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{\Large \bf AS554\quad Irreversible Processes in Plasmas\\
\today
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\section*{Problem Set \#6 (due Apr.~16, 2001)}
\ignore{
\subsection*{Question 1}
Determine the electrical conductivity of a Lorentz plasma. [Take $Z_i$
large so that only electron-ion collisions need to be considered. Write
$f_e=f_M+f_1$ where $f_M$ is a Maxwellian. Linearize the Fokker-Planck
equation (including an external electric field) in $E$, assuming
$f_1=O(E)$. Solve the resulting equation.]
} % end ignore
\subsection*{Question 1: Chapman-Enskog-Braginskii procedure}
In class I wrote down the first order kinetic correction equation
\[
{\partial f_0 \over \partial t} + \vec v \cdot {\partial f_0 \over \partial
\vec x} + {q \over m} \vec E_* \cdot {\partial f_0 \over \partial \vec v} =
- \Omega (\vec v - \vec u) \times \hat{b} \cdot {\partial f_1 \over
\partial \vec v} + \hat{C} f_1
\]
where $\hat{C}$ is the linearized collision operator, $\Omega$ is the
cyclotron frequency, $\hat{b}=\vec B / B$ is the direction of the magnetic
field, and $\vec E_* = \vec E + \vec u \times \vec B /c$ (the electric
field with the dominant $\vec E \times \vec B$ drift component taken out).
$f_0$ is a Maxwellian specified by the density, flow, and temperature
$n(\vec x,t)$, $\vec u(\vec x,t)$, and $T(\vec x,t)$. Expand $\partial
f_0/\partial t$ and $\partial f_0 / \partial \vec x$ explicitly in terms of
gradients of $n$, $\vec u$, and $T$. Then use the lowest order fluid
equations (i.e., Braginskii's fluid equations but with the closure terms
$\vec q$ and $\vec{\vec P}$ set to zero, since they vanish for a Maxwellian
and are thus higher order, and also neglect the collisional friction and
heating terms $\vec R_i$ and $Q_i$ as they are weak for $i-e$ collisions)
to eliminate the time derivatives. Show that the resulting equation can be
written as:
\[
\eqalign{
- \Omega \delta \vec v \times \hat{b} \cdot {\partial f_1 \over
\partial \vec v} + \hat{C} f_1
&=
f_0 \left( {m \over 2} {|\delta \vec v|^2 \over T} - {5 \over 2}
\right) \delta \vec v \cdot \grad \ln T \cr
&+ f_0 {m \over 2 T} \left( \delta \vec v \delta \vec v
- {|\delta \vec v|^2 \vec{\vec 1} \over 3} \right)
: {\vec {\vec W}}
}
\]
where $\vec{\vec{W}} = \grad \vec u + (\grad \vec u)^{\rm T} - (2/3)
{\vec{\vec 1}} \grad \cdot \vec u$, and $\delta \vec v = \vec v - \vec
u(\vec x,t)$. [This result is equivalent to Krommes' Ch.31 Eq. 34
generalized to include a magnetic field, Krommes' Ch. 33 Eq. 54, or
Braginskii's Eq. 4.15.]
Consider the simple limit of a Krook model collision operator $\hat{C} f_1
= - \nu f_1$, no magnetic field, and no gradients in the flow so $\vec{\vec
W}=0$. Calculate the heat flux $\vec q = \int d^3 v f_1 0.5 m |\delta \vec
v|^2 \delta \vec v$ and find the result in the form $\vec q = - n \chi
\grad T$. What value should $\nu$ have to match Braginskii's result for
the ion heat flux in the $\vec B=0$ limit, on p.38 of the Formulary?
\newpage
\subsection*{Question 2}
Using the Rosenbluth potentials, evaluate the collisional velocity-space
flux ${\bf J}^{a/b}$, when $f_b$ is a Maxwellian. Express in terms
of the relaxation rates $\nu_s^{a/b}$, $\nu_\parallel^{a/b}$, and
$\nu_\perp^{a/b}$ (as defined in the NRL Plasma Formulary). Put the answer
in a form similar to the one in the Formulary (under Fokker--Planck
Equation) (but note that the answer in older editions of the Formulary is
wrong!). (I did some of this in class but only for the slowing down
$\nu_s^{a/b}$. You should complete the calculation and do it for all
terms.)
\def\erf{\mathop{\rm erf}\nolimits}
\def\gradv{\nabla_{\bf v}}
%\noindent
Useful formulas:
The Rosenbluth potential form derived in class (which is slightly different
from the form in the Formulary, where the order of one of the derivatives
has been interchanged) can be written as:
\[
{\bf J}^{a/b}= \nu_0^{a/b}v^3
\biggl[{m_a\over m_b}(\gradv h_b)f_a
-{1 \over 2}(\gradv\gradv g_b)\cdot\gradv f_a\biggr]
\]
where
\[
\nu_0^{a/b}={4 \pi e_a^2 e_b^2 \log\Lambda_{ab} \, n_b\over
m_a^2 \, v^3},
\]
(as defined in the NRL Plasma Formulary) and
\[
\gradv^2 h_b= -4 \pi f_b,\qquad \gradv^2 g_b=2 h_b,\qquad
\int h_b\,d^3{\bf v}=1.
\]
Note that in MKS units, $\nu_0^{a/b}$ is:
\[
\nu_0^{a/b}={e_a^2 e_b^2 \log\Lambda_{ab} \, n_b\over
4\pi \epsilon_0^2 m_a^2 v^3},
\]
If $f_b$ is spherically symmetric, $h_b$ and $g_b$ are
found by straightforward integration. For example, if $f_b$ is a Maxwellian
with temperature $T_b$ then
\[
{d h_b\over dv}= - {\psi(x^{a/b}) \over v^2 } =
- {1\over v^2}
{\nu_s^{a/b}\over (1+m_a/m_b)\nu_0^{a/b}}
\]
where $x^{a/b} = v_a^2/(2 T_b/m_b)$ (this is the definition of $x^{a/b}$ as
given on p. 31 of the NRL formulary, though when used in the operator on
p. 34 for collisions with Maxwellian species, it is to be evaluated at $v_a
= v$, where $\vec v$ is the argument of $f_a$ on p.34) and
\[
\psi(x) = {2 \over \sqrt{\pi}} \int_0^x dt \, t^{1/2} e^{-t}
= \erf(\sqrt{x}) - \sqrt{x} {2 \over \sqrt{\pi}} e^{-x}
\]
in terms of the error function
\[
\erf(u)={2\over \sqrt\pi}\int_0^u \exp(-v^2)\,dv,\qquad
\]
\ignore{
\[
\eqalign{
{d h_b\over dv}&= - {1\over v^2}
\bigl(\erf(u)-u\erf'(u)\bigr)=- {1\over v^2}
{\nu_s^{a/b}\over (1+m_a/m_b)\nu_0^{a/b}}\cr
{d h_b\over dv}&={1\over 2}
\biggl(\biggl(2-{1\over u^2}\biggr)\erf(u)+{\erf'(u)\over u}\biggr)
={1\over 2}{\nu_\perp^{a/b}\over\nu_0^{a/b}}\cr
{d^2 h_b\over dv^2}&={1\over v}
\biggl({\erf(u)\over u^2}-{\erf'(u)\over u}\biggr)
={1\over v}{\nu_\parallel^{a/b}\over\nu_0^{a/b}}\cr
\erf(u)&={2\over \sqrt\pi}\int_0^u \exp(-x^2)\,dx,\qquad
\erf'(u)={2\over \sqrt\pi}\exp(-u^2)\cr
u&=v/\sqrt{2T_b/m_b}\cr
}
\]
}
\subsection*{Question 3}
Evaluate this expression for $J^{a/b}$ in a limit appropriate for $i-e$
collisions, $m_a \gg m_b$ (and $x^{a/b} \ll 1$), and ignore collisions with
other ions. This is the limit of ``Brownian motion'', where massive
particles (such as dust particles in air or particles in a colloidal
suspension) move in response to collisions with much lighter particles.
Show that the resulting collision operator can be written in the form
\[
C_{ie}(f_i) = {\partial \over \partial \vec v}
\cdot \left( \nu_{ie} \vec v f_i + D {\partial f_i \over \partial \vec v}
\right)
\]
where the velocity space diffusion coefficient satisfies the Einstein
relation $D=\nu_{ie} T_e/m_i$. (In the previous homework on the
slowing down of beams, the velocity space diffusion due to $i-e$ collisions
was neglected since pitch angle scattering by $i-i$ collisions is the
dominant process.)
Using this expression for the $i-e$ collision operator in
\[
{\partial f_i \over \partial t} = C_{ii}(f_i) + C_{ie}(f_i)
\]
calculate the rate of temperature equilibration, given ions and electrons
initially at temperatures $T_i$ and $T_e$. You can assume that $i-i$
collisions (which are $\sqrt{m_i/m_e}$ stronger than $i-e$ collisions) are
strong enough to force $f_i$ to always be a Maxwellian. Compare your
result with Braginskii's ion heating rate $Q_i$ as given on p.37 of the
Formulary. In what limit will the more general expression for thermal
equilibration on p. 34 of the Formulary agree with your calculation?
\end{document}