Coronal Mass Ejections and Collisionless Shocks (2) - With Problems

In the previous instalment I noted that the dissipation associated with collisionless plasma can be attributed to Landau damping and this may be the most common form to excite a shock associated with CMEs. But even more germane is the two stream instability which 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 to the plasma, the particle's velocity distribution function has a "bump" on its tail:


In the region where the slope is positive (¶f(v)  / ¶ v > 0) there is a greaterI 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 in the positive gradient region are unstable. We expect  v_ph  to occur where the gradient:(¶f(v)/ ¶ v) ê_{wr/ k}  = maximum. Instability can result provided: 

w_e  ²   >   k²v²  or  êkv ê  <   w_e  ^.

Resonant electrons (at v_ph  »  w_e/ k)   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:

S E_tot = ½ E1 _w + E1 _k    =  ½[ ½ e_o | E1|²]  +  ½ r | ṽ|²         

For which the wave energy (E1 _w) is balanced against the kinetic (particle) energy (E1 _k ). Thus, for S E_tot = const. then as ṽ decreases, ½ e_o | E1 |²]   increases. In Landau damping the exact opposite occurs, so as the decrement associated with ¶f(v)/ ¶ v decreases, the wave amplitude (and  E1 _w) decreases, particle kinetic energy increases- i.e. energy is fed to the particles at expense of waves.

For reference, the longest known radio signatures for coronal shocks are the Type II radio bursts. These are narrow band (Dn/n ~ 0.1)  ) radio emissions excited at the local plasma frequency for which the drift rate :

dn/dt ~ ( - 0.2 MHz/s)

Where the negative sign indicates a drift toward lower frequencies. This is as the shock propagates outwards through the corona. Derived measurements disclose a radial velocity on the order of 1000 km/s. In 60 per cent of cases there is a clear harmonic structure with a frequency of occurrence of once every 100 hours. The Type II bursts are accompanied by weak polarization. The original excitation is usually by a fast mode MHD shock though the mechanisms previously considered, i.e. via Landau damping and two stream instability, must also be considered.

If we know the ambient free electron density associated with the event, it is possible to estimate the pure plasma  frequency (n)  using:

1)      n   =   9000  Ö N    or    2) n   =   9  Ö N   

      

In the above (1) is used to get MHz if the units for N are in c.g.s. units, and (2) for S.I. units. Bear in mind:w_e   =   2p n.

It is also useful to note that radio waves cannot propagate through a medium in which the plasma frequency  is greater than the radio frequency.

Problems:

1)For a hydrogen plasma, a lab generated shock of Mach number M= 80  with the temperature at T =   10 ⁵    K is found to have an ion gyroradius of 0.22 cm.  If    Ω _e  =  1.9 x 10 ⁷  /s  what would be the magnitude of the associated magnetic field (B)?

2)Find the associated plasma frequency and use it to obtain the Debye length and plasma parameter if n is a value typical of the solar corona  (10 ¹⁶ /m³). Find also the frequency n of the associated radio emission.

3)If the shock speed is u =  10 ⁷  cm/ sec, estimate the wave number k _w.  

4)Hence, or otherwise, use the equation below to obtain the ion velocity  v _o   if the  frequency w  = 10 ¹⁰ / s

(Note: m is the electron mass, and M =  m_i, the ion mass.)

5)Based on all the above information, estimate the maximum magnetic field B _m using the shock equation.

6) The frequency of a critical layer as a function of electron density is given by:

n   =  e/2p  Ö( N/ ε_o m _e )

Show that this can be reduced to the simplified form:

n   =   9  Ö N

(Hint: ε_o  =  8.85 x  10 ^-12  farad m^-1 )