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 vph in the positive gradient region are unstable. We expect vph to occur where the gradient:(¶f(v)/ ¶ v) êwr/ k = maximum. Instability can result provided:
we 2 > k2v2 or êkv ê < we .
S Etot = ½ E1 w + E1 k = ½[ ½ eo | E1|2] + ½ r | ṽ|2
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:we = 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.
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
m is the electron mass, and M = mi,
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/ εo m e )
Show that this can be reduced to the simplified form:
n = 9 Ö N
(Hint: εo = 8.85 x 10 -12 farad m-1 )