5. Alfvén turbulence
The electro-magnetic modes generated in the cold plasma can be excited by the relativistic particles if they fulfill the Landau-synchrotron resonance condition
Where is the gyro-magnetic frequency of the resonant particle. For a frequency range of we are mainly interested with the excitation of MHD waves by kinetic effects and particularly with the Alfvén waves.
In this case, we will only retain the resonances with and will neglect in regards of the two other terms. Thus resonance with Alfvén waves occurs for . Because this threshold is high, Langmuir turbulence is essential to accelerate particles up to an energy above the threshold.
5.1. Growth rates
The previous remarks allow us a direct derivation of the growth rate of Alfvén waves from Melrose (1968):
where is the Alfvén speed. In the previous expression p must fulfill the resonance condition
The derivation of the growth rate with a mono-energetic distribution function (Eq. (22)) is rather tedious, since contains a factor in addition to the other factor included in the resonance condition. The detailed calculation is reported to the appendix. The final expression can be cast into the form
where . The forward modes () are unstable for .
5.2. Non-linear transfer
We are interested with non-linear quadratic interactions involving Alfvén waves. The second order processes are essentially scattering off ions and three modes coupling. The first process has been already studied by Kaplan and Tsytovich (1973). Akhiezer et al. (1967) have derived the plasmon collision integral involving Alfvén with fast and slow magneto-sound waves. There is no process involving three Alfvén waves since the matrix element vanishes in this case. Moreover in the isothermal plasma considered here there are no coupling with ion-acoustic oscillations.
All these processes imply a formally identical evolution equation for the reduced turbulent spectrum . This spectrum is normalized as
where is the turbulence level.
Including the growth rate this equation takes the form
and is the non-linear kernel of the interaction ( the wave number of the second decay plasmon is characterized by its modulus , its angle relatively to the magnetic field ), and is of order of unity.
These interactions do not favor any particular scale. Thus the kernel of order of 1 can be written as . If we use the new variable in the Eq. (43), we obtain
Let us assume that the growth rate is a power-law function of k say
where . We seek for solutions of the form
where the functions g and h represent the anisotropic contribution to both growth rate and turbulent spectrum and where . Reporting Eqs. (45) and (46) in Eq. (44) we easily find a simple relation between n and m
The isotropic part of this turbulent spectrum can be cast into the following form obtained from (41)
is the instability contribution to the turbulent spectrum.
We have also to consider the inertial contribution, which must cancel the first term of the Eq. (44). The resulting transfer spectrum has a power-law form with an index p.
Note that p is not unique, see for example the solutions in and derived by Mc Ivor (1977) and Achterberg (1979), the first one corresponding to a zero energy flow and the second (Kraichnan spectrum) to a constant energy flow towards the smallest scales. Like in the case of Langmuir wave, mostly backward waves contribute to accelerate particles.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998