When we are talking about any plasma system, the resonance in the system is satisfied by the condition (ω = k · v) without any magnetic field and by (ω − nωc = kzvz ) in a magnetic field. Why it happens so? What is its physical significance? Why only the parallel component contributes in the presence of magnetic field? What are the applications of this phenomenon in different branches of plasma physics starting from the laboratory plasma to the space and astrophysical plasmas? Kindly post your suggestions in the context of your expertise field. Thank you.
The resonance without magnetic field is due to the electric restoring field set up when electrons are collectively displace from the ions in a collisionless plasma (e.g. due the an electromagnetic wave). A magnetic field adds an additional force that changes the resonance frequency, but only if the motion is perpendicular to the magnetic field line. Motion along the field line would have no effect (you might as well have no field at all). My paper Working Paper Excitation of Oscillations of a Magnetized Collisionless Plasma
might be of interest in this respect
The resonance condition (frequency almost matching) is the situation of the maximum energy transfer (dielectric response function tending to infinity) between two or more participating dynamic objects in any medium.
It naturally happens only at the Doppler-shifted frequency tending to zero, thereby making the energy transfer infinitely large.
It is supported well in most of the instability processes in plasmas and fluids, such as the two-stream instability, KHI, RTI, etc.
Basic books on waves and instabilities may be read for further details.
The resonant condition is the situation of the most efficient exchange of energy. This can be understood from the point view of test particle, where the energy change rate of particle depends on E · v. with E being wave electric field. In the absence of magnetic field, there is no intrinsic frequency due to an oscillation of test particle without waves. Therefore, the most efficient exchange of energy or resonant condition of test particle with waves (e.g., the plasma wave in the Landau damping) would be the case when the test particle moves with the wave (v=ω/k) so that it experiences constant (rather than oscillating) electric field from wave. Therefore, the resonant condition is ω = k · v.
On the other hand, there appears an intrinsic oscillation frequency (gyro-frequency) of test particle in the presence of external magnetic field. As a result, even though the test particle moves with the wave so that it experiences constant electric (and magnetic) field, it doesn't exchange energy with wave efficiently due to the oscillating gyro-motion (oscillation of $\mathbf{v}_\perp$). In this case, the resonant condition occurs when the test particle frequency in the magnetic field matches with the frequency of wave experienced by the particle so that when $\mathbf{v}_\perp$ changes direction, the force from wave also changes direction. The computation of the latter is not straightforward. If we do canonical transformation so that the wave phase is the effective time ($\tau=\omega t -k_\perp x-k_\parallel z$), then the frequency of wave experienced by the particle is 1. From the action-angle variables we can find that the frequency of particle oscillating in the magnetic field in the new time framework is $\omega_c/(\gamma\omega-k_\parallel p_z)$. Therefore, the resonant condition finally reads $\omega-k_\parallel v_z=n\omega_c/\gamma$.
Yanzeng Zhang Thank you for such a nice explanation and adding your own flavour to it.
Thanks. Please feel free to let us know if there is any more question.
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