*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*(as compared with those that move faster), leads to an energy transfer from the wave to the particles. In the case of the

_{ph}**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

_{w}**= maximum. Instability can result provided:**

_{r/ k}
w

_{e}^{ 2 }> k^{2}v^{2 }or**ê**kv**ê****<**w_{e}^{ .}_{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 }*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)

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}/m^{3}). 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**if the frequency w = 10**_{ o}**/ s**^{10}

(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**farad m**^{-12}**)**^{-1}
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