Expert: Philip A. Stahl - 4/3/2007

Question
Can you tell me under what conditions the two stream instability occurs for plasmas? I understand that complex roots come out in one set of conditions, what are they?

Answer
Hello,

Two stream instability is a very common phenomenon in plasmas.  It’s usually induced by an energetic particle stream injected in a plasma (for example, into solar coronal loop plasma) or inducing a current in the plasma so different species (ions and electrons) assume different drift velocities. The energy from the particles can lead to plasma wave excitation.

It is believed that the so-called "return beams" said to be observed in certain coronal flares in the pre-flare stage (for example, the 1B/M4 flare of Nov. 5, 1980) arise from this instability.


It can also occur in the magnetosphere (to do with aurorae) under the right conditions.

In finding conditions under which it operates, one considers treating a dispersion relation for plasma waves such that:

 F(x, y) = (me/mi)/ x^2 + 1/ (x^2 - y^2)

 where (me/mi) denotes the electron to ion mass ratio, and we define the variables x, y as follows:

 x = w/ w_e   or the ratio of the plasma frequency to electron plasma frequency

 y = k V_o/ w_e  or the ratio of the product of the wave number k by the electron thermal velocity (V_o) to the electron plasma frequency

 Plotting the graph on the axes:

 F(x, y)
 !
 !
 !- 1
 !
 !
 !
 !-----------------!---------->x
 0                  x=y

 Will yield a bifurcated graph with 4 roots. It will always feature a local minimum F_m such that:

 0 <=  F_m < =  x=y

 When:

 F(x_m, y) = F_m <  1  there will be four real roots

 When F_m >  1 there will be two real and two complex roots

 with suitable approximations, e.g. k^2 V-o^2 <=  w_e^2

 the complex roots are found to be:

 w/w_e =  1/2 [1 +  i(3)^1/2]

 w/w_e = 1/2 [1 -  i(3)^1/2]


 These will give the limits for the instability for when F_m > 1