Generals 2001 II 6A

The Alfven velocity is:

$v_{A}=\frac{B}{\sqrt{4\pi n_{i}m_{i}}}$

The ion sound speed is:

$v_{s}=\sqrt{ZT_{e}/m_{i}}$

So that if $v A ˜ v s$ :

$\frac{B^{2}}{4\pi n_{i}m_{i}}\sim\frac{ZT_{e}}{m_{i}}$

Then:

$\frac{B^{2}}{4\pi n_{i}ZT_{e}}\sim1$

By quasineutrality $n i Z = n e$ :

$\frac{B^{2}}{4\pi n_{e}T}\sim1$

Since $β = 8π n T / B 2$ :

β˜2

For the ion acoustic wave to have landau damping:

v s ˜ v T i

Plugging in:

$\sqrt{\frac{ZT_{e}}{m_{i}}}\sim\sqrt{\frac{T_{i}}{m_{i}}}$

Or:

$\frac{ZT_{e}}{T_{i}}\sim1$