Abstract
Landau & Lifshitz showed that Kolmogorov's
E∼t−10/7 law for the decay of
isotropic
turbulence rests on just two physical ideas: (a) the conservation
of
angular momentum, as expressed by Loitsyansky's integral; and (b)
the removal of energy from the large
scales via the energy cascade. Both Kolmogorov's original analysis
and
Landau &
Lifshitz's reinterpretation in terms of angular momentum are now known
to
be flawed.
The existence of long-range velocity correlations means that Loitsyansky's
integral is
not an exact representation of angular momentum, nor is it strictly conserved.
However, in practice the long-range velocity correlations are weak and
Loitsyansky's
integral is almost constant, so that the Kolmogorov/Landau model provides
a
surprisingly simple and robust description of the decay. In this paper
we redevelop
these ideas in the context of MHD turbulence. We take advantage of the
fact that the
angular momentum of a fluid moving in a uniform magnetic field has particularly
simple properties. Specifically, the component parallel to the magnetic
field is
conserved while the normal components decay exponentially on a time scale
of
τ=ρ/σB2.
We show that the counterpart of Loitsyansky's integral for MHD turbulence
is
∫x2⊥Q⊥dx,
where Qij is the velocity correlation. When
the
long-range correlations
are weak this integral is conserved. This provides an estimate of the rate
of
decay of energy. At low values of magnetic field we recover Kolmogorov's
law.
At high values we find E∼t−1/2,
which is a result derived earlier by Moffatt. We also show that
∫x2⊥Q∥dx
decays exponentially on a time scale of τ. We interpret these results
in terms of
the behaviour of isolated vortices orientated normal and parallel to the
magnetic field.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
74 articles.
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