Looking Again At The Two-Stream Instability Of Plasma Physics

 

Profile for two-stream instability.
 




















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 v_ph (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 v_ph (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     =  [n_e e²/ m_e  ε_o] ^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.

Development:

The electron plasma frequency can be written:

w _e     =  [n_e e²/ m_e  ε_o] ^½

Where n_e is the electron number density, e  is the electron charge,  m_e  is the electron mass and ε_o  is the permittivity of free space.

For our purposes in looking at the 2-stream instability we will change the above form to read:

w _e²     =   4p n _o e²/ m_e  

Similarly, we can write:

w _i²     =   4p n _o e²/ m _i

For the electron ion frequency.

For our purposes also, the Poisson equation, for charges in a vacuum:

Ñ ·E  =  4p r           

Can be rewritten:

ik E =  4p r  

Now, write   in terms of E such that:

ik (E -  4p r / ik ) = ikε E = 0

For cold plasma waves (T _e =T_i  = 0)  we can write:

ik (1 -     w _e ² / w ²  ) E =  0

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/m_e (w  - k V_o ) 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 v_o ) ²  =  0


Note that in the limit, m _i    -> oo  and  w _i -> 0  we have: w  =   k V_o  +  w _e

We can then look at wave numbers k such that  k V_o  =   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

0 =  1 -  w _i  ² / w ²    -    1/(1  -   w  /  w _e ) ²

0   »   1 -  w _i  ² / w ²    -  (1 +     2w / w _e ) 

And:  0 = -  w _i  ² / w ²   -   2w / w _e  

w ³      =  - ½ ( w _i ) ²  (w _e)   

Finally, we can write:

w /w _e  =   ( -1/2) ^1/3   (m _e / m _i) ^1/3       

which represents instability since one of the three values of  (-1) ^1/3   is (1/2) + iÖ 3/ 2.

Thus, in the frame moving with the electrons, the Doppler shifted frequency 
(since   k V_o   =  w _e) :

w’ =   w  - k V_o

Factor into the mix that |  w  |  < <  w _e    and we conclude this is roughly w’ =   w _e  so the electrons are nearly at their natural frequency of oscillation. But there is another way to determine that the dielectric function, i.e.

ε ( w)  =  1 -  w _i  ² / w ²   -  w _e ² /(w  - k v_o ) 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 v_o ) 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 V_o

Yielding an equation which predicts instability whenever:

| k V_o  |    <   w _e