In earlier blog posts we examined the two stream instability, which also has significance for solar and space physics. We saw it can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species (ions and electrons) can have different drift velocities. The energy from the particles can lead to plasma wave excitation
Basically, the two
stream instability can be thought of as the inverse of Landau damping,
where a greater number of particles that move slower than the wave phase
velocity vph (as compared with those that move
faster), leads to an energy transfer from the wave to the
particles. In the case of the two stream instability,
when an electron stream is injected into the plasma, the particle's velocity
distribution function has a "bump" on its tail as shown in Fig. 1.
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 ov)
and perturbed one ( f eb )
applicable to the electron beam (See diagram below ).
In the region where the slope is positive (df v /d
v > 0) there is a greater number of faster i.e. 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 vph (phase
velocity) in the positive gradient region are unstable.
Resonant electrons (at v ph > w e / k) where w e is
the electron plasma frequency, i.e.
w e = [ne e2/
me εo] ½
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) = ½ E1 w + ½ 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.
The electron plasma frequency can be written:
So that: ε = (1 - w e 2 / w 2 )
The dispersion relation here is equivalent to equating ε ( w) to 0.
N.B. A dispersion relation implies that a relationship exists between the plasma frequency w and the wave number k. Now, and this is critical, because we have ε ( w) = 0 this implies:
1 = w e 2 / w 2
Or: w = w e
Now, assembling all the preceding results allows us to write:
ik = 4p e [ (ik n o e)/ m iw 2 - ik n o e/me (w - k Vo ) 2]
Using the earlier equation for the electron and ion plasma frequencies and basic algebra, the energetic reader can satisfy himself that:
ε ( w) = 1 - w i 2 / w 2 - w e 2 /(w - k vo ) 2 = 0
Note that in the limit, m i -> oo and w i -> 0 we have:
We can then look at wave numbers k such that k Vo = w e
and acknowledge that that 2nd term above is much less than unity (to cancel the 1st term the 3rd must be close to unity) whence:
0 = 1 - w i 2 / w 2 - w e 2 /(w - w e ) 2
ε ( w) = 1 - w i 2 / w 2 - w e 2 /(w - k vo ) 2 = 0
yields instability. This can be done via a simple algebraic manipulation, so we transpose the negative terms to the right hand side and let:
F(k, w ) = ε ( w) - 1 = w i 2 / w 2 + w e 2 /(w - k vo ) 2
We can then plot the resulting function vs. the real frequency (at fixed wave number, k) as depicted in my 2nd graph above.
Note where the line at unity intersects the graph of F(k, w) at four different points. In other words, there are four real roots and no instability for the selected value of k. But...what happens when the central minimum of F(k, w) occurs at a value greater than unity? Then, there are only two real roots .
To determine when this happens we determine when: F(k, w ) > 1, or where the function minimum is determined by ¶ F/ ¶ w = 0. We then will find:
w (min) = (m e /m i) 1/2 k Vo
Yielding an equation which predicts instability whenever:
| k Vo | < w e
