Generals 2005 I 3

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[edit] Part 1

We have:

\mathbf{E}+\frac{\mathbf{u}\times\mathbf{B}}{c}=\eta\mathbf{j}+\frac{\mathbf{j}\times\mathbf{B}}{en_{e}c}-\frac{\nabla p_{e}}{en_{e}}


Writing the last two terms, with \mathbf{B}=B, \nabla p_{e}=n_{e}T/h_{p}, \mathbf{j}=\frac{c}{4\pi}\nabla\times\mathbf{B}=B/h_{B}:

\cdots\frac{B^{2}/h}{4\pi en_{e}}-\frac{n_{e}T}{en_{e}h_{p}}


The \mathbf{u}\times\mathbf{B} term is, for \mathbf{u}\approx v_{T}:

\frac{v_{T}B}{c}


So comparing with the \mathbf{j}\times\mathbf{B} term, the latter term will be negligible if:

\frac{v_{Ti}B}{c}\gg\frac{B^{2}/h_{B}}{4\pi en_{e}}


Or:

\frac{v_{Ti}}{c}\gg\frac{B/h_{B}}{4\pi en_{e}}=\frac{\Omega_{i}c}{\omega_{pi}^{2}h_{B}}


So the condition is that:

h_{B}\gg\frac{1}{\rho_{i}}\frac{c^{2}}{\omega_{pi}^{2}}


The second term will be negligible if:

\frac{v_{T}B}{c}\gg\frac{n_{e}T}{en_{e}h_{p}}


Or:

1\gg\frac{Tc}{eh_{p}B}\sqrt{\frac{m_{i}}{T}}=\frac{v_{T}}{\Omega_{i}h_{p}}


So the \nabla p term is negligible if:

h_{p}\gg\frac{v_{T}}{\Omega_{i}}

[edit] Part 2

Along magnetic field lines, the scale length will be λmfp = vT / ν. The relevant collision frequency will be self-collisions (with the like species). The relevant time scale will be ν. Then:

\chi_{e}\sim v_{Te}^{2}/\nu_{ee}=T_{e}/\nu_{ee}m_{e}


\chi_{i}\sim v_{Ti}^{2}/\nu_{ii}=T_{i}/\nu_{ii}m_{i}


The ratio is then:

\frac{\chi_{i}}{\chi_{e}}\sim\frac{T_{i}/\nu_{ii}m_{i}}{T_{e}/\nu_{ee}m_{e}}\sim\frac{\nu_{ee}}{\nu_{ii}}\frac{m_{e}}{m_{i}}\sim\left(\frac{m_{e}}{m_{i}}\right)^{1/2}
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