Space & Atmosphere Research Center ABSOLUTE AND CONVECTIVE
Space & Atmosphere Research Center
ABSOLUTE AND CONVECTIVE INSTABILITIES
OF THE HELIOPAUSE
Rbert Erdlyi and Michael Ruderman
SPARC, Department of Applied Mathematics, University of Sheffield
Hicks Building, Hounsfield Road, S3 7RH Sheffield, England (UK)
[email protected], [email protected]
Abstract Stability of shear flows is of fundamental importance in solar and solar-terrestrial physics. Examples of such flows include plasma
flows, e.g., in the vicinity of the magnetopause of the Earth or planets, the boundaries between fast and slow streams of the solar wind or the
flow in the vicinity of the heliopause. The normal mode analysis is not sufficient to predict if a finite portion of a shear flow looks stable or
unstable. The reason is that this analysis deals with spatially periodic perturbations, while real perturbations are always confined to a finite
region. To study the stability of a shear flow with respect to perturbations finite in space we have to solve an initial-value problem. Then two
scenarios are possible. In the first scenario the initial finite perturbation exponentially grows at any spatial position. Such a type of instability
is called absolute. In the second scenario the initial perturbation also grows exponentially, but it is swept away by the flow from any finite
region so fast that it decays at any fixed spatial position. Such a type of instability is called convective. The classification of absolute and
convective instability is important for the understanding of the physical processes in solar, solar-terrestrial and astrophysical plasmas, and for
the interpretation of in-situ observational data like STEREO.
Our motivations for the present study is to analyse the instabilities of the near
flanks of the heliopause in the model of the solar wind -- interstellar medium
interaction (Fig. 1) first suggested by Baranov et al. 1971. The dynamics of small
localized disturbances is investigated in a KH-type flow in which one of the fluids
is inviscid, but the other one is viscous, and no surface tension is present on the
interface. A zoom of the simplified flank region is shown by Fig. 2.
Fig. 1: The model.
Fig. 2: The flank.
2 Solution to the boundary- and initial value problem
The perturbation interface (x,t) can be formally expressed as an
inverse Laplace-Fourier integral given by
T (k , )
i D(k , ) e dk d .
Here the dispersion function, D(,k),,k),represents the model, whereas
the function T(,k),,k) depends on the initial and external
perturbations. For studying absolute and convective instabilities of,
and signalling in, the model, it is sufficient to treat the asymptotics
of the perturbation interface given above and show that the roots of
T(,k),,k) do not cancel the corresponding contributions.
3 Normal modes are monochromatic disturbances satisfying the
dispersion relation D(,k),,k)=0. Kikina (1967) showed that for any
non-zero value of real k there exists one and only one unstable
normal mode and the growth rate is uniformly bounded the
initial-value problem for localised disturbances is well-posed!
Belov, N.A. & Myasnikov, A.V. Fluid Dyn., 34, 379, 1999
Baranov, V.B. et al. Sov. Phys. Dokl., 15, 791, 1971
Baranov, V.B. et al. Astrophys. Space Sci., 66, 441, 1979
Briggs, R.J. Electron-stream interaction with plasmas, MIT Press, 1964
For modelling the heliopause in the framework of the
stability analysis, we suppose that the linear
perturbations considered possess the characteristic
wavelength which is much smaller than the curvature
radius of the heliopause at the apex point. Then a near
flank of the heliopause can be assumed to be a planar
tangential discontinuity and a local quasi-parallel
stability analysis applied (Fig. 2). In this approach, the
flank of the heliopause is a plane, and the base plasma
flow on both sides of the flank is treated as being open,
space-independent, unidirectional and parallel to this
plane. Restricting our consideration to relatively small
polar angles (<30), where the plasma flow on both ), where the plasma flow on both
sides of the heliopause is strongly subsonic, we can use
the incompressible fluid approximation. The plasma on
both sides of the heliopause is a rarefied gas, and,
hence, effectively no surface tension is present on the
4 Absolute and convective instability
To distinguish between the absolute and convective
instability we have to study the asymtotic behaviour of
(x,t) at a fixed x as t. This analysis has been done
with the use of Briggs method (Briggs, 1964). For
equilibrium values from Baranov et al. (1979) in the
interval 10 30 we found all the instabilities
are convective. These results are in excellent
agreement with the results of numerical studies by
Belov & Myasnikov (1999).
Kikina, N.G. Sov. Phys. Acoustics, 13, 213, 1967
Ruderman, M.S., Brevdo, L. & Erdlyi, R. Phil. Trans.. Roy. Soc. London,
University of Sheffield
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