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.

1 Introduction

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

x, t

1

4

T (k , )

i t

ikx

e

i D(k , ) e dk d .

i

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!

References

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
heliopause.
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,
submitted (2002)
University of Sheffield