16-09-2016, 03:14 PM
The Kelvin–Helmholtz instability in magnetohydrodynamics with finite
Larmor radius effects and application to Venus
1454958350-PlanetaryandSpaceScienceVolume57issue32009doi10.1016j.pss.2008.12.003U.V.AmerstorferTheKelvinHelmholtzinstabi (Size: 141.05 KB / Downloads: 5)
The dissertation deals with the Kelvin–Helmholtz instability in
magnetohydrodynamics (MHD). The instability is studied with
two models: an ideal MHD model and a finite Larmor radius (FLR)
MHD model. The first one uses, as the name already suggests, the
equations of ideal MHD; the second one extends these equations
in order to include the effect of an FLR of the ions (e.g., Braginskii,
1965). In both models compressibility as well as a boundary layer
with finite thickness, which separates the two plasma layers with
different characteristics, are included. The equations are normalized
and linearized for the further numerical treatment. For the
numerical solution to the perturbation equations, the two-step
Lax–Wendroff algorithm is used in the case of the ideal MHD
equations (e.g., Potter, 1973). For the solution to the equations of
the FLR MHD model, where there are partial differential equations
of second order, the Thomas algorithm is utilized (e.g., Press et al.,
1992).
Theoretical studies done with the ideal MHD model within the
thesis show that a dense lower plasma reduces the growth rate of
the Kelvin–Helmholtz instability considerably (Amerstorfer et al.,
2007). This outcome is important when the instability is studied
around unmagnetized planets, where the mass density increases
from the magnetosheath to the ionosphere.
Since often very narrow boundary layers occur in space plasma
configurations, the Kelvin–Helmholtz instability is investigated
within MHD including the FLR of the ions. When the ion Larmor
radius is comparable (or even larger) than the thickness of the
boundary layer, it should not be neglected anymore. Theoretical
investigations show that the growth rate exhibits a dependence on the sign of B X, where X ¼ r v is the vorticity. This means
that, depending upon the magnetic field–velocity configuration,
the growth rate is either larger or smaller than in the pure ideal
MHD case. The growth rate of the Kelvin–Helmholtz instability for
a compressible plasma behaves in the following way. The largest
growth rates are obtained for B X40 and the lowest for B Xo0,
for both, low and high plasma b (in contrast to the incompressible
case, where the behaviour of the growth rate is different for low
and high b, see Huba, 1996). The results obtained with the FLR
MHD model thus suggest that the Kelvin–Helmholtz instability
will develop and evolve differently at a boundary around
unmagnetized planets, depending upon the velocity shear–gyromotion
configuration.
Both theoretical models are applied to the situation around
Venus. The solar wind is diverted around the ionosphere of Venus,
thus creating a boundary layer as well as a velocity shear. Hence,
such a boundary layer should in principal be unstable with regard
to the Kelvin–Helmholtz instability. Waves on the day side of the
boundary might be able to grow and break, forming detached
plasma clouds on the night side. Pioneer Venus Orbiter observed
both phenomena, which could be the different stages of the
instability (Brace et al., 1982). The plasma clouds, which contain
ionospheric particles, could contribute to the loss of particles from
Venus. With this regard, the Kelvin–Helmholtz instability might
thus play an important role.
By considering velocity and density profiles appropriate to
represent the plasma environment around Venus, the ideal MHD
theory is applied to the solar wind interaction with Venus (Biernat
et al., 2007). The development of the Kelvin–Helmholtz instability
in dependence of the ionospheric density is investigated for four
different solar wind conditions. From the results one can conclude
that the plasma conditions around Venus are principally in favour
of the development of the Kelvin–Helmholtz instability, and that
the growth rate decreases substantially with increasing ionospheric
density, which means that the stable and unstable regions
at the boundary seem to be very sensitive to the density ratio
across the boundary layer.
With regard to the solar wind interaction with Venus, where
solar wind protons and oxygen ions form the main species near
the planet, a second ion species is added in the FLR MHD model.
Using parameter values and profiles representing the vicinity of
the magnetopause of Venus, the Kelvin–Helmholtz instability is
studied at this boundary, which is thought to be the stopping
boundary for the solar wind (Zhang et al., 2008). The obtained
growth rates behave differently for different directions of the
velocity shear: For qv0=qx40 the growth rates are larger than for
qv0=qxo0, where v0 is the background flow velocity parallel to
the boundary. Fig. 1 shows a sketch of the situation at Venus.
Comparing the characteristic growth time of the instability
with the characteristic propagation time of the waves, it is found
that both times are of the same order of magnitude, but only for qv0=qx40 the growth time is smaller than the propagation time.
Thus, it is concluded that the FLR of the ions introduces an
asymmetric behaviour of the Kelvin–Helmholtz instability at the
magnetopause of Venus and that the instability is able to develop
on both sides of the boundary, but the excited waves seem to be
able to grow, saturate and possibly break more easily on one side.
Acknowledgements
U.V. Amerstorfer acknowledges financial support by a ‘‘DOC–
fFORTE (Frauen in Forschung und Technologie)’’ scholarship by
the Austrian Academy of Sciences and by the scholarship
‘‘L’ORE´AL O¨ sterreich—For Women in Science’’ in cooperation with
the Austrian UNESCO-Commission and the Austrian Academy of
Sciences, with the support by the Federal Ministry of Science and
Research. She also wants to thank the Space Research Institute for
great support.