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The magnetic Rayleigh-Taylor instability

Chambers, KSS 2012 , 'The magnetic Rayleigh-Taylor instability', PhD thesis, University of Tasmania.

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The Rayleigh-Taylor instability (RTI) arises whenever two fluids with different
densities are arranged such that the heavier fluid sits above the lighter fluid, with
a sharp interface in between. The magnetic Rayleigh-Taylor instability (MRTI)
has the further complication due to the presence of a magnetic field throughout
both media. The two fluids in question may also have differing magnetic
properties, such as the magnetic permeability. When the fluids in consideration
are in fact plasmas comprised of charged particles, induced currents, magnetic
_fields and Lorentz forces can all act in ways that will affect the stability of the
The RTI has widespread applications in atmospheric physics, oceanography,
meteorology, laboratory plasma physics, nuclear reactors, inertial confinement
fusion as well as the field of astrophysics, where the instability plays an important
role in supernova explosions, accretion discs, plasma jets and H II regions
(clouds of gas in which star formation has recently taken place) amongst others.
It is closely related to two other hydrodynamic instabilities, namely the Kelvin-
Helmholtz instability (KHI) and the Richtmyer-Meshkov instability (RMI).
This thesis considers in detail several different flow configurations in which
the RTI arises. A key feature of these configurations is that small wavelike
disturbances to the flow are unstable. These configurations are studied to determine
the behaviour of these unstable flows. A particular focus is the effects
due to the presence of a magnetic field, and the mechanisms that alter the flow
in the magnetic case.
The thesis begins by considering two dimensional planar flow in Cartesian
geometry in which the amplitude of the waves is small compared with their
wavelength. The flow is assumed to be incompressible and inviscid. In this
scenario, linear theory is used to derive a closed form solution for the evolution
of the interface between the two fluids. This solution shows quantitatively how
the position of the interface depends on the ratio of the densities of the two
6 fluids, the wavelength of the disturbance, as well as the strength and direction
of the applied magnetic field.
The unstable nature of the RTI means that after some finite time, the amplitude
of the waves will grow to a size comparable with their wavelength, and
in this scenario, linear theory is not appropriate. For this reason, a non-linear
model is considered, again for two dimensional planar flow in Cartesian geometry.
The flow in this case is considered to be weakly compressible, and
viscous. Results in the non-linear case are obtained by use of a combination
of streamfunction, spectral and finite difference techniques. The results show
qualitatively various non-linear phenomena such as interface roll-up, fingering
and bubble formation. It is shown in particular how different initial conditions
give rise to outcomes that are very different in terms of the geometry of the interface
between the two fluids, primarily the differences between a single mode
disturbance and a multi mode disturbance to the interface at time t = 0.
The final problem studied in this thesis considers two dimensional flow in
circularly symmetric cylindrical geometry. The configuration in this case is
comprised of a heavy fluid surrounding a light fluid, and gravity is directed
radially inwards. A massive object is located at the centre of the light fluid,
and it behaves like a line dipole both for fluid flow and magnetic field strength.
In the non-linear, weakly compressible, viscous regime, the initially circular
interface between the two conducting fluids evolves into plumes, dependent on
the magnetic and fluid dipole strengths and the nature of the initial disturbance
to the interface. A spectral method is presented to solve the time-dependent
interface shapes, and results are presented and discussed. Bipolar solutions are
possible, and these are of particular relevance to astrophysics. The solutions
obtained resemble structures of some HII regions and nebulae.

Item Type: Thesis - PhD
Authors/Creators:Chambers, KSS
Keywords: Fluid insability, outflow, surface roll-up, spectral methods
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