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Towards a finite-time singularity of the Navier-Stokes equations Part 1. Derivation and analysis of dynamical system
© 2018 Cambridge University Press. The evolution towards a finite-time singularity of the Navier-Stokes equations for flow of an incompressible fluid of kinematic viscosity is studied, starting from a finite-energy configuration of two vortex rings of circulation and radius , symmetrically placed on two planes at angles to a plane of symmetry . The minimum separation of the vortices, , and the scale of the core cross-section, , are supposed to satisfy the initial inequalities , and the vortex Reynolds number is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the 'tipping points') is determined solely by the curvature at the tipping points and by and , where is a dimensionless time variable. The Biot-Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot-Savart law breaks down just before this singularity is realised, when and become of order unity. The dynamical system admits 'partial Leray scaling' of just and , and ultimately full Leray scaling of and , conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter ; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot-Savart approach just before the incipient singularity is realised. The Euler flow situation is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case
Convective plan-form two-scale dynamos in a plane layer
We study generation of magnetic fields, involving large spatial scales, by
convective plan-forms in a horizontal layer. Magnetic modes and their growth
rates are expanded in power series in the scale ratio, and the magnetic eddy
diffusivity (MED) tensor is derived for flows, symmetric about the vertical
axis in a layer. For convective rolls magnetic eddy correction is demonstrated
to be always positive. For rectangular cell patterns, the region in the
parameter space of negative MED coincides with that of small-scale magnetic
field generation. No instances of negative MED in hexagonal cells are found. A
family of plan-forms with a smaller symmetry group than that of rectangular
cell patterns has been found numerically, where MED is negative for molecular
magnetic diffusivity over the threshold for the onset of small-scale magnetic
field generation.Comment: Latex. 24 pages with 3 Postscript figures, 19 references. Final
version (expanded Appendix 2, 4 references added, notation changed to a more
"user-friendly"), accepted in Geophysical and Astrophysical Fluid Dynamic
Nonlinearity in a dynamo
Using a rotating flat layer heated from below as an example, we consider
effects which lead to stabilizing an exponentially growing magnetic field in
magnetostrophic convection in transition from the kinematic dynamo to the full
non-linear dynamo. We present estimates of the energy redistribution over the
spectrum and helicity quenching by the magnetic field. We also study the
alignment of the velocity and magnetic fields. These regimes are similar to
those in planetary dynamo simulations.Comment: Accepted to Geophys. Astrophys. Fluid Dyna
Helicity within the vortex filament model
Kinetic helicity is one of the invariants of the Euler equations that is associated with the topology of vortex lines within the fluid. In superfluids, the vorticity is concentrated along vortex filaments. In this setting, helicity would be expected to acquire its simplest form. However, the lack of a core structure for vortex filaments appears to result in a helicity that does not retain its key attribute as a quadratic invariant. By defining a spanwise vector to the vortex through the use of a Seifert framing, we are able to introduce twist and henceforth recover the key properties of helicity. We present several examples for calculating internal twist to illustrate why the centreline helicity alone will lead to ambiguous results if a twist contribution is not introduced. Our choice of the spanwise vector can be expressed in terms of the tangential component of velocity along the filament. Since the tangential velocity does not alter the configuration of the vortex at later times, we are able to recover a similar equation for the internal twist angle to that of classical vortex tubes. Our results allow us to explain how a quasi-classical limit of helicity emerges from helicity considerations for individual superfluid vortex filaments
The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles
The gradient of potential vorticity (PV) is an important quantity because of
the way PV (denoted as ) tends to accumulate locally in the oceans and
atmospheres. Recent analysis by the authors has shown that the vector quantity
\bdB = \bnabla q\times \bnabla\theta for the three-dimensional incompressible
rotating Euler equations evolves according to the same stretching equation as
for \bom the vorticity and \bB, the magnetic field in magnetohydrodynamics
(MHD). The \bdB-vector therefore acts like the vorticity \bom in Euler's
equations and the \bB-field in MHD. For example, it allows various analogies,
such as stretching dynamics, helicity, superhelicity and cross helicity. In
addition, using quaternionic analysis, the dynamics of the \bdB-vector
naturally allow the construction of an orthonormal frame attached to fluid
particles\,; this is designated as a quaternion frame. The alignment dynamics
of this frame are particularly relevant to the three-axis rotations that
particles undergo as they traverse regions of a flow when the PV gradient
\bnabla q is large.Comment: Dedicated to Raymond Hide on the occasion of his 80th birthda
On the effects of turbulence on a screw dynamo
In an experiment in the Institute of Continuous Media Mechanics in Perm
(Russia) an non--stationary screw dynamo is intended to be realized with a
helical flow of liquid sodium in a torus. The flow is necessarily turbulent,
that is, may be considered as a mean flow and a superimposed turbulence. In
this paper the induction processes of the turbulence are investigated within
the framework of mean--field electrodynamics. They imply of course a part which
leads to an enhanced dissipation of the mean magnetic field. As a consequence
of the helical mean flow there are also helical structures in the turbulence.
They lead to some kind of --effect, which might basically support the
screw dynamo. The peculiarity of this --effect explains measurements
made at a smaller version of the device envisaged for the dynamo experiment.
The helical structures of the turbulence lead also to other effects, which in
combination with a rotational shear are potentially capable of dynamo action. A
part of them can basically support the screw dynamo. Under the conditions of
the experiment all induction effects of the turbulence prove to be rather weak
in comparison to that of the main flow. Numerical solutions of the mean--field
induction equation show that all the induction effects of the turbulence
together let the screw dynamo threshold slightly, at most by one per cent,
rise. The numerical results give also some insights into the action of the
individual induction effects of the turbulence.Comment: 15 pages, 7 figures, in GAFD prin
Yoshizawa's cross-helicity effect and its quenching
A central quantity in mean-field magnetohydrodynamics is the mean
electromotive force EMF, which in general depends on the mean magnetic field.
It may however have a part independent of the mean magnetic field. Here we
study an example of a rotating conducting body of turbulent fluid with non-zero
cross-helicity, in which a contribution to the EMF proportional to the angular
velocity occurs (Yoshizawa 1990). If the forcing is helical, it also leads to
an alpha effect, and large-scale magnetic fields can be generated. For not too
rapid rotation, the field configuration is such that Yoshizawa's contribution
to the EMF is considerably reduced compared to the case without alpha effect.
In that case, large-scale flows are also found to be generated.Comment: 10 pages, 8 figures, compatible with published versio
Magnetic helicity fluxes in an alpha-squared dynamo embedded in a halo
We present the results of simulations of forced turbulence in a slab where
the mean kinetic helicity has a maximum near the mid-plane, generating
gradients of magnetic helicity of both large and small-scale fields. We also
study systems that have poorly conducting buffer zones away from the midplane
in order to assess the effects of boundaries. The dynamical alpha quenching
phenomenology requires that the magnetic helicity in the small-scale fields
approaches a nearly static, gauge independent state. To stress-test this steady
state condition we choose a system with a uniform sign of kinetic helicity, so
that the total magnetic helicity can reach a steady state value only through
fluxes through the boundary, which are themselves suppressed by the velocity
boundary conditions. Even with such a set up, the small-scale magnetic helicity
is found to reach a steady state. In agreement with earlier work, the magnetic
helicity fluxes of small-scale fields are found to be turbulently diffusive. By
comparing results with and without halos, we show that artificial constraints
on magnetic helicity at the boundary do not have a significant impact on the
evolution of the magnetic helicity, except that "softer" (halo) boundary
conditions give a lower energy of the saturated mean magnetic field.Comment: 12 pages, 5 figures, submitted to GAF
Kinematic dynamo in spherical Couette flow
We investigate numerically kinematic dynamos driven by flow of electrically
conducting fluid in the shell between two concentric differentially rotating
spheres, a configuration normally referred to as spherical Couette flow. We
compare between axisymmetric (2D) and fully three dimensional flows, between
low and high global rotation rates, between prograde and retrograde
differential rotations, between weak and strong nonlinear inertial forces,
between insulating and conducting boundaries, and between two aspect ratios.
The main results are as follows. Azimuthally drifting Rossby waves arising from
the destabilisation of the Stewartson shear layer are crucial to dynamo action.
Differential rotation and helical Rossby waves combine to contribute to the
spherical Couette dynamo. At a slow global rotation rate, the direction of
differential rotation plays an important role in the dynamo because of
different patterns of Rossby waves in prograde and retrograde flows. At a rapid
global rotation rate, stronger flow supercriticality (namely the difference
between the differential rotation rate of the flow and its critical value for
the onset of nonaxisymmetric instability) facilitates the onset of dynamo
action. A conducting magnetic boundary condition and a larger aspect ratio both
favour dynamo action
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