4. Langmuir turbulence
4.1. Growth rates
The growth rate of Langmuir waves in a magneto-active plasma can be easily derived from a linearized Vlasov equation. Expressed in CGS units the rate becomes
The condition for erenkov resonance in a magnetized medium is
We define µ as the cosine of the pitch-angle of the particle, and the angle between and . The magnetic field modifies the oscillation frequency of Langmuir waves. The dispersion relation of the strongly magnetized Langmuir modes is
where we have taken , and .
The resonance condition (Eq. (16)) leads to
where the thermal speed of ambient electrons of temperature is . The dispersion relation in a magnetized plasma can be approximated as , and
In spherical coordinates we have
Equations (19) and (20) transforms the growth rate as
with the wave number defined as . Here we assume a mono-energetic distribution for the pair plasma in the plasma frame
The Lorentz transformation of the distribution function leads to . As we shall see, the real distribution function is not a delta function but the growth rate is not very sensitive to its exact shape.
Injected in Eq. (21) and using the Lorentz transformations for the momentum the derivation over µ and p reads
We finally obtain
Where selects the pitch-angle of a resonant particle for a given wave number k, and is the pair plasma frequency. The maximum growth rate is obtained for
It is straightforward to see from Eq. (24) that the forward modes () are destabilized () for particles moving within a cone of half opening angle . The backward modes () are destabilized by the particles with a pitch-angle such that .
4.2. Non-linear transfer
In an isothermal plasma the most important interactions mechanisms encountered by Langmuir waves plasma are the induced scattering off ions and/or electrons, and four plasmons interactions. The excitation of ion-acoustic oscillations must be added in a non-isothermal medium (Tsytovich (1977)). In the scattering process the erenkov resonance condition is generalized to the interaction of a particle and the beat of two Langmuir waves. The probability for scattering off electron and ion or ion waves can be derived from an expansion up to a second and third order Vlasov equation. In a field free plasma an estimate of the characteristic time for spectral transfer of energy over an interval for the scattering by the electrons
For a thermal population of ions and isotropic turbulence the transfer can be differential for (where is the Debye wave number). In this case the transfer occurs over an interval . An estimate of the characteristic transfer time is
It clearly appears that the thermal ions will dominate the scattering process for . This relation still holds in a magnetic field. We then focus on the ion scattering in the region of differential transfer, since it is indeed this process which is relevant in the range of resonance with particles. It contributes to the energy redistribution among particles, and to an inverse cascade. The inverse cascade leads to four waves interaction but in a wave number range outside the resonance region, and thus will be disregarded. Thereafter we will use a reduced turbulent spectrum () defined by
Following Tsytovich (1977), we introduce a characteristic wave number for scattering off ions
or in a non-isothermal plasma for scattering off ion acoustic oscillations
The electron temperature can be of order of the Compton temperature .
The kinetic equation describing the evolution of the turbulent spectrum is
The exact treatment of the transfer in the presence of an ambient magnetic field in equipartition with the thermal plasma is rather tedious. The transfer is not differential for and/or that is for where the scattering is mainly due to the thermal electrons. In fact, as seen in the previous section, the growing modes are those with a wave number close to . The energy redistribution in a wide range of k (and a wide angular range) is above all ensured by ion wave scattering and roughly described by Eq. (31). More precisely, the scattering in a strong magnetic field favors the almost elastic backscattering of the primary modes. Those secondary backscattered modes are thus absorbed at a rate almost equal to the growing rate of the primary modes (in absolute value). This is precisely by this means that the beam is re-heated. However the re-heating has a yield slightly reduced by the inverse cascade.
In a first approach we assume that these effects are not strong enough to modify neither well the dispersion relation nor the transfer equation. Works are in progress to treat this question in a more rigorous way.
The resulting spectrum is scale invariant. Adding the wave generation term contribution the stationary turbulent spectrum is obtained via the kinetic equation
For an isotropic turbulence we have
The lower limit of the integral is imposed by in the resonance condition of Langmuir waves. The quantity is the transfer spectrum due to ionic diffusion for . Replacing in Eq. (28) by its expression obtained in equation (20) and performing the integration over k and the turbulent spectrum becomes for
In the derivation of the stationary turbulent spectra we have limited the variation of between beyond which the waves are damped and 1.
The Langmuir modes are damped by Landau effect at the Debye scale (). We can then write , and
Finally the transfer spectrum is
The wave number is the characteristic scale of the turbulent energy transfer. The reduced Langmuir stationary turbulence spectra is given by Fig. 1.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998