In this section we present the spectrum of the MHD surface waves on a single interface as a function of the velocity shear. We resolve the spectrum for the plasma parameter equal to zero with and without a nonuniform transition layer. We restricted the calculations to . This simplifies the picture of the spectrum of the surface modes as function of the velocity shear, since by putting the slow MHD surface mode is excluded from the model. In the presentation of the results is defined as the angle of propagation with respect to the magnetic field, i.e. and . The calculations are performed with and .
When , the slow surface mode is excluded from the configuration, whereas the fast surface mode only exists for a propagation angle different from and . Since the interface is a true discontinuity, there is no Alfvén continuum so that the surface mode cannot be subject to resonant absorption. Any change in the eigenfrequency is due to the presence of the velocity shear.
In Fig. 2 we have plotted the oscillation frequency of the two fast surface modes for two different angles of propagation (Fig. 2a, Fig. 2b) together with the imaginary part of the eigenfrequency (Fig. 2c, Fig. 2d) as function of the velocity shear V. The surface mode with positive frequency is a "forward" propagating wave, while the mode with negative frequency is a "backward" propagating wave. The figures clearly show that when the oscillation frequencies of the "forward" and "backward" propagating fast surface modes merge for increasing velocity shear V, the interface becomes unstable. Hence the merging point indicates the threshold for Kelvin-Helmholz instability.
When we introduce a nonuniform layer, the surface modes with oscillation frequency within the range of the Alfvén continuum resonantly couple to localized Alfvén waves, and, in absence of a velocity shear, they are damped due to the resonant wave excitation.
In these figures we also indicated the upper and lower bounds of the Alfvén continua and which are Doppler shifted due to the presence of the mass flow in the nonuniform layer.
In Fig. 3c and Fig. 3d the corresponding imaginary parts of the eigenfrequencies are plotted as function of the velocity shear V. For both surface modes have their characteristic frequency within the range of the Alfvén continuum (see Figs. 3a and 3b) and hence are damped due to resonant absorption. However when V is increased, the oscillation frequency of the "forward" propagating surface mode shifts out of the Doppler shifted upper Alfvén continuum and the mode becomes undamped, whereas the oscillation frequency of the "backward" propagating surface mode remains for a larger interval for V in the Doppler shifted continuum of "backward" propagating Alfvén modes. In first instance the damping rate increases, reaching a maximum damping rate and then decreases again towards zero at the velocity shear for which the oscillation frequency shifts also out of the Doppler shifted Alfvén continuum.
The most interesting feature in the spectrum appears when the oscillation frequency of the "forward" propagating surface mode shifts into the Doppler shifted continuum of the "backward" propagating Alfvén modes : as long as the oscillation frequency of the "forward" propagating surface mode lies within the range of the Doppler shifted continuum of "backward" propagating Alfvén modes, the surface mode is unstable, i.e. .
Physically this means that the flow acts at the resonant layer as a source of energy from which the surface mode gains energy. From a mathematical point of view, the reason lies in the fact that , which causes the jump in the Poynting flux across the resonant surface to be negative. The x -component of the modal energy flux, averaged over the oscillation period, as seen by an observer fixed relative to some constant flow (which may be taken as the plasma flow V) is given by (see e.g. Adam 1978)
where the asterisk denotes the complex conjugate. Note that the modal energy flux, averaged over the oscillation period, exponentially increases or decreases depending on the overstability or damping of the mode. With the analytical solution (11) inside the dissipative resonance layer, we can calculate the jump in the energy flux across the dissipative layer as :
Using the asymptotic expansion of the function as (see e.g. Goossens, Ruderman, & Hollweg 1995),
we can reduce the jump in the modal energy flux across the dissipative resonance layer approximately to
When this jump in energy flux is positive, there is an energy transfer from the fast surface mode in the Alfven resonance layer and hence the surface mode is damped. However when the jump in energy flux across the resonance layer is negative, this means that there is an energy flow from the resonance layer towards the fast surface mode, which is unstable in this case. This happens only when the oscillation frequency of the "forward" propagating surface mode lies in the Doppler shifted continuum of "backward" propagating Alfvén modes. It is easy to check that the formula (16) is consistent with the Fig. 3c and Fig. 3d.
When the oscillation frequencies of both the "forward" and "backward" propagating surface modes shift out of the lower Doppler shifted Alfvén continuum, they merge leading to Kelvin-Helmholz instability.
© European Southern Observatory (ESO) 1998
Online publication: March 23, 1998