Expert: Philip A. Stahl - 5/10/2009

Question
I am interested in the sort of plasma instabilities that occur in solar conditions (say before solar flares) and what their thresholds are for occurrence. For example, when does the ion acoustic instability occur? What velocity exceeds what factor or other velocity to make it happen? What about the beam instability, two-stream instability? The Buneman instability? Any help is appreciated!

Answer
Hello,

Yes, there are a number of instabilities associated with solar plasma, mostly defined in terms of specific velocities. One of the more important of these is known as the electron drift velocity (v(d)):

v(d) =   I/ n e

where I is the current, n the number density (per cubic meter, for example) and e the electron charge. Often it is defined instead based no current density (J) where: J = I/A  (current in amps flowing through a magnetic flux tube or channel of some cross-section area A)

So:  v(d) =   J/ A n e

In many other cases, where J, I are not so easily obtained,  v(d) will be the electron drift velocity in a combined magnetic and gravitational field:

v(d) =  m_e g_S c/  eB

where B is the magnetic induction, e the electronic charge, g_S = 273 m/s^2 the gravitational acceleration on the Sun and m_e the electron mass.


In the case of the ion-acoustic instability, the threshhold for its onset is when:

v(d) >  43 v(s)

that is, the drift velocity is at least 43 times the ion sound speed, where:

v(s) = (kT/ m_i)^1/2

where k is the Boltzmann constant, T the temperature and m_i the ion mass.

Sen and White in a 1972 paper dealing with the role of the Hall effect in flares, showed that the two-stream instability is incepted when the drift velocity:

v(d) >  v_(i)th

where v_(i)th is the ion-thermal velocity:

v_(i)th  = 2 (kT/m_i)^1/2   (rms value = {3kT/m_i)^1/2


Note also that the "plasma beam instability" is just the finite temperature analog for the two stream instability. In other words, if one has a "two stream instability" one de facto also has the beam instability.

In two-stream instability, when an electron flow is suddenly injected into a plasma – say for a coronal loop  – the particles’ (Maxwellian) velocity distribution acquires a “bump” on its "tail" (higher velocity end of the distribution), consistent with two streams- an unperturbed one f_o(v)   and perturbed one (f_eb ) applicable to the electron beam.   

In the region where the slope is positive (df(v)/d v > 0) there is a greater number of faster than slower particles so a greater amount of energy is transferred from particles to associated (e.g. Alfven) waves.  Since f_eb contains more fast than slow particles a wave is excited, and there is inverse Landau damping such that plasma oscillations with v_ph (phase velocity) in the positive gradient region are unstable.

Resonant electrons (at v_ph  >  w_e/ k, w_e the electron plasma freqquency, k  the wave number)   are the first to be affected by the local wave-particle interactions and have distributions altered by the wave electric field, E1, such that the total energy balance:

E1(TOT) = ˝ E1_ w + ˝ E1_ k    

referencing the wave and kinetic (particle) contributions respectively.

Thus, for  E1(TOT) = const. then as the electron velocity decreases, the particle kinetic energy decreases and the wave energy density increases. In Landau damping the exact opposite occurs, so the gradient df(v)/d v decreases, and with it the wave amplitude, while the particle kinetic energy increases- i.e. wave energy lost is fed to the particles (electrons) which gain energy.

Lastly, for the Buneman instability, the relevant condition is:

v(d) >  v(e)_th

where v(e)_th is the electron thermal velocity, v(e)_th = (2kT_e/m_e)^1/2

where T_e is the electron temperature and k the Boltzmann constant.

The rms (root mean square) value for v(e)_th is  (3kT_e/m_e)^1/2

Hppefully this answer sheds some light.