k,t)\ge \frac{\vartheta}{Uk_d}k^{-1}.$$ This is consistent with recent numerical studies and with Batchelor's theory that predicts a $k^{-1}$ spectrum (with a slightly different proportionality constant) for the viscous-convective range, which could be identified with $(k_d,k_\kappa)$. 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 $\ge2$, including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, $\Theta(k,t)$ and $\vartheta$ are the kinetic energy spectrum and flux, respectively.Comment: 6 journal pages, 1 "cartoon" figure, to appear in PR

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

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

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 $$\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi) =\mu\Delta\psi+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $\Psi_1=/2$ and $\Psi_2=/2$ (kinetic energy), where $$ denotes a spatial average. The energy density $\Psi_2$ is bounded and its spectrum $\Psi_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $\Psi_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ 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

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 on the inverse energy transfer, specifically on the growth rate \d\ell/\dt of the characteristic length scale $\ell$ 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 $\ell_s$ represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential energy dominated regime ($\ell\ge\ell_s\gg\lambda^{-1}$), 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.Comment: 8 pages, 2 figures, accepted for publication in JF