6 research outputs found
Stochastic attractors for shell phenomenological models of turbulence
Recently, it has been proposed that the Navier-Stokes equations and a
relevant linear advection model have the same long-time statistical properties,
in particular, they have the same scaling exponents of their structure
functions. This assertion has been investigate rigorously in the context of
certain nonlinear deterministic phenomenological shell model, the Sabra shell
model, of turbulence and its corresponding linear advection counterpart model.
This relationship has been established through a "homotopy-like" coefficient
which bridges continuously between the two systems. That is, for
one obtains the full nonlinear model, and the corresponding linear
advection model is achieved for . In this paper, we investigate the
validity of this assertion for certain stochastic phenomenological shell models
of turbulence driven by an additive noise. We prove the continuous dependence
of the solutions with respect to the parameter . Moreover, we show the
existence of a finite-dimensional random attractor for each value of
and establish the upper semicontinuity property of this random attractors, with
respect to the parameter . This property is proved by a pathwise
argument. Our study aims toward the development of basic results and techniques
that may contribute to the understanding of the relation between the long-time
statistical properties of the nonlinear and linear models
Current status of turbulent dynamo theory: From large-scale to small-scale dynamos
Several recent advances in turbulent dynamo theory are reviewed. High
resolution simulations of small-scale and large-scale dynamo action in periodic
domains are compared with each other and contrasted with similar results at low
magnetic Prandtl numbers. It is argued that all the different cases show
similarities at intermediate length scales. On the other hand, in the presence
of helicity of the turbulence, power develops on large scales, which is not
present in non-helical small-scale turbulent dynamos. At small length scales,
differences occur in connection with the dissipation cutoff scales associated
with the respective value of the magnetic Prandtl number. These differences are
found to be independent of whether or not there is large-scale dynamo action.
However, large-scale dynamos in homogeneous systems are shown to suffer from
resistive slow-down even at intermediate length scales. The results from
simulations are connected to mean field theory and its applications. Recent
work on helicity fluxes to alleviate large-scale dynamo quenching, shear
dynamos, nonlocal effects and magnetic structures from strong density
stratification are highlighted. Several insights which arise from analytic
considerations of small-scale dynamos are discussed.Comment: 36 pages, 11 figures, Spa. Sci. Rev., submitted to the special issue
"Magnetism in the Universe" (ed. A. Balogh