10 research outputs found
On the elimination of the sweeping interactions from theories of hydrodynamic turbulence
In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian
same time correlation tensors are equal. This statement allows us to transform
the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic
turbulence back to the Eulerian representation. We define a hierarchy of
homogeneity symmetries between incremental homogeneity and global homogeneity.
It is shown that both the elimination of the sweeping interactions and the
derivation of the 4/5-law require a homogeneity assumption stronger than
incremental homogeneity but weaker than global homogeneity. The
quasi-Lagrangian transformation, on the other hand, requires an even stronger
homogeneity assumption which is many-time rather than one-time but still weaker
than many-time global homogeneity. We argue that it is possible to relax this
stronger assumption and still preserve the conclusions derived from theoretical
work based on the quasi-Lagrangian transformation.Comment: v1: submitted to Physica D. v2: major revisions; resubmitted to
Physica D. v3: minor revisions requested by referee
Locality and stability of the cascades of two-dimensional turbulence
We investigate and clarify the notion of locality as it pertains to the
cascades of two-dimensional turbulence. The mathematical framework underlying
our analysis is the infinite system of balance equations that govern the
generalized unfused structure functions, first introduced by L'vov and
Procaccia. As a point of departure we use a revised version of the system of
hypotheses that was proposed by Frisch for three-dimensional turbulence. We
show that both the enstrophy cascade and the inverse energy cascade are local
in the sense of non-perturbative statistical locality. We also investigate the
stability conditions for both cascades. We have shown that statistical
stability with respect to forcing applies unconditionally for the inverse
energy cascade. For the enstrophy cascade, statistical stability requires
large-scale dissipation and a vanishing downscale energy dissipation. A careful
discussion of the subtle notion of locality is given at the end of the paper.Comment: v2: 23 pages; 4 figures; minor revisions; resubmitted to Phys. Rev.
Recent Developments in Understanding Two-dimensional Turbulence and the Nastrom-Gage Spectrum
Two-dimensional turbulence appears to be a more formidable problem than
three-dimensional turbulence despite the numerical advantage of working with
one less dimension. In the present paper we review recent numerical
investigations of the phenomenology of two-dimensional turbulence as well as
recent theoretical breakthroughs by various leading researchers. We also review
efforts to reconcile the observed energy spectrum of the atmosphere (the
spectrum) with the predictions of two-dimensional turbulence and
quasi-geostrophic turbulence.Comment: Invited review; accepted by J. Low Temp. Phys.; Proceedings for
Warwick Turbulence Symposium Workshop on Universal features in turbulence:
from quantum to cosmological scales, 200
Multi-locality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence
Using the fusion rules hypothesis for three-dimensional and two-dimensional
Navier-Stokes turbulence, we generalize a previous non-perturbative locality
proof to multiple applications of the nonlinear interactions operator on
generalized structure functions of velocity differences. We shall call this
generalization of non-perturbative locality to multiple applications of the
nonlinear interactions operator "multilocality". The resulting cross-terms pose
a new challenge requiring a new argument and the introduction of a new fusion
rule that takes advantage of rotational symmetry. Our main result is that the
fusion rules hypothesis implies both locality and multilocality in both the IR
and UV limits for the downscale energy cascade of three-dimensional
Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy
cascade of two-dimensional Navier-Stokes turbulence. We stress that these
claims relate to non-perturbative locality of generalized structure functions
on all orders, and not the term by term perturbative locality of diagrammatic
theories or closure models that involve only two-point correlation and response
functions.Comment: 25 pages, 24 figures, resubmitted to Physical Review
Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?
In systems governing two-dimensional turbulence, surface quasi-geostrophic
turbulence, (more generally -turbulence), two-layer quasi-geostrophic
turbulence, etc., there often exist two conservative quadratic quantities, one
``energy''-like and one ``enstrophy''-like. In a finite inertial range there
are in general two spectral fluxes, one associated with each conserved
quantity. We derive here an inequality comparing the relative magnitudes of the
``energy'' and ``enstrophy'' fluxes for finite or infinitesimal dissipations,
and for hyper or hypo viscosities. When this inequality is satisfied, as is the
case of 2D turbulence,where the energy flux contribution to the energy spectrum
is small, the subdominant part will be effectively hidden. In sQG turbulence,
it is shown that the opposite is true: the downscale energy flux becomes the
dominant contribution to the energy spectrum. A combination of these two
behaviors appears to be the case in 2-layer QG turbulence, depending on the
baroclinicity of the system.Comment: 23 pages; accepted at Discrete and Continuous Dynamical Systems B;
Major revisio