44 research outputs found

### On the Two-point Correlation of Potential Vorticity in Rotating and Stratified Turbulence

A framework is developed to describe the two-point statistics of potential
vorticity in rotating and stratified turbulence as described by the Boussinesq
equations. The Karman-Howarth equation for the dynamics of the two-point
correlation function of potential vorticity reveals the possibility of
inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and
Reynolds number parameters. For the case of large Rossby and Froude numbers,
and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3
prefactor is derived for the third-order mixed correlation between potential
vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law
for the third-order velocity structure function in turbulence theory.Comment: 10 pages, to appear in Journal of Fluid Mechanics (2006

### Anisotropic small-scale constraints on energy in rotating stratified turbulence

Author's version issued as working paper on Arxiv.orgRapidly rotating, stably stratified three-dimensional inviscid flows conserve both energy and potential enstrophy. We show that in such flows, the forward cascade of potential enstrophy imposes anisotropic constraints on the wavenumber distribution of kinetic and potential energy. The horizontal kinetic energy is suppressed in the large, nearly horizontal wave modes, and should decay with the horizontal wavenumber as $k_h^{-3}$. The potential energy is suppressed in the large, nearly vertical wave modes, and should decay with the vertical wavenumber as $k_z^{-3}$. These results augment the only other exact prediction for the scaling of energy spectra due to constraints by potential enstrophy obtained by Charney (J. Atmos. Sci. 28, 1087 (1971)), who showed that in the quasi-geostrophic approximation for rotating stratified flows, the energy spectra must scale isotropically with total wavenumber as $k^{-3}$. We test our predicted scaling estimates using resolved numerical simulations of the Boussinesq equations in the relevant parameter regimes, and find reasonable agreement

### Sign-symmetry of temperature structure functions

New scalar structure functions with different sign-symmetry properties are
defined. These structure functions possess different scaling exponents even
when their order is the same. Their scaling properties are investigated for
second and third orders, using data from high-Reynolds-number atmospheric
boundary layer. It is only when structure functions with disparate
sign-symmetry properties are compared can the extended self-similarity detect
two different scaling ranges that may exist, as in the example of convective
turbulence.Comment: 18 pages, 5 figures, accepted for publication in Physical Review

### Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence

We present data from high-resolution numerical simulations of the
Navier-Stokes-$\alpha$ and the Leray-$\alpha$ models for two-dimensional
turbulence. It was shown previously (Lunasin et al., J. Turbulence, 8, (2007),
751-778), that for wavenumbers $k$ such that $k\alpha\gg 1$, the energy
spectrum of the smoothed velocity field for the two-dimensional
Navier-Stokes-$\alpha$ (NS-$\alpha$) model scales as $k^{-7}$. This result is
in agreement with the scaling deduced by dimensional analysis of the flux of
the conserved enstrophy using its characteristic time scale. We therefore
hypothesize that the spectral scaling of any $\alpha$-model in the sub-$\alpha$
spatial scales must depend only on the characteristic time scale and dynamics
of the dominant cascading quantity in that regime of scales. The data presented
here, from simulations of the two-dimensional Leray-$\alpha$ model, confirm our
hypothesis. We show that for $k\alpha\gg 1$, the energy spectrum for the
two-dimensional Leray-$\alpha$ scales as $k^{-5}$, as expected by the
characteristic time scale for the flux of the conserved enstrophy of the
Leray-$\alpha$ model. These results lead to our conclusion that the dominant
directly cascading quantity of the model equations must determine the scaling
of the energy spectrum.Comment: 11 pages, 4 figure

### Cascade time-scales for energy and helicity in homogeneous isotropic turbulence

We extend the Kolmogorov phenomenology for the scaling of energy spectra in
high-Reynolds number turbulence, to explicitly include the effect of helicity.
There exists a time-scale $\tau_H$ for helicity transfer in homogeneous,
isotropic turbulence with helicity. We arrive at this timescale using the
phenomenological arguments used by Kraichnan to derive the timescale $\tau_E$
for energy transfer (J. Fluid Mech. {\bf 47}, 525--535 (1971)). We show that in
general $\tau_H$ may not be neglected compared to $\tau_E$, even for rather low
relative helicity. We then deduce an inertial range joint cascade of energy and
helicity in which the dynamics are dominated by $\tau_E$ in the low wavenumbers
with both energy and helicity spectra scaling as $k^{-5/3}$; and by $\tau_H$ at
larger wavenumbers with spectra scaling as $k^{-4/3}$. We demonstrate how,
within this phenomenology, the commonly observed ``bottleneck'' in the energy
spectrum might be explained. We derive a wavenumber $k_h$ which is less than
the Kolmogorov dissipation wavenumber, at which both energy and helicity
cascades terminate due to dissipation effects. Data from direct numerical
simulations are used to check our predictions.Comment: 14 pages, 5 figures, accepted to Physical Review