180 research outputs found
On the spectrum of the magnetohydrodynamic mean-field alpha^2-dynamo operator
The existence of magnetohydrodynamic mean-field alpha^2-dynamos with
spherically symmetric, isotropic helical turbulence function alpha is related
to a non-self-adjoint spectral problem for a coupled system of two singular
second order ordinary differential equations. We establish global estimates for
the eigenvalues of this system in terms of the turbulence function alpha and
its derivative alpha'. They allow us to formulate an anti-dynamo theorem and a
non-oscillation theorem. The conditions of these theorems, which again involve
alpha and alpha', must be violated in order to reach supercritical or
oscillatory regimes.Comment: 35 pages, 4 figures, to be published in SIAM J. Math. Anal
Inertial Range Scaling, Karman-Howarth Theorem and Intermittency for Forced and Decaying Lagrangian Averaged MHD in 2D
We present an extension of the Karman-Howarth theorem to the Lagrangian
averaged magnetohydrodynamic (LAMHD-alpha) equations. The scaling laws
resulting as a corollary of this theorem are studied in numerical simulations,
as well as the scaling of the longitudinal structure function exponents
indicative of intermittency. Numerical simulations for a magnetic Prandtl
number equal to unity are presented both for freely decaying and for forced two
dimensional MHD turbulence, solving directly the MHD equations, and employing
the LAMHD-alpha equations at 1/2 and 1/4 resolution. Linear scaling of the
third-order structure function with length is observed. The LAMHD-alpha
equations also capture the anomalous scaling of the longitudinal structure
function exponents up to order 8.Comment: 34 pages, 7 figures author institution addresses added magnetic
Prandtl number stated clearl
Astrophysical turbulence modeling
The role of turbulence in various astrophysical settings is reviewed. Among
the differences to laboratory and atmospheric turbulence we highlight the
ubiquitous presence of magnetic fields that are generally produced and
maintained by dynamo action. The extreme temperature and density contrasts and
stratifications are emphasized in connection with turbulence in the
interstellar medium and in stars with outer convection zones, respectively. In
many cases turbulence plays an essential role in facilitating enhanced
transport of mass, momentum, energy, and magnetic fields in terms of the
corresponding coarse-grained mean fields. Those transport properties are
usually strongly modified by anisotropies and often completely new effects
emerge in such a description that have no correspondence in terms of the
original (non coarse-grained) fields.Comment: 88 pages, 26 figures, published in Reports on Progress in Physic
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
Shell Models of Magnetohydrodynamic Turbulence
Shell models of hydrodynamic turbulence originated in the seventies. Their
main aim was to describe the statistics of homogeneous and isotropic turbulence
in spectral space, using a simple set of ordinary differential equations. In
the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged
based on the same principles as their hydrodynamic counter-part but also
incorporating interactions between magnetic and velocity fields. In recent
years, significant improvements have been made such as the inclusion of
non-local interactions and appropriate definitions for helicities. Though shell
models cannot account for the spatial complexity of MHD turbulence, their
dynamics are not over simplified and do reflect those of real MHD turbulence
including intermittency or chaotic reversals of large-scale modes. Furthermore,
these models use realistic values for dimensionless parameters (high kinetic
and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing
extended inertial range and accurate dissipation rate. Using modern computers
it is difficult to attain an inertial range of three decades with direct
numerical simulations, whereas eight are possible using shell models. In this
review we set up a general mathematical framework allowing the description of
any MHD shell model. The variety of the latter, with their advantages and
weaknesses, is introduced. Finally we consider a number of applications,
dealing with free-decaying MHD turbulence, dynamo action, Alfven waves and the
Hall effect.Comment: published in Physics Report
Why dynamos are prone to reversals
In a recent paper (Phys. Rev. Lett. 94 (2005), 184506; physics/0411050) it
was shown that a simple mean-field dynamo model with a spherically symmetric
helical turbulence parameter alpha can exhibit a number of features which are
typical for Earth's magnetic field reversals. In particular, the model produces
asymmetric reversals, a positive correlation of field strength and interval
length, and a bimodal field distribution. All these features are attributable
to the magnetic field dynamics in the vicinity of an exceptional point of the
spectrum of the non-selfadjoint dynamo operator. The negative slope of the
growth rate curve between the nearby local maximum and the exceptional point
makes the system unstable and drives it to the exceptional point and beyond
into the oscillatory branch where the sign change happens. A weakness of this
reversal model is the apparent necessity to fine-tune the magnetic Reynolds
number and/or the radial profile of alpha. In the present paper, it is shown
that this fine-tuning is not necessary in the case of higher supercriticality
of the dynamo. Numerical examples and physical arguments are compiled to show
that, with increasing magnetic Reynolds number, there is strong tendency for
the exceptional point and the associated local maximum to move close to the
zero growth rate line. Although exemplified again by the spherically symmetric
alpha^2 dynamo model, the main idea of this ''self-tuning'' mechanism of
saturated dynamos into a reversal-prone state seems well transferable to other
dynamos. As a consequence, reversing dynamos might be much more typical and may
occur much more frequently in nature than what could be expected from a purely
kinematic perspective.Comment: 11 pages, 10 figure
Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of incompressible electrically conducting fluid
We introduce six families of three-dimensional space-periodic steady
solenoidal flows, whose kinetic helicity density is zero at any point. Four
families are analytically defined. Flows in four families have zero helicity
spectrum. Sample flows from five families are used to demonstrate numerically
that neither zero kinetic helicity density, nor zero helicity spectrum prohibit
generation of large-scale magnetic field by the two most prominent dynamo
mechanisms: the magnetic -effect and negative eddy diffusivity. Our
computations also attest that such flows often generate small-scale field for
sufficiently small magnetic molecular diffusivity. These findings indicate that
kinetic helicity and helicity spectrum are not the quantities controlling the
dynamo properties of a flow regardless of whether scale separation is present
or not.Comment: 37 pages, 11 figures, 54 reference
Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of incompressible electrically conducting fluid
We introduce six families of three-dimensional space-periodic steady
solenoidal flows, whose kinetic helicity density is zero at any point. Four
families are analytically defined. Flows in four families have zero helicity
spectrum. Sample flows from five families are used to demonstrate numerically
that neither zero kinetic helicity density, nor zero helicity spectrum prohibit
generation of large-scale magnetic field by the two most prominent dynamo
mechanisms: the magnetic -effect and negative eddy diffusivity. Our
computations also attest that such flows often generate small-scale field for
sufficiently small magnetic molecular diffusivity. These findings indicate that
kinetic helicity and helicity spectrum are not the quantities controlling the
dynamo properties of a flow regardless of whether scale separation is present
or not.Comment: 37 pages, 11 figures, 54 reference
The Lagrangian-averaged model for magnetohydrodynamics turbulence and the absence of bottleneck
We demonstrate that, for the case of quasi-equipartition between the velocity
and the magnetic field, the Lagrangian-averaged magnetohydrodynamics
alpha-model (LAMHD) reproduces well both the large-scale and small-scale
properties of turbulent flows; in particular, it displays no increased
(super-filter) bottleneck effect with its ensuing enhanced energy spectrum at
the onset of the sub-filter-scales. This is in contrast to the case of the
neutral fluid in which the Lagrangian-averaged Navier-Stokes alpha-model is
somewhat limited in its applications because of the formation of spatial
regions with no internal degrees of freedom and subsequent contamination of
super-filter-scale spectral properties. No such regions are found in LAMHD,
making this method capable of large reductions in required numerical degrees of
freedom; specifically, we find a reduction factor of 200 when compared to a
direct numerical simulation on a large grid of 1536^3 points at the same
Reynolds number.Comment: 22 pages, 9 figures; accepted Phys.Rev.
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