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Astron. Astrophys. 332, 786-794 (1998)

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7. Negative energy waves

To illustrate the correspondence between the overstabilities found here and the negative energy waves, we replace the inhomogeneous transition layer in our model, as described by Fig. 1, by a tangential discontinuity. For the present discussion, it is also instructive to redraw Fig. 2b (e.g. [FORMULA]) in the reference frame in which the region at the left of the discontinuity is static and the region at the right has a mass flow [FORMULA]. For [FORMULA], the dispersion relation in this reference frame is for the two surface modes

[EQUATION]

where

[EQUATION]

Following Cairns (1979), a wave mode is a negative energy wave when

[EQUATION]

The coefficient [FORMULA] has to be taken either 1 or -1 so that in the case of absence of a flow the energy of the modes is positive. After some algebra, we obtain for the inequality (18)

[EQUATION]

Using the values of the eigenfrequencies of the two surface modes in Fig. 4 in absence of a velocity shear, we conclude that [FORMULA] has to be -1.

[FIGURE] Fig. 4. The redraw of Fig. 2b in the reference frame in which the region at the left of the discontinuity is static. The gray lines indicate the upper and lower boundaries of the Alfvén continua.

In Fig. 5 we plot [FORMULA] as function of velocity shear for the eigenfrequencies of the surface modes corresponding to the upper branch and lower branch of Fig. 4. In this figure we also indicate the critical velocity [FORMULA], for which the forward propagating surface mode becomes a backward propagating one, with a vertical black line. The velocity shear for which the eigenfrequency corresponding to the upper branch enters the interval [FORMULA], is indicated with a vertical gray line.

[FIGURE] Fig. 5. [FORMULA] as function of velocity shear for the eigenfrequencies of the surface modes corresponding to the upper branch and lower branch of Fig. 4. The vertical black line indicates the critical velocity [FORMULA] whereas the vertical gray line indicates the velocity shear for which the eigenfrequency of the negative energy wave enters the lower Alfvén continuum.

Hence in case of a thin nonuniform transition layer the surface mode corresponding to the upper branch, which resonantly couples to a localized Alfvén continuum mode, is a negative energy wave for [FORMULA] and thus becomes unstable due to the presence of dissipation by resonant absorption.

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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