8. Summary and discussion
In this paper we investigated how a velocity shear alters the spectrum of the MHD surface modes on a single interface. The single interface is modeled by two semi-infinite uniform regions separated by a nonuniform layer. Due to the presence of nonuniformity a surface mode with characteristic frequency within the range of one of the continuous spectra (associated with the nonuniform layer) may resonantly couple to localized Alfvén and/or slow continuum modes.
To illustrate the combined effect of the velocity shear and the resonant absorption process on the spectrum of the MHD surface modes, we considered in a first step a cold plasma (), so that the slow waves are absent. This simplifies the picture of the spectrum of the surface modes as function of the velocity shear.
In the case of a true discontinuity or the single interface, we showed that Kelvin-Helmholz instability occurs when the oscillation frequencies of the "forward" and "backward" propagating surface modes merge for increasing velocity shear. The merging point then indicates the threshold of Kelvin-Helmholz instability for the single interface.
In the case of a nonuniform layer, the continuous spectra of "forward" and "backward" propagating Alfvén modes are Doppler shifted due to the presence of the mass flow. When the oscillation frequency of the "forward" ("backward") propagating surface mode lies in the continuum of the "forward" ("backward") propagating Alfvén modes, the surface modes are damped due to the resonant wave excitation. However, when the oscillation frequency of the "forward" propagating fast surface mode lies in the frequency range of the "backward" propagating Alfvén waves, which is only possible due to the presence of the velocity shear, the surface mode is unstable, i.e. the surface mode gains energy from the flow at the resonance layer.
In a reference frame moving with the mass flow, it can clearly be seen that whereas the "backward" propagating surface mode is accelerated, the "forward" propagating surface mode is retarded. Hence a sufficiently large flow changes the sign of propagation. In this case this surface mode becomes a negative energy wave when the shear velocity is larger than a certain critical velocity (determined by the condition in Figs. 3a and 3b) which turns out to be much smaller than the threshold for Kelvin-Helmholz instability. Note that . With the negative energy, we mean the "linear" part that is associated with the linearized equations and that is of course not the total energy (see Ostrovskii et al. 1986).
Ostrovskii et al. (1986) have shown in hydrodynamics that negative energy waves are unstable when dissipation is present. Later on Ryutova (1988), Hollweg et al. (1990) and Ruderman & Goossens (1995) found the same instabilities of negative energy waves in different dissipative systems. The introduction of wave energy dissipation due to either sound-wave radiation, resonant absorption or viscosity makes negative energy waves unstable. Hence a presence of any dissipation in the system leads to amplification of the wave. The conservation of energy, however, is not violated, because energy moves from the flow into the waves.
The interpretation in terms of negative energy waves can be worked further out analytically when the thickness of the nonuniform transition layer is assumed to be small compared to the wave length of the surface mode (Erdélyi et al. 1998).
The most important feature of this instability is that its threshold velocity is smaller than the threshold velocity for the onset of the Kelvin-Helmholz instability. In this paper we restricted the calculations to the case in order to give a clear picture of the nature of the resonant flow instability. In contrast, the analysis of the solutions inside the dissipative (Alfvén and cusp) resonance layer is performed in a general way, so that extension to parametric studies of more complicated configuration is straightforward.
© European Southern Observatory (ESO) 1998
Online publication: March 23, 1998