Plasma Derivations

February 15, 2014

Chapter 1General Plasma

1.1 Deﬁnition of a plasma

A plasma is, at a basic level, an ionized gas. When electrons are stripped from ions, the gas becomes a quasi-neutral system of two or more interpenetrating ﬂuids.

In a gas, microscopic behavior is governed by collisions between atoms or molecules. The macroscopic behavior is determined by taking volumes with large numbers of particles within and taking statistical averages over known particle distributions (i.e. a Maxwellian). A volume element of ﬂuid must be small relative to the system size and contain a large number of particles for this to be valid.

In a plasma, electrons and ions still undergo collisions, but additionally they interact via the long-range Coulomb force. This allows the plasma to screen both DC and AC ﬁelds, as we will see in subsequent sections. This introduces two more conditions:

1. ${\lambda }_{De}\ll L$
The Debye length, or DC electric ﬁeld screening length, must be much smaller than the system size for standard plasma behavior.
2. ${N}_{D}=\frac{4\pi }{3}n{\lambda }_{De}^{3}\gg 1$
The number of particles within a Debye sphere, as deﬁned above, must be large. This plus $1$ is analogous to the condition placed on ideal gases.
3. ${\omega }_{pe}\tau \gg 1$
If ${\omega }_{pe}$ is a typical frequency for plasma oscillations, as governed by the Coulomb force, then the ’plasma frequency’ must be much greater than a typical frequency for hydrodynamic behavior $\omega =1∕\tau$ which could be taken as a collision frequency with neutral atoms. Thus we arrive at this condition.

1.2 Debye Length

Here we seek to deﬁne the typical length scale for DC electric ﬁeld screening using simple arguments.

In an inﬁnite homogenous plasma, we can take the electron distribution function as

 ${f}_{e}\left(u\right)=A{e}^{-\left(m{u}^{2}∕2\right)∕{k}_{B}T}$ (1.1)

where $A$ is a constant of proportionality, $u$ is the velocity of a particle, ${T}_{e}$ is the temperature, and ${k}_{B}$ is Boltzmann’s constant. In the presence of an electrostatic potential, $\varphi$, this becomes

 ${f}_{e}\left(u\right)=A{e}^{-\left(m{u}^{2}∕2+q\varphi \right)∕{k}_{B}T}$ (1.2)

where $q$ is the electron charge $q=-e$. This is somewhat intuitive and also can be derived rigorously by taking the partition function for Boltzmann statistics and using $q\varphi$ as the ‘chemical potential’ $\mu$. This derivation omitted here for simplicity.

Now consider the eﬀect of a small deviation in density between electrons and ions, using the Poisson equation:

 ${\nabla }^{2}\varphi =-\frac{e}{{𝜖}_{0}}\left({n}_{i}-{n}_{e}\right)$ (1.3)

where we have taken $Z=1$, a hydrogen plasma, for simplicity. Next we assume that only the electrons move in response to the applied potential, which is reasonable given the small electron mass ${m}_{e}∕{m}_{i}\ll 1$. In this case the ion density is simply the equilibrium value ${n}_{0}$ and we can rewrite Poisson’s equation as

 ${\nabla }^{2}\varphi =-\frac{{n}_{0}e}{{𝜖}_{0}}\left(1-{n}_{e}∕{n}_{0}\right).$ (1.4)

We can calculate the perturbed electron density from the distribution functions by taking the ratio of Eq 1.2 to Eq 1.1:

 $\frac{{n}_{e}}{{n}_{0}}={e}^{e\varphi ∕{k}_{B}T}.$ (1.5)

Substituting into the Poisson equation,

 ${\nabla }^{2}\varphi =-\frac{{n}_{0}e}{{𝜖}_{0}}\left(1-{e}^{e\varphi ∕{k}_{B}T}\right).$ (1.6)

In general this cannot be solved analytically, but in the limit $e\varphi ∕{k}_{B}T\ll 1$ we can Taylor expand:

 ${e}^{e\varphi ∕{k}_{B}T}\approx 1+\frac{e\varphi }{{k}_{B}T}+...$ (1.7)

which we put into the Poisson equation to get:

 ${\nabla }^{2}\varphi =\frac{{n}_{0}e}{{𝜖}_{0}}\frac{e\varphi }{{k}_{B}T}$ (1.8)

 ${\nabla }^{2}\varphi =\frac{{n}_{0}{e}^{2}}{{𝜖}_{0}{k}_{B}T}\varphi$ (1.9)

which has solutions

 $\varphi \left(\stackrel{\to }{r}\right)={\varphi }_{0}{e}^{-|\stackrel{\to }{r}-\stackrel{\to }{{r}_{0}}|∕{\lambda }_{De}},$ (1.10)

with

 ${\lambda }_{De}=\sqrt{\frac{{𝜖}_{0}{k}_{B}T}{{n}_{0}{e}^{2}}}$ (1.11)

which is the electron Debye length. Rigorously, the ion motion must be treated too, in which case

 $\frac{1}{{\lambda }_{D}}=\frac{1}{{\lambda }_{De}}+\frac{1}{{\lambda }_{Di}}.$ (1.12)

Physically, the Debye length represents the distance an electrostatic ﬁeld can penetrate into a plasma before it is screened. Electrons and ions move to oppose the imposed ﬁeld. At zero temperature the screening is perfect, but at ﬁnite temperature the Debye length is ﬁnite due to thermal ions and electrons having enough energy to sample the screened potential. The density dependence is logical - if there are more available charges, the screening length with be short.

1.3 Plasma Frequency

In the previous section we considered the plasma DC response, now we consider the AC response. A simple intuitive approach is taken here, which is rigorously conﬁrmed later.

If we consider a high-frequency oscillation, we can consider the ion inertia to be inﬁnite.

A simple intuitive system is as follows. Consider semi-inﬁnite slabs of rigid electron and ion ﬂuids, with ﬁnite length $L$ in the $x$ direction. Take the ion slab as stationary and the electron slab as displaced by a distance $x$. There will be a restoring Coulomb force between the two slabs.

First we need the ﬁeld of a charge slab. Using Poisson’s equation,

 ${\nabla }^{2}\varphi =-\rho ∕{𝜖}_{0}$ (1.13)

the charge density is $\rho =ne$ with ${n}_{i}={n}_{e}=n$ taken for simplicity (Z=1 hydrogen plasma). Using Stokes’ Theorem, and $\stackrel{\to }{E}=-\nabla \varphi$,

 ${\int }_{S}\stackrel{\to }{E}\cdot \stackrel{\to }{dA}={\rho }_{enc}∕{𝜖}_{0}$ (1.14)

for a Gaussian pillbox, the ﬁeld outside a single slab is,

 $E=\frac{nq}{{𝜖}_{0}}x.$ (1.15)

Now we consider the restoring force on the electron slab,

 $F=mẍ=-e\stackrel{\to }{E}=-\frac{n{e}^{2}}{{𝜖}_{0}}x.$ (1.16)

which simpliﬁes to

 $ẍ=-\frac{n{e}^{2}}{{m}_{e}{𝜖}_{0}}x$ (1.17)

which of course has well-known oscillatory solutions of the form

 $x\left(t\right)={x}_{0}+A{e}^{-i{\omega }_{pe}t}$ (1.18)

with characteristic frequency

 ${\omega }_{pe}=\sqrt{\frac{n{e}^{2}}{{m}_{e}{𝜖}_{0}}}.$ (1.19)

which is the electron plasma frequency.

At lower frequencies, the ion plasma frequency is also important, which is obtained by taking ${m}_{e}\to {m}_{i}$.

Chapter 2Single Particle Motion

2.1 Cyclotron Motion

First, we consider the motion of a single charged particle in constant uniform magnetic ﬁeld - the cyclotron motion. So we take $\stackrel{\to }{E}=0$ and $\stackrel{\to }{B}={B}_{z}ẑ$ where we can choose the magnetic ﬁeld along the $z$ axis without loss of generality. Then the Lorentz Force,

 $\stackrel{\to }{F}=q\left(\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}\right)=q\stackrel{\to }{v}×\stackrel{\to }{B}$ (2.1)

$\begin{array}{rcll}mẍ& =& q{v}_{y}{B}_{z}& \text{(2.2)}\text{}\text{}\\ mÿ& =& -q{v}_{x}{B}_{z}& \text{(2.3)}\text{}\text{}\\ m\stackrel{̈}{z}& =& 0& \text{(2.4)}\text{}\text{}\end{array}$

The $z$ equation, of course, is trivial: constant motion in the $ẑ$ direction. The other two axes can be addressed by taking the derivative of the $y$ equation of motion to substitute in from the $x$:

 $m\stackrel{̈}{{v}_{y}}=-q\stackrel{̇}{{v}_{x}}{B}_{z}=-q{B}_{z}\frac{q{v}_{y}{B}_{z}}{m},$ (2.5)

which simpliﬁes to

 $\stackrel{̈}{{v}_{y}}=-{\left(\frac{q{B}_{z}}{m}\right)}^{2}{v}_{y}=-{\omega }_{c}^{2}{v}_{y},$ (2.6)

where we have deﬁned the cyclotron frequency ${\omega }_{c}=q{B}_{z}∕m$. This clearly allows oscillatory solutions; without loss of generality we take

 ${v}_{y}\left(t\right)={v}_{\perp }cos{\omega }_{c}t$ (2.7)

combined with the original equations of motion, and using the initial conditions ${E}_{\perp }=\frac{1}{2}m{v}_{\perp }^{2}$ we arrive at the $x$ velocity

 ${v}_{x}\left(t\right)={v}_{\perp }sin{\omega }_{c}t.$ (2.8)

Clearly the total perpendicular kinetic energy is conserved, ${v}_{x}^{2}+{v}_{y}^{2}={v}_{\perp }^{2}$, as expected since the magnetic force does no work.

Next we consider the particle position. We simply need to integrate the velocity equations once more, which gives:

$\begin{array}{rcll}x\left(t\right)& =& {x}_{0}-\frac{{v}_{\perp }}{{\omega }_{c}}cos{\omega }_{c}t& \text{(2.9)}\text{}\text{}\\ y\left(t\right)& =& {y}_{0}+\frac{{v}_{\perp }}{{\omega }_{c}}sin{\omega }_{c}t& \text{(2.10)}\text{}\text{}\end{array}$

The average position $\left({x}_{0},{y}_{0}\right)$ is generally referred to as the ‘guiding center’ position. The quantity

 ${r}_{L}=\frac{{v}_{\perp }}{{\omega }_{c}}=\frac{m{v}_{\perp }}{q{B}_{z}}$ (2.11)

is the Larmor radius, which gives the size of gyrations due to the Lorentz force. These equations describe the motion of any charged particle in a constant and uniform magnetic ﬁeld. The motion is generally helical gyrations about the magnetic ﬁeld lines.

2.2 $\stackrel{\to }{E}×\stackrel{\to }{B}$ Drift

Next we consider the motion in constant and uniform $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ ﬁelds.

First, consider the situation where $\stackrel{\to }{E}\parallel \stackrel{\to }{B}$. Clearly this will lead to continuous acceleration along the magnetic ﬁeld lines where $\stackrel{\to }{v}×\stackrel{\to }{B}$ is zero. This is not generally an interesting phenomenon, except in the case of runaway electrons, which will be considered later.

So for now, we consider the case $\stackrel{\to }{E}\perp \stackrel{\to }{B}$. Without loss of generality we take $\stackrel{\to }{E}={E}_{x}\stackrel{̂}{x}$ and $\stackrel{\to }{B}={B}_{z}ẑ$. In this case the general Lorentz force,

 $\stackrel{\to }{F}=q\left(\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}\right),$ (2.12)

reduces to equations of motion,

$\begin{array}{rcll}mẍ& =& q{E}_{x}+q{v}_{y}{B}_{z}& \text{(2.13)}\text{}\text{}\\ mÿ& =& -q{v}_{x}{B}_{z}& \text{(2.14)}\text{}\text{}\\ m\stackrel{̈}{z}& =& 0.& \text{(2.15)}\text{}\text{}\end{array}$

Once again, the equation of motion along the magnetic ﬁeld lines is trivial. Focusing on the $x-y$ plane, we start by taking the derivative of the $y$ equation,

 $m\stackrel{̈}{{v}_{y}}=-q{B}_{z}ẍ$ (2.16)

which allows us to substitute the ﬁrst equation of motion, giving,

 $m\stackrel{̈}{{v}_{y}}=-q{B}_{z}\left(q{E}_{x}+q{v}_{y}{B}_{z}\right)∕m.$ (2.17)

Rearranging terms, we get that,

 $\stackrel{̈}{{v}_{y}}=-\frac{{q}^{2}{B}_{z}^{2}}{{m}^{2}}\left({E}_{x}∕{B}_{z}+{v}_{y}\right)$ (2.18)

This clearly suggests a coordinate transformation,

 ${v}_{y}^{\prime }={v}_{y}+{E}_{x}∕{B}_{z}$ (2.19)

in which case the equation of motion becomes

 $m\stackrel{̈}{{v}_{y}^{\prime }}=-{\omega }_{c}^{2}{v}_{y}^{\prime }$ (2.20)

which is simply cyclotron motion in the primed coordinate system. But now that we have introduced the primed system, we note that it is drifting relative to the original reference frame with a constant velocity:

 ${v}_{D}={E}_{x}∕{B}_{z}$ (2.21)

More generally, if we allow $\stackrel{\to }{E}$ to lie anywhere in the $xy$ plane, we get that

 ${\stackrel{\to }{v}}_{E×B}=\frac{\stackrel{\to }{E}×\stackrel{\to }{B}}{{B}^{2}}$ (2.22)

where this is called the $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift for obvious reasons. We note the important result that this drift does not depend on the particle mass or the particle charge, which means that in a plasma electrons and ions will drift with the same velocity and the same direction. The $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift can therefore cause net motion of the plasma, and in certain situations can be problematic for plasma conﬁnement in magnetic fusion devices.

2.3 $\stackrel{\to }{F}×\stackrel{\to }{B}$ Drift

We observe that in the derivation of the $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift, there is nothing ‘special’ about the electrostatic force

 $\stackrel{\to }{F}=q\stackrel{\to }{E}$ (2.23)

and we notice that we can substitute $\stackrel{\to }{E}=\stackrel{\to }{F}∕q$ in the previous result to get that

 ${\stackrel{\to }{v}}_{F}=\frac{1}{q}\frac{\stackrel{\to }{F}×\stackrel{\to }{B}}{{B}^{2}}$ (2.24)

the general force drift. We note that in the general case there is a dependence on the particle charge $q$, meaning that electrons and ions will drift in opposite directions due to the ‘general force’.

2.4 $\nabla B$ Drift

Up until now, we have only considered uniform ﬁelds. That will now change. Take $\stackrel{\to }{E}=0$ and non-uniform but constant $\stackrel{\to }{B}={B}_{z}\left(x\right)ẑ$. The magnetic ﬁeld points in the $z$ direction but has a gradient in the $x$ direction (taken without loss of generality). As usual we start with the Lorentz force,

 $\stackrel{\to }{F}=q\left(\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}\right)$ (2.25)

which gives equations of motion

$\begin{array}{rcll}mẍ& =& q{v}_{y}{B}_{z}\left(x\right)& \text{(2.26)}\text{}\text{}\\ mÿ& =& -q{v}_{x}{B}_{z}\left(x\right)& \text{(2.27)}\text{}\text{}\\ m\stackrel{̈}{z}& =& 0& \text{(2.28)}\text{}\text{}\end{array}$

Again, the $z$ equation of motion is trivially solved. Focusing on the $xy$ plane, the equations of motion are now more complicated than previously due to the position dependence of the magnetic ﬁeld strength.

If we take the limit ${r}_{L}\ll |B|∕\nabla B$, or that the particle Larmor radius is much smaller than the gradient length scale for changes in $B$, then we can Taylor expand the magnetic ﬁeld around the particle guiding center position:

 ${B}_{z}\left(x\right)\approx {B}_{z}\left({x}_{0}\right)\left(1+\frac{1}{{B}_{0}}\frac{\partial {B}_{z}}{\partial x}\left(x-{x}_{0}\right)\right)$ (2.29)

substituting this into the $y$ equation of motion,

 $mÿ=-q{v}_{x}{B}_{z}\left({x}_{0}\right)\left(1+\frac{1}{{B}_{0}}\frac{\partial {B}_{z}}{\partial x}\left(x-{x}_{0}\right)\right)$ (2.30)

In general this would be very diﬃcult to solve, but if we treat it perturbatively and use the general cyclotron motion as a 0th order solution, we can rewrite this as

 $mÿ=-q{v}_{\perp }cos\left({\omega }_{c}t\right)\left({B}_{z}\left({x}_{0}\right)+\frac{\partial {B}_{z}}{\partial x}{r}_{L}cos{\omega }_{c}t\right).$ (2.31)

Now consider averaging over a gyration. The ﬁrst term averages to $0$, and the second term becomes (using ${cos}^{2}\to 1∕2$ over a whole gyration)

 ${F}_{y}=-\frac{q{v}_{\perp }{r}_{L}}{2}\frac{\partial {B}_{z}}{\partial x}$ (2.32)

using our previous deﬁnitions of ${r}_{L}$,

 ${F}_{y}=-\frac{q{v}_{\perp }^{2}}{2{\omega }_{c}}\frac{\partial {B}_{z}}{\partial x}$ (2.33)

Combined with our result for the general force drift, we can simply write the drift due to the magnetic ﬁeld gradient as

 ${v}_{\nabla B}=\mp \frac{{v}_{\perp }^{2}}{2{\omega }_{c}}\frac{\nabla B×\stackrel{\to }{B}}{{B}^{2}}$ (2.34)

where we have made the generalization to arbitrary directions of the magnetic ﬁeld gradient by intuition. We note that the $\nabla B$ drift depends on both the particle charge and mass, which means that electrons and ions will drift in opposite directions at potentially diﬀerent rates due to magnetic ﬁeld gradients. The sign of the above equation is the sign of the charge. If the species temperatures are equal (${T}_{e}={T}_{i}$) then $m{v}_{\perp }^{2}$ is equal and the electrons and ions drift in opposite directions but at the same rate.

2.5 Curvature/Gravitational Drift

We can use our generalized force drift equation to examine two other interesting situations.

First, consider the eﬀect of a gravitational ﬁeld on the plasma. In terrestrial experiments there is an unavoidable force

 $\stackrel{\to }{F}=m\stackrel{\to }{g}$ (2.35)

which leads to a drift velocity

 ${\stackrel{\to }{v}}_{g}=\frac{m}{q}\frac{\stackrel{\to }{g}×\stackrel{\to }{B}}{{B}^{2}}$ (2.36)

So a gravitational ﬁeld will induce a drift, which depends on both the particle mass and charge. In real experiment, however, this is usually neglected due to the smallness of $g$ relative to other forces (i.e. the electromagnetic force).

Now consider a curved plasma. If the magnetic ﬁeld lines are bent with some curvature radius, how will that aﬀect the particle motion?

The simple answer is to consider the particles as primarily streaming along the ﬁeld lines. If the ﬁeld lines are bent into a circle, as in a tokamak, then the curvature is essentially equivalent to considering the motion in a rotating reference frame. This induces a centrifugal force opposite the direction of curvature and perpendicular to the magnetic ﬁeld:

 ${\stackrel{\to }{F}}_{cf}=m\stackrel{\to }{\Omega }×\left(\stackrel{\to }{r}×\stackrel{\to }{\Omega }\right)=m{v}_{\parallel }^{2}\frac{\stackrel{\to }{{R}_{c}}}{{R}_{c}^{2}}$ (2.37)

this is written by inspection and intuition but could be derived rigorously. The velocity parallel to the magnetic ﬁeld lines is denoted by ${v}_{\parallel }$. If we substitute into the general force drift equation, then we get

 ${\stackrel{\to }{v}}_{c}=\frac{m{v}_{\parallel }^{2}}{q{R}_{c}^{2}}\frac{\stackrel{\to }{{R}_{c}}×\stackrel{\to }{B}}{{B}^{2}}.$ (2.38)

The curvature is perpendicular to the radius of curvature vector and the magnetic ﬁeld. Its magnitude depends on the radius of curvature, the magnetic ﬁeld, and particle info. In particular we note that it is proportional to the ratio $m∕q$ which means that electrons and ions will drift in opposite directions and diﬀering rates (unless ${T}_{e}={T}_{i}$).

2.6 Polarization Drift

Consider a time-varying electric ﬁeld given by

 $\stackrel{\to }{E}={E}_{0}{e}^{-i\omega t}\stackrel{̂}{x}$ (2.39)

Also allow a constant uniform magnetic ﬁeld $\stackrel{\to }{B}={B}_{z}ẑ$, so that the equations of motion become

$\begin{array}{rcll}mẍ& =& q{v}_{y}{B}_{z}+q{E}_{0}{e}^{-i\omega t}& \text{(2.40)}\text{}\text{}\\ mÿ& =& -q{v}_{x}{B}_{z}& \text{(2.41)}\text{}\text{}\\ m\stackrel{̈}{z}& =& 0& \text{(2.42)}\text{}\text{}\end{array}$

as usual the $z$ equation is trivial. Taking a derivative of the ﬁrst equation,

 $\stackrel{̈}{{v}_{x}}=-{\omega }_{c}^{2}\left({v}_{x}\mp \frac{i\omega }{{\omega }_{c}}\frac{\stackrel{̃}{{E}_{x}}}{{B}_{z}}\right)$ (2.43)

where we have used Fourier analysis $Ė=-i\omega E$ and the over-tilde denotes time varying quantities. If we deﬁne two drift velocities

$\begin{array}{rcll}{v}_{p}& =& \frac{i\omega }{{\omega }_{c}}\frac{\stackrel{̃}{{E}_{x}}}{{B}_{z}}& \text{(2.44)}\text{}\text{}\\ {v}_{E}& =& \frac{\stackrel{̃}{{E}_{x}}}{{B}_{z}}.& \text{(2.45)}\text{}\text{}\end{array}$

The latter drift is simply the $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift but with a time-varying electric ﬁeld; the drift velocity will vary sinusoidally with the imposed ﬁeld as we might expect. The ﬁrst drift is the so-called polarization drift. Rewriting the equations of motion,

 $\stackrel{̈}{{v}_{x}}=-{\omega }_{c}^{2}\left({v}_{x}-{v}_{p}\right),$ (2.46)

the motion in the $\stackrel{̂}{x}$ direction is simply a superposition of the cyclotron motion and the polarization drift. In the $ŷ$ direction,

 $\stackrel{̈}{{v}_{y}}=-{\omega }_{c}^{2}\left({v}_{y}-{v}_{E}\right)$ (2.47)

the motion is a superposition of cyclotron motion and the $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift.

In the case of an electric ﬁeld which cannot be simply decomposed into Fourier components, we can rewrite a more general expression by inspection,

 ${\stackrel{\to }{v}}_{p}=\frac{1}{{\omega }_{c}B}\frac{\partial \stackrel{\to }{E}}{\partial t}$ (2.48)

2.7.1 $\mu$

This is the ﬁrst adiabatic invariant. We want to start from the action integral, in general form,

 $\oint pdq$ (2.49)

which will be an adiabatic invariant of the motion (in certain limits). First we consider the cyclotron motion of a particle, so let us take $p=m{v}_{\perp }{r}_{L}$ and $q=𝜃$ and integrate over one gyration

 $\oint pdq=\oint m{v}_{\perp }{r}_{L}d𝜃=2\pi m{v}_{\perp }{r}_{L}$ (2.50)

Now we need to rewrite this quantity as

 $4\pi \frac{m}{|q|}×\frac{m{v}_{\perp }^{2}}{2B}$ (2.51)

The ﬁrst part is constant if the charge-to-mass ratio is not changing over the motion (a good assumption). In that case, the second part of the above equation is conserved, leading to

 $\mu \equiv \frac{m{v}_{\perp }^{2}}{2B}$ (2.52)

This quantity $\mu$ is an invariant of the motion. However, we have assumed that pure cyclotron motion is a good approximation of the particle motion over short time scales, or equivalently, we have assumed slow motion $\omega \ll {\omega }_{c}$. In cases where this is not true, then $\mu$ is not an invariant of the motion.

2.8 Magnetic Mirror Machine

The adiabatic invariant $\mu$ can be directly applied to a machine of fusion (historical) interest - the magnetic mirror.

Consider a linear cylindrical machine where the magnetic ﬁeld is weakest at the center and reaches a peak ﬁeld value near the ends. This could be arranged, for example, by a two-coil conﬁguration.

A particle near the center of the machine, i.e. a low-ﬁeld region, is heading towards the high-ﬁeld region. The problem is to deﬁne when the particle is conﬁned. The initial particle kinetic energy can be decomposed into parallel and perpendicular components which will satisfy

 ${v}_{\perp }={v}_{0}sin𝜃$ (2.53)

 ${v}_{\parallel }={v}_{0}cos𝜃$ (2.54)

where $𝜃$ is the pitch angle relative to the magnetic ﬁeld. Based on the previous derivation, in the slow motion limit, we will have the adiabatic invariant of the motion

 $\mu =\frac{m{v}_{\perp }^{2}}{2B}.$ (2.55)

We can see from the deﬁnition of $\mu$ that as $B$ increases, the particle’s perpendicular velocity must increase to conserve $\mu$. For a given particle’s kinetic energy, then, there is a maximum value of $B$ that can be reached before the particle is reﬂected. Consider the extremes of the motion, and setting ${\mu }_{i}={\mu }_{f}$ we get

 $\frac{{v}_{\perp i}^{2}}{{B}_{i}}=\frac{{v}_{\perp f}^{2}}{{B}_{f}}$ (2.56)

or equivalently,

 $\frac{{B}_{f}}{{B}_{i}}=\frac{{v}_{\perp f}}{{v}_{\perp i}}$ (2.57)

using ${v}_{\perp f}={v}_{0}$ and the pitch angle deﬁnition,

 $\frac{{B}_{f}}{{B}_{i}}=\frac{1}{{sin}^{2}𝜃}.$ (2.58)

Next we deﬁne the ‘mirror ratio’ of the experimental machine as the ratio of minimum to maximum ﬁeld strength, $R\equiv {B}_{max}∕{B}_{min}>1$. In this case there is a critical pitch angle for reﬂection,

 ${sin}^{2}{𝜃}_{c}=\frac{{B}_{min}}{{B}_{max}}.$ (2.59)

Any particles with $𝜃>{𝜃}_{c}$ are conﬁned, while particles with $𝜃<{𝜃}_{c}$ are not conﬁned. This leads to a ‘loss cone’ in phase space.

The mirror machine was originally proposed as a scheme for fusion energy. Unfortunately, the loss cone particle loss is too extreme for an eﬃcient machine. Coulomb scattering continually scatters particles into the loss cone, and they are lost out the ends of the machine. A rigorous derivation ﬁnds that the maximum theoretical $Q$ value for the mirror machine is $1.1$. While this is actually slightly greater than unity, a real machine will not achieve the theoretical result, and even if it did a $Q=1.1$ machine will never be economical for energy generation.

2.9 Runaway Electrons

Consider a single electron moving in a tokamak. In today’s machines, large transformers are used to inductively drive a toroidal current in the machine. In the (realistic) event that the plasma has ﬁnite resistivity, this will create a non-zero toroidal EMF. The value of this ﬁeld is not important for the problem; we simply consider the case of a toroidal electric ﬁeld.

The electron equation of motion will be

 $m\stackrel{̇}{\stackrel{\to }{v}}=q\stackrel{\to }{E}-\nu {v}_{e}$ (2.60)

where $\nu$ is the collision frequency for momentum transfer, generally $\nu ={\nu }_{ei}+{\nu }_{ee}$. It is convenient to introduce the form ${w}_{e}={v}_{e}∕{v}_{Te}$ where the denominator is the electron thermal velocity ${v}_{Te}=\sqrt{2{T}_{e}∕{m}_{e}}$. If we rewrite the electron equation of motion as,

 $\frac{d{w}_{e}}{dt}=\stackrel{̂}{{E}_{\parallel }}-\frac{{\nu }_{r}}{{w}_{e}^{2}},$ (2.61)

then the parts of the right hand side are normalized to

 $\stackrel{̂}{{E}_{\parallel }}=\frac{e|{E}_{\parallel }|}{\sqrt{2{m}_{e}{T}_{e}}}$ (2.62)

and

 ${\nu }_{R}=\frac{3}{8\sqrt{2}\pi }\frac{n{e}^{4}log\Lambda }{{𝜖}_{0}^{2}\sqrt{{m}_{e}}{T}_{e}^{3∕2}},$ (2.63)

which we will not derive here. The important part is that with the velocity normalization we have done, neither $\stackrel{̂}{{E}_{\parallel }}$ or ${\nu }_{R}$ depend on ${w}_{e}$. We can then immediately see that for

 $\stackrel{̂}{{E}_{\parallel }}\ge {\nu }_{R}∕{w}_{e}^{2}$ (2.64)

or

 ${w}_{e}^{2}\ge {\nu }_{R}∕\stackrel{̂}{{E}_{\parallel }}$ (2.65)

the electron will continuously accelerate. This is the runaway condition. In particular, we can write down a ‘critical velocity’

 ${w}_{c}=\sqrt{{\nu }_{R}∕\stackrel{̂}{{E}_{\parallel }}},$ (2.66)

where all electrons starting with a velocity greater than ${w}_{c}$ will continuously accelerate. This phenomenon arises because the electron collision frequency for momentum transfer is proportional to $1∕{v}^{3}$ so that fast electrons collide very rarely. Runaway electrons are potentially problematic for tokamak systems.

Chapter 3MHD

3.1 2-ﬂuid derivation

3.1.1 Momentum Equation

This is more a physical argument than a strict derivation. We start from the well-known Navier-Stokes equation, i.e. the momentum equation for ordinary hydrodynamics:

 $\rho \left[\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}\right]=-\nabla p-\rho \nu {\nabla }^{2}\stackrel{\to }{u},$ (3.1)

where $\rho$ is the mass density, $\stackrel{\to }{u}$ is the ordinary ﬂuid velocity, and $p$ is the scalar pressure. $\nu$ in this equation is the ﬂuid kinematic viscosity.

Right oﬀ the bat, we know to make the substitution $\rho \to {m}_{j}{n}_{j}$ for plasma species $j$. The kinematic viscosity term $\rho \nu {\nabla }^{2}\stackrel{\to }{u}$ is absorbed with the scalar pressure term into a tensor pressure term, giving us:

 ${m}_{j}{n}_{j}\left[\frac{\partial \stackrel{\to }{{u}_{j}}}{\partial t}+\left({\stackrel{\to }{u}}_{j}\cdot \nabla \right)\stackrel{\to }{{u}_{j}}\right]=-\nabla \cdot P+...$ (3.2)

where the bold-face $P$ denotes a tensor, and the ellipses are included to show that we are still missing terms in this equation.

The ﬁrst obvious addition that must be included when going from ordinary hydrodynamics to plasma magnetohydrodynamics is the eﬀect of the Lorentz force. While ordinary ﬂuids are composed of charge neutral particles, plasmas of course consist of electrons and ions and the ﬂuid species $j$ generally has non-zero charge ${q}_{j}$. In this case each particle feels a Lorentz force due to the local ﬁelds:

 $F={q}_{j}\left(\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}\right)$ (3.3)

which is generalized to a force on a ﬂuid element by multiplying by the particle density ${n}_{j}$ and allowing the particle velocity to go to the average ﬂuid velocity. This term is added to the momentum equation to obtain

 ${m}_{j}{n}_{j}\left[\frac{\partial \stackrel{\to }{{u}_{j}}}{\partial t}+\left({\stackrel{\to }{u}}_{j}\cdot \nabla \right)\stackrel{\to }{{u}_{j}}\right]={n}_{j}{q}_{j}\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{j}×\stackrel{\to }{B}\right)-\nabla \cdot P+...$ (3.4)

There is one more eﬀect we must include. In ordinary hydrodynamics the ﬂuid motion is determined by collisions within the ﬂuid, and by external forces represented by the pressure term. But in a plasma, multiple ﬂuids can be interpenetrating, most obviously the electron and ion ﬂuids which must be co-located to preserve charge quasi-neutrality. This allows for a transfer of momentum between ﬂuids via collisions, which will generally obey a relation

 ${m}_{j}{n}_{j}{\nu }_{ji}\left(\stackrel{\to }{{u}_{j}}-\stackrel{\to }{{u}_{i}}\right)$ (3.5)

for collisions between ﬂuids $i$ and $j$, with collision frequency for momentum transfer ${\nu }_{ji}={\nu }_{ij}{m}_{i}∕{m}_{j}$. This is added to the momentum equation to give us its ﬁnal two-ﬂuid form:

 ${m}_{j}{n}_{j}\left[\frac{\partial \stackrel{\to }{{u}_{j}}}{\partial t}+\left({\stackrel{\to }{u}}_{j}\cdot \nabla \right)\stackrel{\to }{{u}_{j}}\right]={n}_{j}{q}_{j}\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{j}×\stackrel{\to }{B}\right)-\nabla \cdot P-{m}_{j}{n}_{j}{\nu }_{ji}\left({\stackrel{\to }{u}}_{j}-{\stackrel{\to }{u}}_{i}\right).$ (3.6)

If there are more than two ﬂuid species (i.e. in a multi ion species plasma) then we must sum the last term with $i$ going over all other species in the plasma, though primarily we will be interested in electron+ion two-ﬂuid plasmas.

3.1.2 Continuity Equation

In ordinary hydrodynamics the continuity equation is:

 $\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \stackrel{\to }{u}\right)=0.$ (3.7)

If we simply make the same substitutions, $\stackrel{\to }{u}\to {\stackrel{\to }{u}}_{j}$ and $\rho \to {n}_{j}{m}_{j}$ we get that

 $\frac{\partial {n}_{j}}{\partial t}+\nabla \cdot \left({n}_{j}{\stackrel{\to }{u}}_{j}\right)=0.$ (3.8)

3.1.3 Energy Equation

This is not as often quoted in ﬂuid mechanics. Basically, we must think about how the internal energy of a ﬂuid element changes. It can change via the ﬁrst-order ﬂow of the ﬂuid, via compressive ($PdV$) work, and via heat conduction. Additionally, there may be any number of ‘external’ sources or sinks of energy.

The average internal energy of a particle in an ideal gas, equilibrated to a Maxwellian, can be given by $\left(3∕2\right){k}_{B}T$. The overall internal energy is the particle density multiplied by this quantity. Changes in density are included in the $PdV$ term; here we consider the change in temperature due to the ﬂow. We can simply write this via the convective derivative as:

 $\frac{3}{2}{n}_{j}\left(\frac{\partial }{\partial t}+{\stackrel{\to }{u}}_{j}\cdot \nabla \right){T}_{j}.$ (3.9)

Next, we consider the compressive work done on (or by) a ﬂuid element, which is simply the pressure multiplied by the divergence of the ﬂuid velocity:

 ${p}_{j}\nabla \cdot {\stackrel{\to }{u}}_{j}$ (3.10)

Finally, we can simply write down the heat ﬂux in terms of the thermal conductivity and temperature gradient,

 $\nabla \cdot \stackrel{\to }{q}$ (3.11)

with $\stackrel{\to }{q}=-\kappa \cdot \nabla T$.

We combine these various terms together, with the total change in internal energy from the three equal to the external sources or sinks of energy:

 $\frac{3}{2}{n}_{j}\left(\frac{\partial }{\partial t}+{\stackrel{\to }{u}}_{j}\cdot \nabla \right){T}_{j}+{p}_{j}\nabla \cdot {\stackrel{\to }{u}}_{j}+\nabla \cdot \stackrel{\to }{q}={S}_{j}$ (3.12)

where ${S}_{j}$ represents several sources and sinks of energy:

• Fusion heating power
• External heating (RF, NB, ...)
• Ohmic heating
• Bremsstrahlung losses
• Energy equilibration between plasma species (ion-electron)
• ...

3.1.4 Maxwell’s Equations

For completeness we have to include Maxwell’s equations for electromagnetism to the ﬂuid equations above, since the electric and magnetic ﬁelds must be determined self-consistently in MHD. In plasma physics we prefer to use the vacuum equations with plasma serving as the source terms. Anyways, in SI units the four Maxwell equations are:

$\begin{array}{rcll}\nabla \cdot \stackrel{\to }{E}& =& \frac{1}{{𝜖}_{0}}\rho & \text{(3.13)}\text{}\text{}\\ \nabla \cdot \stackrel{\to }{B}& =& 0& \text{(3.14)}\text{}\text{}\\ \nabla ×\stackrel{\to }{E}& =& -\frac{\partial \stackrel{\to }{B}}{\partial t}& \text{(3.15)}\text{}\text{}\\ \nabla ×\stackrel{\to }{B}& =& \frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}+{\mu }_{0}\stackrel{\to }{j}& \text{(3.16)}\text{}\text{}\end{array}$

where $\rho$ is the total charge density:

 $\rho \equiv \sum _{i}{n}_{i}{q}_{i}$ (3.17)

and $j$ is the total current density:

 $\stackrel{\to }{j}\equiv \sum _{i}{n}_{i}{q}_{i}{\stackrel{\to }{u}}_{i}$ (3.18)

3.2 Single-ﬂuid MHD

In many plasmas, the generalized two-ﬂuid system described above can be simpliﬁed considerably to a one-ﬂuid model. The goal of this section is to present a derivation of the one-ﬂuid model with a discussion of when it is applicable. For simplicity we will use a single-species $Z=1$ plasma with the usual assumptions.

First, we need to deﬁne the single-ﬂuid variables which will be used throughout this section:

$\begin{array}{rcll}\rho & =& {m}_{i}{n}_{i}+{m}_{e}{n}_{e}& \text{(3.19)}\text{}\text{}\\ \stackrel{\to }{v}& =& \frac{1}{\rho }\left({m}_{i}{n}_{i}{\stackrel{\to }{u}}_{i}+{m}_{e}{n}_{e}\stackrel{\to }{{u}_{e}}\right)& \text{(3.20)}\text{}\text{}\\ \stackrel{\to }{j}& =& e\left({n}_{i}{\stackrel{\to }{u}}_{i}-{n}_{e}{\stackrel{\to }{u}}_{e}\right)& \text{(3.21)}\text{}\text{}\end{array}$

We start oﬀ by writing the momentum equations for both electron and ion species. For simplicity we take the pressure tensor as isotropic, meaning that the pressure appears only as the usual scalar pressure. The viscosity terms are small if the ion Larmor radius is much smaller than scale lengths for typical variations, for instance this condition can be written as ${r}_{Li}\ll \rho ∕\nabla \rho$ using the density gradient length scale.

We also drop the convective terms $\left({\stackrel{\to }{u}}_{j}\cdot \nabla \right){\stackrel{\to }{u}}_{j}$. Chen describes this as ‘hard to justify’. I think that the best justiﬁcation is that in motions where the ﬂuid velocity is ‘small’, and implicitly the 0th-order velocity is zero, then this convective term is second-order in the small velocity and can be ignored. A valid question is ‘small relative to what?’ If we consider the ratio of the convective term to the partial time derivative:

 $\left|\frac{\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}}{\partial \stackrel{\to }{u}∕\partial t}\right|\approx \frac{ik{u}^{2}}{i\omega u}\approx \frac{u}{\omega ∕k}$ (3.22)

If the plasma response is, say, acoustic (the slowest case), then $\omega ∕k\sim {c}_{s}$ and the convective term is negligible if the ﬂuid motion is suﬃciently sub-sonic.

In any event, we can now write the ion momentum equation as

 ${m}_{i}{n}_{i}\frac{\partial \stackrel{\to }{{u}_{i}}}{\partial t}={n}_{i}e\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{i}×\stackrel{\to }{B}\right)-\nabla {p}_{i}+{m}_{i}{n}_{i}{\nu }_{ie}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)$ (3.23)

and similarly for the electrons:

 ${m}_{e}{n}_{e}\frac{\partial \stackrel{\to }{{u}_{e}}}{\partial t}=-{n}_{e}e\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{e}×\stackrel{\to }{B}\right)-\nabla {p}_{e}-{m}_{e}{n}_{e}{\nu }_{ei}\left({\stackrel{\to }{u}}_{e}-{\stackrel{\to }{u}}_{i}\right)$ (3.24)

where in the collisional momentum transfer term (the last one) we note that ${\nu }_{ie}={\nu }_{ei}{m}_{e}∕{m}_{i}$, which leads to the physically intuitive result that momentum lost by one species is gained by the other (terms are equal and opposite).

We start oﬀ by taking the sum of the two momentum equations, which will lead to the single-ﬂuid equation of motion. We get that,

 $n\left({m}_{i}\frac{\partial \stackrel{\to }{{u}_{i}}}{\partial t}+{m}_{e}\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}\right)=en\left(\stackrel{\to }{{u}_{i}}-\stackrel{\to }{{u}_{e}}\right)×\stackrel{\to }{B}-\nabla \left({p}_{i}+{p}_{e}\right)$ (3.25)

The left-hand side is simply the time derivative of the single-ﬂuid velocity multiplied by the density. The electric ﬁeld terms have canceled. The magnetic ﬁeld part of the Lorentz force reduces to using the single-ﬂuid total current. We take the single-ﬂuid total pressure as the sum of the individual ion and electron pressures, which is sensible. By taking the sum, the collision term has also canceled (what is lost by one ﬂuid is gained by the other). We are thus left with the single-ﬂuid equation of motion:

 $\rho \frac{\partial \stackrel{\to }{v}}{\partial t}=\stackrel{\to }{j}×\stackrel{\to }{B}-\nabla p.$ (3.26)

In some cases a gravitational force term $+\rho \stackrel{\to }{g}$ is added to the right hand side, which is not included in this derivation but can be seen via physical intuition.

To get to the various Ohm’s Laws, we must take the less-obvious step of calculating the diﬀerence of the ion and electron momentum equations. But in particular, we multiply the ion equation by the electron mass and vice versa, so that the diﬀerence becomes

$\begin{array}{rcll}{m}_{i}{m}_{e}n\frac{\partial }{\partial t}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)& =& en\left({m}_{i}+{m}_{e}\right)\stackrel{\to }{E}+en\left({m}_{e}{\stackrel{\to }{u}}_{i}+{m}_{i}{\stackrel{\to }{u}}_{e}\right)×\stackrel{\to }{B}& \text{}\\ & -& {m}_{e}\nabla {p}_{i}+{m}_{i}\nabla {p}_{e}-\left({m}_{i}+{m}_{e}\right)n{\nu }_{ei}\left({\stackrel{\to }{u}}_{e}-{\stackrel{\to }{u}}_{i}\right),& \text{(3.27)}\text{}\text{}\end{array}$

which is kind of a mess but simpliﬁes considerably. The left hand side can be written in terms of the current,

 ${m}_{i}{m}_{e}n\frac{\partial }{\partial t}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)=\frac{{m}_{i}{m}_{e}n}{e}\frac{\partial }{\partial t}\left(\frac{\stackrel{\to }{j}}{n}\right).$ (3.28)

The electric ﬁeld term can easily be simpliﬁed by recognizing that it is just $e\rho \stackrel{\to }{E}$. The magnetic ﬁeld term is trickier. We have to write:

$\begin{array}{rcll}{m}_{e}{\stackrel{\to }{u}}_{i}+{m}_{i}{\stackrel{\to }{u}}_{e}& =& {m}_{i}{\stackrel{\to }{u}}_{i}+{m}_{e}\stackrel{\to }{{u}_{e}}+{m}_{i}\left({\stackrel{\to }{u}}_{e}-{\stackrel{\to }{u}}_{i}\right)+{m}_{e}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)& \text{}\\ & =& \frac{\rho \stackrel{\to }{v}}{n}-\left({m}_{i}-{m}_{e}\right)\frac{\stackrel{\to }{j}}{ne}& \text{(3.29)}\text{}\text{}\end{array}$

Then the entire magnetic ﬁeld term is:

 $e\rho \stackrel{\to }{v}×\stackrel{\to }{B}-\left({m}_{i}-{m}_{e}\right)\stackrel{\to }{j}×\stackrel{\to }{B}$ (3.30)

The collision term (momentum transfer) is re-written in terms of the resistivity $\eta ={m}_{e}{\nu }_{ei}∕n{e}^{2}$ so that it becomes

 $\left({m}_{i}+{m}_{e}\right)ne\eta \stackrel{\to }{j}$ (3.31)

Combining all of these together we get

$\begin{array}{rcll}\frac{{m}_{i}{m}_{e}n}{e}\frac{\partial }{\partial t}\left(\frac{\stackrel{\to }{j}}{n}\right)& =& e\rho \stackrel{\to }{E}+e\rho \stackrel{\to }{v}×\stackrel{\to }{B}-\left({m}_{i}-{m}_{e}\right)\stackrel{\to }{j}×\stackrel{\to }{B}& \text{}\\ & -& {m}_{i}\nabla {p}_{e}+{m}_{e}\nabla {p}_{i}-\left({m}_{i}+{m}_{e}\right)ne\eta \stackrel{\to }{j}& \text{(3.32)}\text{}\text{}\end{array}$

An immediate and obvious simpliﬁcation is to take the limit ${m}_{e}\ll {m}_{i}$ which means that the terms $\left({m}_{i}±{m}_{e}\right)\to {m}_{i}$ and ${m}_{i}n\to \rho$. We also observe that, since ${p}_{j}={n}_{j}{k}_{B}{T}_{j}$ the electron and ion pressures are ${p}_{i}\sim {p}_{e}$ for ${T}_{i}\sim {T}_{e}$ in which case ${m}_{e}\nabla {p}_{i}\ll {m}_{i}\nabla {p}_{e}$ so we can drop the ion pressure gradient term. We can also rearrange terms and divide by $\rho e$ to get that

 $\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}-\eta \stackrel{\to }{j}=\frac{1}{e\rho }\left[\frac{{m}_{i}{m}_{e}n}{e}\frac{\partial }{\partial t}\left(\frac{\stackrel{\to }{j}}{n}\right)-{m}_{i}\nabla {p}_{e}+{m}_{i}\stackrel{\to }{j}×\stackrel{\to }{B}\right]$ (3.33)

We then generally take the limit of slow motions, i.e. where inertial (cyclotron motion) eﬀects are unimportant. Explicitly this is the limit $\stackrel{\to }{v}∕a\ll {\omega }_{ci}$ where $a$ is the plasma scale size. We can get this result by considering the ratio of the current time derivative term to the $\stackrel{\to }{j}×\stackrel{\to }{B}$ term within square brackets:

 $\left|\frac{\left({m}_{i}{m}_{e}n∕e\right)\partial ∕\partial t\left(\stackrel{\to }{j}∕n\right)}{{m}_{i}\stackrel{\to }{j}×\stackrel{\to }{B}}\right|\approx \frac{{m}_{e}\omega j}{ejB}\approx \frac{\omega }{{\omega }_{ce}}$ (3.34)

So, if $\omega \ll {\omega }_{ce}$ we can neglect the time derivative term.

In any event, this limit reduces the equation to:

 $\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}=\eta \stackrel{\to }{j}+\frac{1}{en}\left[\stackrel{\to }{j}×\stackrel{\to }{B}-\nabla {p}_{e}\right].$ (3.35)

This is known as the Generalized Ohm’s Law. The $\stackrel{\to }{j}×\stackrel{\to }{B}$ term is the Hall current term. In many physical systems it turns out that the Hall current and pressure gradient terms are small, in which case this expression reduces to the Resistive MHD Ohm’s Law:

 $\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}=\eta \stackrel{\to }{j}$ (3.36)

In some plasmas the resistivity is small enough to be neglected (i.e. high temperature and low density). In this case the resistive Ohm’s law reduces to the Ideal MHD Ohm’s Law:

 $\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}=0$ (3.37)

In summary, the single-ﬂuid MHD model is:

$\begin{array}{rcll}\rho \frac{\partial \stackrel{\to }{v}}{\partial t}& =& \stackrel{\to }{j}×\stackrel{\to }{B}-\nabla p& \text{(3.38)}\text{}\text{}\\ \frac{\partial \rho }{\partial t}& =& -\nabla \cdot \left(\rho \stackrel{\to }{v}\right)& \text{(3.39)}\text{}\text{}\\ \stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}& =& \eta \stackrel{\to }{j}+\frac{1}{ne}\left[\stackrel{\to }{j}×\stackrel{\to }{B}+\nabla {p}_{e}\right]\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{(Generalized)}}& \text{(3.40)}\text{}\text{}\\ & =& \eta \stackrel{\to }{j}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{(Resistive)}}& \text{(3.41)}\text{}\text{}\\ & =& 0\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{(Ideal)}}& \text{(3.42)}\text{}\text{}\end{array}$

with the various options given for Ohm’s Law (more complex ones can be derived but are not included).

3.3 Pinches (MHD Equilibria)

In this section we consider a few cases of MHD equilibria analysis, using the single-ﬂuid MHD theory just derived. First, we note that in equilibrium several simpliﬁcations can be made to the single-ﬂuid equations, in particular by taking time derivatives to zero (no change in an equilibrium situation). For simplicity we take ideal MHD.

The continuity equation can be rewritten as:

 $\frac{d\rho }{dt}+\rho \nabla \cdot \stackrel{\to }{v}=0$ (3.43)

in equilibria this implies that $\nabla \cdot \stackrel{\to }{v}=0$, which means also that $\nabla \cdot \stackrel{\to }{j}$ = 0. There are no particle or current sources/sinks in equilibrium solutions in our simple model (which does not include, for instance, current drive or neutral-beam heating).

The momentum equation becomes simply:

 $\nabla p=\stackrel{\to }{j}×\stackrel{\to }{B}$ (3.44)

The complete set of equations which deﬁne simple MHD equilibrium systems is combined with two of Maxwells’ equations; in total we have that:

$\begin{array}{rcll}\nabla p& =& \stackrel{\to }{j}×\stackrel{\to }{B}& \text{(3.45)}\text{}\text{}\\ \nabla ×\stackrel{\to }{B}& =& {\mu }_{0}\stackrel{\to }{j}& \text{(3.46)}\text{}\text{}\\ \nabla \cdot \stackrel{\to }{B}& =& 0& \text{(3.47)}\text{}\text{}\end{array}$

One important qualitative feature of solutions to this set of equations is that both the lines of force ($\stackrel{\to }{B}$ ﬁeld lines) and the current $\stackrel{\to }{j}$ must lie in a plane perpendicular to the pressure gradient. The rest of this section is devoted to identifying a few useful solutions of the MHD equilibrium problem.

3.3.1 $𝜃$ Pinch

Consider a cylindrical plasma which has a current ﬂow in the azimuthal ($\stackrel{̂}{𝜃}$) direction, thus the name. This can be created via the diamagnetic eﬀect of plasmas by an imposed axial ﬁeld. In any event, the unknowns are $p=p\left(r\right)$, ${B}_{z}={B}_{z}\left(r\right)$ and ${j}_{𝜃}={j}_{𝜃}\left(r\right)$ with the total ﬁeld and current given by $\stackrel{\to }{B}={B}_{z}\left(r\right)ẑ$ and $\stackrel{\to }{j}={j}_{𝜃}\left(r\right)\stackrel{̂}{𝜃}$.

By symmetry the condition $\nabla \cdot \stackrel{\to }{B}=0$ is automatically satisﬁed. Ampère’s Law reduces to

 $-\frac{d{B}_{z}}{dr}={\mu }_{0}{j}_{𝜃}.$ (3.48)

And the pressure balance equation becomes

 $\frac{dp}{dr}={j}_{𝜃}{B}_{z}=-\frac{1}{{\mu }_{0}}{B}_{z}\frac{d{B}_{z}}{dr}=-\frac{1}{2{\mu }_{0}}\frac{d}{dr}\left({B}_{z}^{2}\right)$ (3.49)

this can be rewritten, combining the derivatives with respect to r, to get that

 $\frac{d}{dr}\left(p+\frac{{B}_{z}^{2}}{2{\mu }_{0}}\right)=0$ (3.50)

which can be trivially integrated, yielding the pressure balance equation:

 $p\left(r\right)+\frac{{B}_{z}^{2}\left(r\right)}{2{\mu }_{0}}=\frac{{B}_{0}^{2}}{2{\mu }_{0}}$ (3.51)

where we have automatically solved for the constant of integration by noting that the ﬁeld ${B}_{z}$ must become the vacuum value for radii outside the plasma where $p=0$, i.e. $r>a$.

3.3.2 Z Pinch

We now consider the complementary system, the Z pinch. In this case, the current ﬂows along the axis of a cylindrical plasma $\stackrel{\to }{j}={j}_{z}\left(r\right)ẑ$ which creates an azimuthal magnetic ﬁeld $\stackrel{\to }{B}={B}_{𝜃}\left(r\right)\stackrel{̂}{𝜃}+{B}_{0}ẑ$. There is generally assumed to be an imposed magnetic ﬁeld along the axis as well, given by the ${B}_{0}$ term. The current in this pinch could be driven by a set of electrodes at either end of the plasma.

Once again we have the condition $\nabla \cdot \stackrel{\to }{B}$ trivially satisﬁed by the geometry. And Ampère’s Law becomes:

 ${\mu }_{0}{j}_{z}=\frac{1}{r}\frac{d}{dr}\left(r{B}_{𝜃}\right).$ (3.52)

Combining this with the pressure balance equation,

 $\frac{dp}{dr}=-{j}_{z}{B}_{𝜃}=-\frac{{B}_{𝜃}}{{\mu }_{0}r}\frac{d}{dr}\left(r{B}_{𝜃}\right).$ (3.53)

If we expand the $r{B}_{𝜃}$ derivative we get that:

 $\frac{dp}{dr}+\frac{1}{2{\mu }_{0}}\frac{d}{dr}\left({B}_{𝜃}^{2}\right)+\frac{{B}_{𝜃}^{2}}{{\mu }_{0}r}=0.$ (3.54)

which can be simpliﬁed by combining the $d∕dr$ terms:

 $\frac{d}{dr}\left[p+\frac{{B}_{𝜃}^{2}}{2{\mu }_{0}}\right]+\frac{{B}_{𝜃}^{2}\left(r\right)}{{\mu }_{0}r}=0$ (3.55)

At this point one must typically assume a current proﬁle to solve analytically for the pressure proﬁle.

3.3.3 Screw Pinch

A screw pinch is a linear combination of $𝜃-$ and Z- pinch, named because the magnetic ﬁeld lines wrap around the cylindrical plasma like the threads on a screw (or the stripes on a barber’s pole). The deﬁning pressure balance equation can be written by combining Eqs. 3.50 and 3.55:

 $\frac{d}{dr}\left[p+\frac{{B}_{z}^{2}}{2{\mu }_{0}}+\frac{{B}_{𝜃}^{2}}{2{\mu }_{0}}\right]+\frac{{B}_{𝜃}^{2}\left(r\right)}{{\mu }_{0}r}=0$ (3.56)

The equation has three unknowns - $p\left(r\right),{B}_{z}\left(r\right),{B}_{𝜃}\left(r\right)$. In general solutions are obtained by assuming proﬁles for two of the three; MHD will then deﬁne the third.

3.4 Diamagnetic Drift

In our previous discussion of particle drifts, we left out one important drift - the diamagnetic drift. This is because the diamagnetic drift arises as a ﬂuid eﬀect and not from single-particle motion. We start out with the momentum equation for an arbitrary ﬂuid (omitting subscripts for clarity):

 $mn\left[\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}\right]=nq\left(\stackrel{\to }{E}+\stackrel{\to }{u}×\stackrel{\to }{B}\right)-\nabla p.$ (3.57)

First, consider the eﬀect of the time derivative term. Let $\partial ∕\partial t\to -i\omega$ and take the ratio of the time derivative term to the $\stackrel{\to }{u}×\stackrel{\to }{B}$ term considering only the motion perpendicular to the magnetic ﬁeld:

 $\left|\frac{i\omega mn{v}_{\perp }}{qn{v}_{\perp }B}\right|=\frac{\omega }{{\omega }_{c}}$ (3.58)

For drifts slow compared to the cyclotron motion, $\omega \ll {\omega }_{c}$ we can neglect the time derivative term.

What about the term $\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}$ ? Observe that if the drift velocity we will derive ends up being $\stackrel{\to }{{v}_{d}}\perp \nabla \stackrel{\to }{u}$ then this term will be 0. Therefore we will assume oﬀ the bat that this is the case, and at the end of the derivation we can check to make sure that this assumption is valid.

So what we have is that for slow drifts with the drift velocity perpendicular to the gradient,

 $0=nq\left(\stackrel{\to }{E}+\stackrel{\to }{u}×\stackrel{\to }{B}\right)-\nabla p.$ (3.59)

If we take the equation’s cross product with $\stackrel{\to }{B}$,

 $0=nq\left(\stackrel{\to }{E}×\stackrel{\to }{B}+\left(\stackrel{\to }{u}×\stackrel{\to }{B}\right)×\stackrel{\to }{B}\right)-\nabla p×\stackrel{\to }{B}.$ (3.60)

Using the vector identity $\stackrel{\to }{A}×\left(\stackrel{\to }{B}×\stackrel{\to }{C}\right)=\left(\stackrel{\to }{A}\cdot \stackrel{\to }{C}\right)\stackrel{\to }{B}-\left(\stackrel{\to }{A}\cdot \stackrel{\to }{B}\right)\stackrel{\to }{C}$ for the middle term:

 $0=nq\left(\stackrel{\to }{E}×\stackrel{\to }{B}+\left(\stackrel{\to }{{u}_{\perp }}\cdot \stackrel{\to }{B}\right)\stackrel{\to }{B}-{\stackrel{\to }{u}}_{\perp }{B}^{2}\right)-\nabla p×\stackrel{\to }{B}.$ (3.61)

By deﬁnition of course ${\stackrel{\to }{u}}_{\perp }\cdot \stackrel{\to }{B}=0$. If we take this equation, rearrange and solve for $\stackrel{\to }{{u}_{\perp }}$ we get that:

 ${\stackrel{\to }{u}}_{\perp }=\frac{\stackrel{\to }{E}×\stackrel{\to }{B}}{{B}^{2}}-\frac{\nabla p×\stackrel{\to }{B}}{nq{B}^{2}}.$ (3.62)

The ﬁrst term is the familiar $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift. The second term is the Diamagnetic Drift:

 $\stackrel{\to }{{v}_{D}}=-\frac{\nabla p×\stackrel{\to }{B}}{nq{B}^{2}}.$ (3.63)

We observe that the diamagnetic drift is indeed perpendicular to the pressure gradient, as we had assumed before. Due to the dependence on $q$, electrons and ions will drift in opposite directions and thus the diamagnetic current can induce azimuthal currents in cylindrical plasmas. For $Z=1$ and $p=nkT$, we can write the diamagnetic current as

 ${\stackrel{\to }{j}}_{D}=ne\left({\stackrel{\to }{u}}_{Di}-{\stackrel{\to }{u}}_{De}\right)=\left(k{T}_{i}+k{T}_{e}\right)\frac{\stackrel{\to }{B}×\nabla n}{{B}^{2}}.$ (3.64)

3.5 Drift Waves

The diamagnetic drift of the previous section can lead to an unstable situation in cylindrical or toroidal plasmas, leading to so-called drift waves.

The basic idea is as follows. We have a plasma magnetically conﬁned by the 0th order $B$ ﬁeld, shown as out of the page (or in the $ẑ$ direction). Due to the density gradient as shown, the diamagnetic drift can set up electric ﬁelds if there is a perturbation in the isobaric surface (shown in red). This electric ﬁeld causes $\stackrel{\to }{E}×\stackrel{\to }{B}$ drifts, shown as ${\stackrel{\to }{v}}_{1}$. A detailed analysis follows.

The zeroth-order drifts are due to the diamagnetic drift:

 ${\stackrel{\to }{v}}_{e0}={\stackrel{\to }{v}}_{De}=-\frac{{k}_{B}{T}_{e}}{e{B}_{0}}\frac{{n}_{0}^{\prime }}{{n}_{0}}ŷ$ (3.65)

and also for the ions. Since the electrons can ﬂow along the magnetic ﬁeld, and we assume that this wave is slow relative to the electron motion, we can use the Boltzmann relation for the electron density:

 $\frac{{n}_{1}}{{n}_{0}}=\frac{e{\varphi }_{1}}{{k}_{B}{T}_{e}}$ (3.66)

which is to ﬁrst order, in the limit $e\varphi \ll {k}_{B}{T}_{e}$. Where the plasma is perturbed to higher pressure (higher density), as in the top of the ﬁgure, the potential is positive. The electric ﬁeld thus points from peak to trough of the wave, as shown.

We can now write the ﬁrst-order velocity due to the $\stackrel{\to }{E}×\stackrel{\to }{B}$ drift as:

 ${v}_{1x}=\frac{{E}_{y}}{{B}_{0}}=-\frac{i{k}_{y}{\varphi }_{1}}{{B}_{0}}$ (3.67)

which can be obtained from $\stackrel{\to }{E}=-\nabla \varphi =-ik\varphi$. We can also use a sort of ‘continuity equation’ for guiding centers to write:

 $\frac{\partial {n}_{1}}{\partial t}=-{v}_{1x}\frac{\partial {n}_{0}}{\partial x}$ (3.68)

Doing some Fourier analysis on this continuity equation,

 $-i\omega {n}_{1}=-ik{v}_{1x}{n}_{0}=-\frac{i{k}_{y}{\varphi }_{1}}{{B}_{0}}{n}_{0}^{\prime }$ (3.69)

Where ${n}_{0}^{\prime }$ represents the density perturbation due to the warped isobar. Using the Boltzmann relation from earlier we can also write (separately) that

 $-i\omega {n}_{1}=-i\omega {n}_{0}\frac{e{\varphi }_{1}}{{k}_{B}{T}_{e}}$ (3.70)

Equation the RHS of these two gives:

 $\frac{{k}_{y}{\varphi }_{1}{n}_{0}^{\prime }}{{B}_{0}}=\frac{\omega {n}_{0}e{\varphi }_{1}}{{k}_{B}{T}_{e}}$ (3.71)

 $\frac{\omega }{{k}_{y}}=\frac{{k}_{B}{T}_{e}}{e{B}_{0}}\frac{{n}_{0}^{\prime }}{{n}_{0}}={v}_{De}$ (3.72)

The waves travel at the electron diamagnetic drift velocity. This is the behavior in the azimuthal direction. There is a small component ${k}_{z}\ll {k}_{y}$ which causes these perturbations to propagate. Along the way we implicitly assumed that electron currents cannot simply ﬂow along ${B}_{0}$ to neutralize the electric ﬁeld; this means that the plasma must be resistive (and these are sometimes called ‘resistive drift waves’).

We have also neglected the instability analysis. One can see via qualitative arguments that this situation is very similar to the gravitational instability, and thus it can be argued that the perturbations in isobars tend to grow. The full analysis requires treating the polarization drift and non-uniform $\stackrel{\to }{E}$ drift as well.

3.6 Two-stream Instability

We wish to consider the following scenario: an inﬁnite uniform plasma, where the ions are stationary in the lab frame but the electrons have a non-zero uniform ${\stackrel{\to }{u}}_{e0}$. The plasma is unmagnetized (${B}_{0}=0$) and cold (${T}_{i}={T}_{e}=0$) for simplicity. The ﬂuid momentum equation is:

 $mn\left[\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}\right]=nq\left(\stackrel{\to }{E}+\stackrel{\to }{u}×\stackrel{\to }{B}\right)-\nabla p-mn\nu \left(\stackrel{\to }{u}-\stackrel{\to }{{u}_{0}}\right)$ (3.73)

In a 1-D ﬂow the $\stackrel{\to }{u}×\stackrel{\to }{B}$ ﬁrst-order terms are zero for the electrons (ions have no ﬁrst-order contribution from this term). Further, the $\nabla p$ terms are zero because we assumed a cold plasma. The collisional momentum transfer term also goes to $0$ because we note that $\nu \propto {T}^{3∕2}$.

For the ions, $\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}$ contains no ﬁrst-order terms since ${\stackrel{\to }{u}}_{0}=0$. For the electrons, a term $\left({\stackrel{\to }{u}}_{0}\cdot \nabla \right){\stackrel{\to }{u}}_{1}$ remains. For simplicity we take $Z=1$. We thus get an ion equation:

 ${m}_{i}n\frac{\partial {\stackrel{\to }{u}}_{i1}}{\partial t}=ne{\stackrel{\to }{E}}_{1},$ (3.74)

and an electron equation:

 ${m}_{e}n\left[\frac{\partial {\stackrel{\to }{u}}_{e1}}{\partial t}+\left({\stackrel{\to }{u}}_{e0}\cdot \nabla \right){\stackrel{\to }{u}}_{e1}\right]=-en{\stackrel{\to }{E}}_{1}.$ (3.75)

Assuming electrostatic waves of the form:

 ${\stackrel{\to }{E}}_{1}={E}_{1}{e}^{-i\left(kx-\omega t\right)}\stackrel{̂}{x},$ (3.76)

we can use Fourier analysis to simplify the ion equation to:

 $-i\omega {m}_{i}n{\stackrel{\to }{u}}_{i1}=ne\stackrel{\to }{{E}_{1}},$ (3.77)

 ${\stackrel{\to }{u}}_{i1}=\frac{ie}{\omega {m}_{i}}{E}_{1}\stackrel{̂}{x}.$ (3.78)

And the electron equation similarly:

 $-i\omega {m}_{e}n{\stackrel{\to }{u}}_{e1}+ik{m}_{e}n{u}_{e0}{\stackrel{\to }{u}}_{e1}=-en{\stackrel{\to }{E}}_{1}$ (3.79)

 ${\stackrel{\to }{u}}_{e1}=-\frac{ie}{{m}_{e}\left(\omega -k{u}_{0}\right)}{E}_{1}\stackrel{̂}{x}.$ (3.80)

Next we use the ﬂuid equation of continuity, which is generally:

 $\frac{\partial n}{\partial t}+\nabla \cdot \left(n\stackrel{\to }{u}\right)=0.$ (3.81)

So for the ions, employing our previous assumptions and Fourier analysis,

 $-i\omega {n}_{i1}+ik{n}_{0}{u}_{i1}=0$ (3.82)

 ${n}_{i1}=\frac{k{n}_{0}}{\omega }{u}_{i1}=\frac{iek{n}_{0}}{{m}_{i}{\omega }^{2}}{E}_{1}$ (3.83)

where we have used the previously-derived relation between ${u}_{i1}$ and ${E}_{1}$. Following through the same steps for the electrons:

 $\frac{\partial {n}_{e1}}{\partial t}+{n}_{0}\nabla \cdot {\stackrel{\to }{u}}_{e1}+\left({\stackrel{\to }{u}}_{e0}\cdot \nabla \right){n}_{e1}=0$ (3.84)

 $-i\omega {n}_{e1}+ik{n}_{0}{u}_{e1}+ik{n}_{e1}{u}_{e0}=0$ (3.85)

 ${n}_{e1}=\frac{k{n}_{0}}{\omega -k{u}_{e0}}{u}_{e1}=\frac{iek{n}_{0}}{{m}_{e}{\left(\omega -k{u}_{e0}\right)}^{2}}{E}_{1}.$ (3.86)

We need one more equation to close these relations. We note that this is a high-frequency oscillations so we can use Poisson’s equation to eliminate the electric ﬁeld:

 $\nabla \cdot \stackrel{\to }{E}=\frac{e}{{𝜖}_{0}}\left({n}_{i}-{n}_{e}\right).$ (3.87)

Using the ﬁrst order density perturbations we have already derived on the right side, and Fourier analysis again on the left side, we get:

 $ik{E}_{1}=\frac{e}{{𝜖}_{0}}\frac{iek{n}_{0}}{1}{E}_{1}×\left(\frac{1}{{m}_{i}{\omega }^{2}}-\frac{1}{{m}_{e}{\left(\omega -k{u}_{e0}\right)}^{2}}\right)$ (3.88)

We can now eliminate the electric ﬁeld, and recognizing the plasma frequency ${\omega }_{pe}=n{e}^{2}∕{m}_{e}{𝜖}_{0}$ on the right hand side,

 $1={\omega }_{pe}^{2}\left[\frac{{m}_{e}∕{m}_{i}}{{\omega }^{2}}-\frac{1}{{\left(\omega -k{u}_{e0}\right)}^{2}}\right].$ (3.89)

An important question is whether the plasma is stable or unstable to perturbations. If we deﬁne the dimensionless variables $x=\omega ∕{\omega }_{pe}$ and $y=k{u}_{e0}∕{\omega }_{pe}$ we can deﬁne:

 $F\left(x,y\right)\equiv \frac{{m}_{e}∕{m}_{i}}{{x}^{2}}+\frac{1}{{\left(x-y\right)}^{2}}=1$ (3.90)

There are singularities at $x=0$ and $x=y$. But we note that solutions to $F=1$ are solutions to the dispersion relation. If $y$ is suﬃciently large then there are 4 real roots for $x$ and thus $\omega$, which all correspond to stable though oscillatory solutions. On the other hand, if there are only two real solutions for $\omega$ then there are necessarily two complex, one of which will be a damped oscillation and the other will be unstable (growing). This occurs for suﬃciently small values of $y$, or more physically, for small $k{u}_{e0}$ or large-wavelength perturbations.

3.7 Flux Freezing

This is an important question in the dynamics of magnetized ideal plasmas. We want to know how plasma elements move with respect to the magnetic ﬁeld lines, or vice versa. Consider the magnetic ﬂux through a surface of plasma:

 $\psi \left(t\right)=\int \stackrel{\to }{B}\cdot \stackrel{̂}{n}dS$ (3.91)

where $\stackrel{̂}{n}$ is the unit vector normal to the surface $S$. We now want to explore how the magnetic ﬂux through the plasma changes with time. The time derivative of the previous equation can be written as:

 $\frac{d\psi }{dt}=\int \frac{\partial \stackrel{\to }{B}}{\partial t}\cdot \stackrel{̂}{n}dS+\oint \stackrel{\to }{B}×{\stackrel{\to }{u}}_{\perp }\cdot d\stackrel{\to }{l}.$ (3.92)

The left hand side is a simple time derivative. The right hand side has been decomposed: ﬂux can change due to a changing magnetic ﬁeld (the ﬁrst term) or due to motion of the plasma surface relative to the magnetic ﬁeld (the second term).

 $\nabla ×\stackrel{\to }{E}=-\frac{\partial \stackrel{\to }{B}}{\partial t},$ (3.93)

and the ideal Ohm’s Law,

 $\stackrel{\to }{E}=-\stackrel{\to }{{v}_{\perp }}×\stackrel{\to }{B},$ (3.94)

 $\frac{d\psi }{dt}=\oint \left[\left({\stackrel{\to }{v}}_{\perp }-{\stackrel{\to }{u}}_{\perp }\right)×\stackrel{\to }{B}\right]\cdot d\stackrel{\to }{l}.$ (3.95)

The important result is that for ${\stackrel{\to }{v}}_{\perp }={\stackrel{\to }{u}}_{\perp }$ then the ﬂux is unchanging with time. Physically this occurs when the magnetic ﬁeld lines are moving with the plasma, and we call this ”frozen in ﬂux”. The obvious analogy is a superconductor. It is a well-known result that magnetic ﬂux cannot penetrate a superconducting volume. If it did, and the ﬂux was changing in time (as it must) then it would induce an EMF in the superconductor, which would cause an inﬁnite current due to the zero resistivity.

A real plasma will have ﬁnite resistivity but in many scenarios the resistivity is small enough that ideal MHD is a decent approximation. This is particularly true in certain high-temperature magnetically conﬁned plasmas and in very low-density space plasmas.

3.8 MHD Instabilities

First we discuss the classiﬁcation schemes generally used for MHD instabilities, then discuss qualitatively a few important examples.

3.8.1 Classiﬁcation

First an instability is generally described as internal or external mode. An internal mode instability is one in which the plasma surface does not move. Generally these aﬀect transport and impose operational limits but do not impact conﬁnement. Conversely, an external mode instability is one in which the surface of the plasma moves. These can cause loss of conﬁnement.

Next we classify the source of the instability. Generally a non-equilibrium current ﬂows, which modiﬁes the MHD. If a current ﬂows perpendicular the magnetic ﬁeld, these instabilities are often called pressure-driven since $\nabla p={\stackrel{\to }{J}}_{\perp }×\stackrel{\to }{B}$. On the other hand, if a parallel current ﬂows to drive the instability we call it a current-driven mode.

Finally, the wall characteristics are often important. We can see this simply as follows: If the plasma surface moves towards the wall in an ideal MHD system (frozen ﬂux), the magnetic ﬂux between the plasma surface and wall will be compressed. This will aﬀect the plasma motion and feed back. If the wall is resistive then the compressed ﬂux will ohmically dissipate, but if it is superconducting then this cannot happen. We can therefore discuss no wall, conducting wall, or superconducting wall conﬁgurations (though the last is operationally diﬃcult).

3.8.2 Qualitative Descriptions

Z-pinch interchange / sausage

We start with a qualitative description of the Z-pinch interchange or ‘sausage’ instability, which is depicted in Fig. 3.2. As a reminder, the 0th order current ﬂows axially and thus left to right (or vice versa) in the ﬁgure. Consider a perturbation where the surface of the plasma is rippled such that the radius varies with axial position, in particular shown for ${r}_{1}>{r}_{2}>{r}_{3}$. In this case we know that ${B}_{1}<{B}_{2}<{B}_{3}$ which implies that the magnetic conﬁnement is higher at $3$ than at $1$. This causes the plasma to expand at $1$ and contract further at $3$.

Z-pinch twisting instability

Continuing with the Z-pinch equilibrium conﬁnement scheme, consider the scenario in Fig. 3.3. In this case the cylindrical symmetry is broken by allowing the axis (dotted line) to be twisted. Since the current is ﬂowing axially in the plasma, this creates a magnetic ﬁeld scheme as shown in blue. At the top of the indicated twist, the azimuthal magnetic ﬁeld is stronger than the equilibrium value. Below the twist it is weaker. This creates a force on the plasma that tends to reinforce the kink because the conﬁnement is coming from the MHD equilibrium term $\nabla p=\stackrel{\to }{j}×\stackrel{\to }{B}$.

Screw pinch kink instability

This conﬁguration is very similar to the previous. We simply note that a screw pinch suﬀers from the same sensitivity to kinking or twisting as the Z-pinch. See Fig. 3.4. The mechanism is exactly the same as in the previous section, but now we have to remember that there is a potentially large axial/toroidal ﬁeld ${B}_{\varphi }$ as well as the ${B}_{𝜃}$ ﬁeld.

Favorable vs unfavorable curvature

The previous sections raise a general question - when is curvature of the plasma surface favorable versus unfavorable? Consider Fig. 3.5. The equilibrium surface of the plasma is denoted in black. We follow the general coloring scheme (plasma is red, magnetic ﬁeld is blue) used in the previous sections. Consider a perturbation from the equilibrium, as shown.

The general picture follows again from considering the $\nabla p=\stackrel{\to }{j}×\stackrel{\to }{B}$ conﬁnement in MHD equilibria systems. When the plasma surface is perturbed as shown on the left of Fig. 3.5, it will tend to grow because the conﬁnement is not good at the peak. More generally, it turns out that when the surface of the plasma is curved towards the plasma then the surface is unstable, so we call that ‘unfavorable’ curvature. Conversely, when the curvature is away from the plasma the system is well-conﬁned and we thus call that ‘favorable’ curvature.

Gravitational instability

This instability is named the ‘gravitational’ instability but really can occur for any non-electromagnetic force applied to the plasma.

Consider the plasma conﬁguration shown in Fig. 3.6. The plasma is conﬁned by a magnetic ﬁeld, shown as out of the page, with a well-deﬁned surface horizontal on the page. There is a vertical force away from the plasma surface denoted by $\stackrel{\to }{g}$. We take $\stackrel{\to }{g}$ to be any generalized non-electromagnetic force. Gravity is one potential application.

As we saw in the single particle motion derivations, there is a generalized force drift which can be written:

 ${\stackrel{\to }{v}}_{D}=\frac{1}{q}\frac{\stackrel{\to }{g}×\stackrel{\to }{B}}{{B}^{2}}.$ (3.96)

We can immediately see that the electrons and ions drift horizontally on the page due to the force $\stackrel{\to }{g}$, and in opposite directions as denoted by ${\stackrel{\to }{v}}_{Di}$ and ${\stackrel{\to }{v}}_{De}$ in the ﬁgure.

Now we have to consider what will happen if the surface of the plasma is perturbed. Consider Fig. 3.7, in which we have assumed an imposed sinusoidal horizontal perturbation in the plasma surface. The imposed force $\stackrel{\to }{g}$ is still vertically down and $\stackrel{\to }{{B}_{0}}$ is still out of the page. Because of the $\stackrel{\to }{g}×{\stackrel{\to }{B}}_{0}$ drift, the ions tend to ‘pile up’ on the left side of a plasma protrusion while the electrons pile up on the right hand side of a protrusion. This creates a ﬁrst-order electric ﬁeld ${E}_{1}$ which is horizontal in this scheme.

On the right where the plasma is protruding from its equilibrium position, the ﬁrst-order drift ${\stackrel{\to }{E}}_{1}×{\stackrel{\to }{B}}_{0}$ points down, as shown. As we know the $E×B$ drift aﬀects ions and electrons equally, and thus this creates a net force on the plasma that reinforces the perturbation. In cases where the plasma is depressed from its equilibrium location the ﬁrst-order electric ﬁeld is reversed in direction, and thus the ${\stackrel{\to }{E}}_{1}×{\stackrel{\to }{B}}_{0}$ drift also reinforces the perturbation. Therefore the plasma conﬁguration explored in this section is unstable to perturbations in the surface as shown.

It is worth noting that this can be thought of as a magnetically-conﬁned plasma version of the well-known hydrodynamic Rayleigh-Taylor instability. In this case the plasma is a ‘heavy’ ﬂuid supported by the magnetic ﬁeld, a ‘light’ ﬂuid.

Drift instability / wave

Qualitatively one can arrive at the same sort of instability without an imposed external force $\stackrel{\to }{g}$ if instead there is a density gradient. Recall that the diamagnetic drift is deﬁned as:

 ${\stackrel{\to }{v}}_{D}=\frac{\nabla p×\stackrel{\to }{B}}{nq{B}^{2}}.$ (3.97)

If instead of the external force, the plasma in Fig. 3.6 had a vertical density gradient (high density at the top) then the zero order $\stackrel{\to }{g}×\stackrel{\to }{{B}_{0}}$ are instead replaced by the electron and ion diamagnetic drifts. The remaining analysis is exactly the same.

This is a very important instability because in cylindrical systems, there is necessarily a radial pressure gradient and thus the plasma is susceptible to drift instabilities. In a cylindrical plasma this is sometimes referred to as the ﬂute instability instead of the drift instability since the resulting plasma surface looks like a ﬂuted column.

In toroidal systems the same instability occurs. The result of detailed analysis is that there is a small component along the toroidal axis. In this case the perturbations curve slightly and wrap around the plasma like screw threads or stripes on a barber’s pole. This causes the perturbations to propagate and they are called true ‘drift waves’.

Weibel Instability

The ﬁnal instability we consider is the Weibel instability. This situation is illustrated in Fig. 3.8 and described here. Consider a plasma where the electron temperature is much hotter in one direction than the other two. This can arise in magnetically conﬁned systems (where $\perp$ and $\parallel$ directions have diﬀerent behavior) or in laser plasmas, particularly in direct laser illumination between the ablation surface and the critical surface.

Anyways, in the ﬁgure we take ${T}_{y}\gg {T}_{x},{T}_{z}$. as shown. Consider a randomly-arising magnetic ﬁeld in the $x-z$ plane as shown in blue. Due to the ﬁeld, electrons will tend to curve as shown for a few diﬀerent locations. A detailed analysis shows that the electrons tend to curve to form current sheets in the vertical ($ŷ$) direction. These current sheets end up reinforcing the imposed ﬁeld. The plasma is therefore unstable to randomly-arising magnetic ﬁelds in the $x-z$ plane.

Chapter 4Waves

4.1 Gas Dynamics Sound Waves

Before we get into plasma physics, we start with ordinary gas dynamics. Starting with the Navier-Stokes equation:

 $\rho \left[\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}\right]=-\nabla p-\rho \nu {\nabla }^{2}\stackrel{\to }{u},$ (4.1)

we can derive sound waves as follows. Neglecting kinematic viscosity ($\nu \to 0$), assuming ${\stackrel{\to }{u}}_{0}=0$ and taking small waves (so keep only ﬁrst order terms), we get that

 $\rho \frac{\partial \stackrel{\to }{u}}{\partial t}=-\nabla p.$ (4.2)

Using Fourier analysis,

 $-i\omega \rho u=-ikp.$ (4.3)

We know that the group velocity of waves is $\partial \omega ∕\partial k$ so we can rewrite this as

 $\omega =kp∕u\rho .$ (4.4)

We will also need to use the continuity equation:

 $\frac{\partial \rho }{\partial t}+\nabla \cdot \rho \stackrel{\to }{u}=0,$ (4.5)

 $ik\rho u-i\omega \rho =0,$ (4.6)

 $u=\omega ∕k.$ (4.7)

Using this with the momentum equation derived dispersion relation, we get that

 $w=k\sqrt{p∕\rho },$ (4.8)

which directly implies

 ${v}_{g}={c}_{s}=\sqrt{\frac{p}{\rho }}.$ (4.9)

We implicitly assumed $\gamma =1$ earlier, a more general result is that

 ${c}_{s}=\sqrt{\frac{\gamma p}{\rho }}$ (4.10)

4.2 Electromagnetic Waves

Before we introduce plasma eﬀects, it is useful to derive electromagnetic waves without plasma ﬁrst. First we should write down Maxwell’s equations:

$\begin{array}{rcll}\nabla \cdot \stackrel{\to }{E}& =& \rho ∕{𝜖}_{0}& \text{(4.11)}\text{}\text{}\\ \nabla \cdot \stackrel{\to }{B}& =& 0& \text{(4.12)}\text{}\text{}\\ \nabla ×\stackrel{\to }{E}& =& -\frac{\partial \stackrel{\to }{B}}{\partial t}& \text{(4.13)}\text{}\text{}\\ \nabla ×\stackrel{\to }{B}& =& {\mu }_{0}\stackrel{\to }{j}+\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}& \text{(4.14)}\text{}\text{}\end{array}$

In a vacuum we can immediately drop the source terms

$\begin{array}{rcll}\nabla \cdot \stackrel{\to }{E}& =& 0& \text{(4.15)}\text{}\text{}\\ \nabla \cdot \stackrel{\to }{B}& =& 0& \text{(4.16)}\text{}\text{}\\ \nabla ×\stackrel{\to }{E}& =& -\frac{\partial \stackrel{\to }{B}}{\partial t}& \text{(4.17)}\text{}\text{}\\ \nabla ×\stackrel{\to }{B}& =& \frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}& \text{(4.18)}\text{}\text{}\end{array}$

For plane waves the ﬁrst two equations will be automatically satisﬁed. Taking the curl of Faraday’s law,

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)=-\frac{\partial }{\partial t}\left(\nabla ×\stackrel{\to }{B}\right),$ (4.19)

and combining with Ampère’s law,

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)=-\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{E}}{\partial {t}^{2}}.$ (4.20)

If we assume plane wave Fourier solutions of the form

 $\stackrel{\to }{E}=E{e}^{-i\left(\omega t-kz\right)}\stackrel{̂}{x}$ (4.21)

and use the vector identity

 $\nabla ×\left(\nabla ×\stackrel{\to }{A}\right)=\nabla \left(\nabla \cdot \stackrel{\to }{A}\right)-{\nabla }^{2}\stackrel{\to }{A}$ (4.22)

We get that

 ${\nabla }^{2}\stackrel{\to }{E}=\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{E}}{\partial {t}^{2}}$ (4.23)

which reduces to, using Fourier analysis,

 $w=ck$ (4.24)

which leads to the group and phase velocities:

 ${v}_{g}\equiv \frac{\partial \omega }{\partial k}=c$ (4.25)

 ${v}_{p}\equiv \frac{\omega }{k}=c$ (4.26)

In media we generally deﬁne the index of refraction,

 $n=ck∕\omega$ (4.27)

and the group / phase velocities become

 ${v}_{g}=c∕n$ (4.28)

 ${v}_{p}=c∕n$ (4.29)

4.3 Polarization

In the derivation of waves we neglected the geometry of the system, which leads to a discussion of the wave polarization. Starting oﬀ with the Fourier component solution for the electric ﬁeld,

 $\stackrel{\to }{E}=E{e}^{-i\left(\omega t-kz\right)}\stackrel{̂}{x}$ (4.30)

we note that the solutions here are waves with $\stackrel{\to }{E}\parallel \stackrel{̂}{x}$ but the wave propagation direction is $\parallel ẑ$. We can also ﬁnd the magnetic ﬁeld for this conﬁguration, it is obtained from

 $\nabla ×\stackrel{\to }{B}=\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}$ (4.31)

and the solution is

 $\stackrel{\to }{B}=\frac{E\omega }{{c}^{2}k}{e}^{-i\left(\omega t-kz\right)}ŷ=B{e}^{-i\left(\omega t-kz\right)}ŷ$ (4.32)

with $B=E∕c$. So to summarize, in this nice plane polarized solution we have $\stackrel{\to }{E}\parallel \stackrel{̂}{x}$, $\stackrel{\to }{B}\parallel ŷ$, and ﬁnally $\stackrel{\to }{k}\parallel ẑ$. The choice of coordinate system does not cause loss of generality. It is important to note that we will always have $\stackrel{\to }{E}×\stackrel{\to }{B}\parallel \stackrel{\to }{k}$ in a vacuum.

We could have chosen a wave solution in which the electric ﬁeld vector rotated in the $xy$ plane as it propagated along $z$. In this case the magnetic ﬁeld vector will also have to rotate. These are circularly or elliptically polarized solutions. We can characterize these solutions by taking the ratio

 $i{E}_{x}∕{E}_{y}$ (4.33)

when this $=1$, the wave is right-hand circularly polarized. When it is $=-1$ the wave is left-hand circularly polarized instead.

4.4 Electrostatic Plasma Oscillations

We start oﬀ with the MHD picture of an electrostatic plasma oscillation. We will see at the end why it is an ’oscillation’ and not a wave. We need to start with the electron momentum equation:

 ${m}_{e}{n}_{e}\left[\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}+\left({\stackrel{\to }{u}}_{e}\cdot \nabla \right){\stackrel{\to }{u}}_{e}\right]=-{n}_{e}e\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{e}×\stackrel{\to }{B}\right)-\nabla {p}_{e}-mn\nu \left({\stackrel{\to }{u}}_{e}-{\stackrel{\to }{u}}_{i}\right).$ (4.34)

We need to make several simpliﬁcations and approximations in this section to make the problem tractable. First, we take ${T}_{e}=0$ meaning that ${p}_{e}=0$. Next, we neglect collisions, $\nu =0$. Next we take ${\stackrel{\to }{u}}_{e0}=\stackrel{\to }{{E}_{0}}={\stackrel{\to }{B}}_{0}=0$ and keep only ﬁrst-order quantities:

 $m{n}_{0}\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}=-{n}_{0}e\stackrel{\to }{E}$ (4.35)

for convenience we have dropped the $e$ subscripts. If we assume sinusoidally oscillatory solutions, then we can use Fourier analysis to get that

 $-i\omega m{n}_{0}{\stackrel{\to }{u}}_{1}=-{n}_{0}e{\stackrel{\to }{E}}_{1}$ (4.36)

 $\omega m{u}_{1}=-ie{E}_{1}$ (4.37)

We have also assumed plane waves. This is the basis of our dispersion relation.

Next we can invoke the continuity equation:

 $\frac{\partial n}{\partial t}+\nabla \cdot \left(n\stackrel{\to }{u}\right)=0.$ (4.38)

Using the same set of simpliﬁcations, we get:

 $-i\omega {n}_{1}+ik{n}_{0}{u}_{1}=0.$ (4.39)

We can use this to rewrite the dispersion relation to eliminate $u$:

 ${u}_{1}=\frac{\omega {n}_{1}}{k{n}_{0}}$ (4.40)

So that the dispersion relation becomes:

 $\frac{{\omega }^{2}m{n}_{1}}{k{n}_{0}}=-ie{E}_{1}.$ (4.41)

Next we need to eliminate both ${E}_{1}$ and ${n}_{1}$. We can do this by the use of Poisson’s equation. As we know we have to be careful in plasma physics as to when we can invoke Poisson’s equation, but in this case since we assumed high frequency waves and that the ion inertia is inﬁnite, there must be a resulting electric ﬁeld.

 $\nabla \cdot \stackrel{\to }{E}=\frac{e}{{𝜖}_{0}}\left({n}_{i}-{n}_{e}\right)$ (4.42)

Using ${n}_{e}={n}_{0}+{n}_{1}$ and ${n}_{i}={n}_{0}$ plus our usual set of assumptions, this simpliﬁes to

 $ik{E}_{1}=-e{n}_{1}∕{𝜖}_{0}$ (4.43)

 ${E}_{1}=-\frac{e{n}_{1}}{ik{𝜖}_{0}}$ (4.44)

Using this result the dispersion relation becomes

 $\frac{{\omega }^{2}m{n}_{1}}{k{n}_{0}}=\frac{i{e}^{2}{n}_{1}}{ik{𝜖}_{0}}$ (4.45)

which can be rewritten as

 ${\omega }^{2}=\frac{{n}_{0}{e}^{2}}{{m}_{e}{𝜖}_{0}}={\omega }_{pe}^{2}$ (4.46)

These disturbances do not propagate, $k=0$, but instead are stationary oscillatory solutions with a given frequency, known as the plasma frequency:

 ${\omega }_{pe}=\sqrt{\frac{{n}_{e}{e}^{2}}{{m}_{e}{𝜖}_{0}}}$ (4.47)

4.5 Electron Plasma Waves

The electron oscillations we derived in the previous result can become true waves (i.e. $k\ne 0$) when there is a ﬁnite electron temperature. They are called Electron Plasma Waves, or sometimes Bohm-Gross Waves or Langmuir Waves. Revisiting the electron momentum equation, with $e$ subscripts omitted:

 $mn\left[\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}\right]=-ne\left(\stackrel{\to }{E}+\stackrel{\to }{u}×\stackrel{\to }{B}\right)-\nabla p-mn\nu \left(\stackrel{\to }{u}-\stackrel{\to }{{u}_{i}}\right)$ (4.48)

Again we neglect collisions, and take plane waves with small perturbations (1st order terms only) but keep the pressure gradient, and take ${u}_{0}={E}_{0}={B}_{0}=0$:

 $mn\frac{\partial {u}_{1}}{\partial t}=-ne{E}_{1}-\nabla p$ (4.49)

Generally we must make an assumption about the plasma equation of state, for example adiabatic:

 $p\propto {\rho }^{\gamma }\to \frac{\nabla p}{p}=\gamma \frac{\nabla n}{n}$ (4.50)

if we then substitute the ideal gas law $p=nkT$ into the above, and in this case, if we take 1-D isothermal compression and expansion of the plasma then $\gamma =3$.

 $\nabla p=\gamma kT\nabla n=3kT\nabla {n}_{1}$ (4.51)

since there is no gradient in the equilibrium density ${n}_{0}$. Plugging this into the momentum equation, taking ${k}_{B}T\to T$ for simplicity of notation,

 $mn\frac{\partial {u}_{1}}{\partial t}=-ne{E}_{1}-3T\nabla {n}_{1},$ (4.52)

which we can use Fourier analysis to simplify further to:

 $-i\omega mn{u}_{1}=-ne{E}_{1}-3ikT{n}_{1}.$ (4.53)

We need two relations between ${u}_{1},{E}_{1},{n}_{1}$ to eliminate them. First we use the continuity equation:

 $\frac{\partial n}{\partial t}+\nabla \cdot \left(n\stackrel{\to }{u}\right)=0$ (4.54)

which, with ﬁrst-order terms Fourier analyzed:

 $-i\omega {n}_{1}+ik{n}_{0}{u}_{1}=0.$ (4.55)

Solving for ${u}_{1}$:

 ${u}_{1}=\frac{\omega {n}_{1}}{k{n}_{0}}$ (4.56)

Next, we can use the Poisson equation (valid because these are fast oscillations, and ion momentum can be approximated as inﬁnite), to get a relation between $n$ and $E$. To start:

 $\nabla \cdot \stackrel{\to }{E}=\frac{e}{{𝜖}_{0}}\left({n}_{i}-{n}_{e}\right)$ (4.57)

with ${n}_{e}={n}_{0}+{n}_{1}$ and ${n}_{i}={n}_{0}$, and using $\nabla \to ik$,

 $ik{E}_{1}=-e{n}_{1}∕{𝜖}_{0}$ (4.58)

or solving for ${E}_{1}$:

 ${E}_{1}=-\frac{e{n}_{1}}{ik{𝜖}_{0}}$ (4.59)

Using these expressions for ${u}_{1}$ (continuity eq) and ${E}_{1}$ (Poisson eq) the momentum equation becomes:

 $i\omega m{n}_{0}\frac{\omega {n}_{1}}{k{n}_{0}}=-ne\frac{e{n}_{1}}{ik{𝜖}_{0}}+3ikT{n}_{1}$ (4.60)

 ${\omega }^{2}=\frac{{n}_{0}{e}^{2}}{m{𝜖}_{0}}+\frac{3{n}_{0}T}{m}{k}^{2}$ (4.61)

using the deﬁnition of the thermal velocity ${v}_{th}=\sqrt{2T∕m}$, we get the electron plasma wave dispersion relation:

 ${\omega }^{2}={\omega }_{pe}^{2}+\frac{3}{2}{v}_{th}^{2}{k}^{2}.$ (4.62)

We can see immediately that as $T\to 0$, then ${v}_{th}\to 0$ and this dispersion relation reduces the electron oscillation found previously. Furthermore, we note that propagation requires ${\omega }^{2}>{\omega }_{pe}^{2}$ for the electron plasma wave.

4.6 Ion Acoustic Waves

Next we consider the ion acoustic wave, which as we will see is the analog of traditional hydrodynamic sound waves. Since the ions are involved, it is by necessity a slow wave. We therefore take $n={n}_{e}={n}_{i}$, since the electron response will be very fast compared to the wave. This requires that we do not use Poisson’s equation in the derivation!

Starting with the ion momentum equation:

 ${m}_{i}n\left[\frac{\partial {\stackrel{\to }{u}}_{i}}{\partial t}+\left({\stackrel{\to }{u}}_{i}\cdot \nabla \right){\stackrel{\to }{u}}_{i}\right]=ne\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{i}×\stackrel{\to }{B}\right)-\nabla {p}_{i}$ (4.63)

I note that we have omitted the collision term. Taking ${u}_{0}={E}_{0}={B}_{0}=0$, keeping only ﬁrst-order terms, and using $\stackrel{\to }{E}=-\nabla \varphi$, we simplify this to:

 ${m}_{i}n\frac{\partial {\stackrel{\to }{u}}_{1}}{\partial t}=-ne\nabla {\varphi }_{1}-\gamma {T}_{i}\nabla {n}_{1}.$ (4.64)

Doing our traditional simpliﬁcation with Fourier analysis $\partial ∕\partial t\to -i\omega$ and $\nabla \to ik$, and taking plane waves, we get that:

 $-i\omega {m}_{i}{n}_{0}{u}_{1}=-ik{n}_{0}e{\varphi }_{1}-ik\gamma {T}_{i}{n}_{1}.$ (4.65)

To simplify this further requires that we obtain relations between ${u}_{1}$, ${\varphi }_{1}$, and ${n}_{1}$ so that they can be eliminated.

First, we use the continuity equation as always:

 $\frac{\partial n}{\partial t}+\nabla \cdot \left(n\stackrel{\to }{u}\right)=0,$ (4.66)

 $i\omega {n}_{1}+ik{n}_{0}{u}_{1}=0,$ (4.67)

 ${u}_{1}=\frac{\omega {n}_{1}}{k{n}_{0}}$ (4.68)

Next we note that in the presence of an electric potential, the electrons can be written as obeying the Boltzmann distribution:

 ${n}_{e}={n}_{0}{e}^{e\varphi ∕kT}={n}_{0}\left(1+\frac{e\varphi }{T}+...\right)$ (4.69)

since the electron response is much faster than changes in $\varphi$. In the last step we used the assumption $e\varphi \ll T$ to Taylor expand the exponential. Recognizing the contribution from the perturbation in density, ${n}_{1}$, we write:

 ${n}_{1}={n}_{0}\frac{e{\varphi }_{1}}{{T}_{e}}.$ (4.70)

Or equivalently, we can eliminate $\varphi$ using:

 ${\varphi }_{1}=\frac{{n}_{1}{T}_{e}}{e{n}_{0}}$ (4.71)

Now we can use these two results to solve the momentum equation for the dispersion relation.

 $-i\omega {m}_{i}{n}_{0}\frac{\omega {n}_{1}}{k{n}_{0}}=-ik{n}_{0}e\frac{{n}_{1}{T}_{e}}{e{n}_{0}}-ik\gamma {T}_{i}{n}_{1}.$ (4.72)

We can eliminate ${n}_{1}$ now and further simplify algebraically to get that:

 ${\omega }^{2}={k}^{2}\left(\frac{{T}_{e}+\gamma {T}_{i}}{{m}_{i}}\right)={c}_{s}^{2}{k}^{2}$ (4.73)

with the equivalent of the sound speed in a plasma being:

 ${c}_{s}=\sqrt{\frac{{T}_{e}+\gamma {T}_{i}}{{m}_{i}}}.$ (4.74)

We note that the electrons essentially have ${\gamma }_{e}=1$ while the ion contribution is the typical one and actually will reduce to the hydrodynamic result in that limit.

4.7 Electromagnetic Waves in Plasmas

Next we turn to the critical question of electromagnetic waves in plasmas. In vacuum, of course, this would be a typical light wave propagating at the speed of light, but in a plasma the $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ ﬁelds can induce plasma ‘sources’ of charge and current that feed back to the wave. The plasma response can be written in terms of the induced charge and current densities:

$\begin{array}{rcll}\rho & =& \frac{e}{{𝜖}_{0}}\left({n}_{i}-{n}_{e}\right)& \text{(4.75)}\text{}\text{}\\ \stackrel{\to }{j}& =& e\left({n}_{i}{\stackrel{\to }{u}}_{i}-{n}_{e}\stackrel{\to }{{u}_{e}}\right)& \text{(4.76)}\text{}\text{}\end{array}$

and thus Maxwell’s equations are:

$\begin{array}{rcll}\nabla \cdot \stackrel{\to }{E}& =& \frac{e}{{𝜖}_{0}}\left({n}_{i}-{n}_{e}\right)& \text{(4.77)}\text{}\text{}\\ \nabla \cdot \stackrel{\to }{B}& =& 0& \text{(4.78)}\text{}\text{}\\ \nabla ×\stackrel{\to }{E}& =& -\frac{\partial \stackrel{\to }{B}}{\partial t}& \text{(4.79)}\text{}\text{}\\ \nabla ×\stackrel{\to }{B}& =& {\mu }_{0}e\left({n}_{i}{\stackrel{\to }{u}}_{i}-{n}_{e}\stackrel{\to }{{u}_{e}}\right)+\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}& \text{(4.80)}\text{}\text{}\end{array}$

We start by taking the curl of Faraday’s Law:

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)=-\frac{\partial }{\partial t}\left(\nabla ×\stackrel{\to }{B}\right),$ (4.81)

and substitute in Ampère’s Law to get the dispersion relation:

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)=-\frac{\partial }{\partial t}\left[{\mu }_{0}e\left({n}_{i}{\stackrel{\to }{u}}_{i}-{n}_{e}\stackrel{\to }{{u}_{e}}\right)+\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}\right].$ (4.82)

Of course we know our vector identity:

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)=\nabla \left(\nabla \cdot \stackrel{\to }{E}\right)-{\nabla }^{2}\stackrel{\to }{E}.$ (4.83)

Unlike the vacuum scenario, $\nabla \cdot \stackrel{\to }{E}\ne 0$. We use this result with our proto-dispersion-relation equation, collecting ‘vacuum’ terms on the left and ‘plasma’ terms on the right to get:

 $\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{E}}{\partial {t}^{2}}-{\nabla }^{2}\stackrel{\to }{E}=-\nabla \left(\nabla \cdot \stackrel{\to }{E}\right)-{\mu }_{0}e\frac{\partial }{\partial t}\left({n}_{i}{\stackrel{\to }{u}}_{i}-{n}_{e}{\stackrel{\to }{u}}_{e}\right).$ (4.84)

At this point it is useful to note that if we set ${n}_{e}={n}_{i}=0$ we recover the vacuum result ${\omega }^{2}={c}^{2}{k}^{2}$. If we assume transverse waves, then $\nabla \cdot \stackrel{\to }{E}=i\stackrel{\to }{k}\cdot \stackrel{\to }{E}=0$, and

 $\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{E}}{\partial {t}^{2}}-{\nabla }^{2}\stackrel{\to }{E}=-{\mu }_{0}e\frac{\partial }{\partial t}\left({n}_{i}{\stackrel{\to }{u}}_{i}-{n}_{e}{\stackrel{\to }{u}}_{e}\right).$ (4.85)

We know that light waves are very fast, so we can treat the ion inertia as inﬁnite and consider the current as coming from the electron response only. In this case we have to solve the electron equation of motion, i.e. the momentum equation:

 ${m}_{e}{n}_{e}\left[\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}+\left({\stackrel{\to }{u}}_{e}\cdot \nabla \right){\stackrel{\to }{u}}_{e}\right]=-{n}_{e}e\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{e}×\stackrel{\to }{B}\right)-\nabla {p}_{e}$ (4.86)

At this point we assume ${T}_{e}=0\to {p}_{e}=0$, and take the case ${\stackrel{\to }{u}}_{0}={\stackrel{\to }{E}}_{0}={\stackrel{\to }{B}}_{0}=0$. Furthermore, assuming that the wave-induced perturbation is small and keeping only ﬁrst-order terms, we get that:

 ${m}_{e}{n}_{0}\frac{\partial {\stackrel{\to }{u}}_{1}}{\partial t}=-{n}_{0}e{\stackrel{\to }{E}}_{1}.$ (4.87)

Using Fourier analysis:

 $-i\omega {m}_{e}{n}_{0}{\stackrel{\to }{u}}_{1}=-{n}_{0}e{\stackrel{\to }{E}}_{1},$ (4.88)

 ${\stackrel{\to }{u}}_{1}=\frac{e}{i\omega {m}_{e}}{\stackrel{\to }{E}}_{1}.$ (4.89)

We can substitute this result into the dispersion relation to get:

 $\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{E}}{\partial {t}^{2}}-{\nabla }^{2}\stackrel{\to }{E}={\mu }_{0}e{n}_{0}\frac{e}{i\omega {m}_{e}}\frac{\partial \stackrel{\to }{E}}{\partial t}.$ (4.90)

Now we have to use Fourier analysis on the whole thing, $\partial ∕\partial t\to -i\omega$ and $\nabla \to ik$ to get:

 $-\frac{{\omega }^{2}}{{c}^{2}}\stackrel{\to }{E}+{k}^{2}\stackrel{\to }{E}=-\frac{{\mu }_{0}{n}_{0}{e}^{2}}{{m}_{e}}\stackrel{\to }{E}$ (4.91)

We can now cancel the electric ﬁeld, use ${𝜖}_{0}{\mu }_{0}={c}^{2}$ and ${\omega }_{pe}=\sqrt{n{e}^{2}∕{m}_{e}{𝜖}_{0}}$, and rearrange to get the dispersion relation for EM waves in plasmas:

 ${\omega }^{2}={c}^{2}{k}^{2}+{\omega }_{pe}^{2}.$ (4.92)

The most important thing to note from this dispersion relation is that if ${\omega }^{2}<{\omega }_{pe}^{2}$, then $k$ is imaginary and the wave is evanescent. We call this a cutoﬀ for the wave. It is also useful to calculate the group velocity:

 ${v}_{g}\equiv \frac{\partial \omega }{\partial k}$ (4.93)

If we directly take the derivative $\partial ∕\partial k$ of the dispersion relation, we get that:

 $2\omega {v}_{g}=2{c}^{2}k,$ (4.94)

or

 ${v}_{g}={c}^{2}k∕\omega \left(=nc\right)$ (4.95)

If we do some algebraic manipulation of the original dispersion relation,

 ${c}^{2}{k}^{2}={\omega }^{2}\left(1-{\omega }_{pe}^{2}∕{\omega }^{2}\right)$ (4.96)

or

 $\frac{k}{\omega }=\frac{1}{c}\sqrt{1-\frac{{\omega }_{pe}^{2}}{{\omega }^{2}}}$ (4.97)

using this we get a group velocity of:

 ${v}_{g}=c\sqrt{1-\frac{{\omega }_{pe}^{2}}{{\omega }^{2}}}$ (4.98)

For propagating waves, we must have $\omega >{\omega }_{pe}$ which nicely gives us the result that ${v}_{g}. We also note that, since ${\omega }_{pe}\propto \sqrt{{n}_{e}}$, the wave propagation depends on the plasma density in this case.

4.8 Electromagnetic Ion Waves

In this section we consider two electromagnetic ion waves: the Alfvén wave and the magnetosonic wave.

4.8.1 Alfvén Wave

This is a fundamental plasma wave. We wish to consider low-frequency ion oscillations in the presence of a magnetic ﬁeld. We must consider a magnetized plasma with non-zero ${\stackrel{\to }{B}}_{0}$, which we take as ${\stackrel{\to }{B}}_{0}={B}_{0}ẑ$. The geometry of this wave is $\stackrel{\to }{k}\parallel {\stackrel{\to }{B}}_{0}$, ${\stackrel{\to }{E}}_{1},{\stackrel{\to }{j}}_{1}\perp {\stackrel{\to }{B}}_{0}$, and $\stackrel{\to }{v},{\stackrel{\to }{B}}_{1}$ perpendicular to both ${\stackrel{\to }{B}}_{0}$ and ${\stackrel{\to }{j}}_{1}$. By convention we take ${\stackrel{\to }{E}}_{1}\parallel \stackrel{̂}{x}$ without loss of generality.

Starting with Maxwell:

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)=-\stackrel{\to }{k}\left(\stackrel{\to }{k}\cdot {\stackrel{\to }{E}}_{1}\right)+{k}^{2}{\stackrel{\to }{E}}_{1}=\frac{{\omega }^{2}}{{c}^{2}}{\stackrel{\to }{E}}_{1}+\frac{i\omega }{{𝜖}_{0}{c}^{2}}{\stackrel{\to }{j}}_{1}$ (4.99)

Since $\stackrel{\to }{k}\cdot {\stackrel{\to }{E}}_{1}=0$ by geometry, the only non-trivial equation is for the $x$ direction:

 ${𝜖}_{0}\left({\omega }^{2}-{c}^{2}{k}^{2}\right){E}_{1}=-i\omega {n}_{0}e\left({u}_{ix}-{u}_{ex}\right)$ (4.100)

We have to calculate the ion response from the momentum equation.

 ${m}_{i}{n}_{i}\left[\frac{\partial {\stackrel{\to }{u}}_{i}}{\partial t}+\left({\stackrel{\to }{u}}_{i}\cdot \nabla \right){\stackrel{\to }{u}}_{i}\right]={n}_{i}e\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{i}×\stackrel{\to }{B}\right)-\nabla p$ (4.101)

For simplicity we take the zero temperature limit, and use $\stackrel{\to }{E}=-\nabla \varphi$. We also neglect second-order terms. This gives us

 ${m}_{i}\frac{\partial {\stackrel{\to }{u}}_{i1}}{\partial t}=-e\nabla {\varphi }_{1}+e{\stackrel{\to }{u}}_{i}×{\stackrel{\to }{B}}_{0}.$ (4.102)

Splitting into $x$ and $y$ components, and using Fourier analysis, we get

$\begin{array}{rcll}-i\omega {m}_{i}{u}_{ix}& =& -ike{\varphi }_{1}+e{u}_{iy}{B}_{0}& \text{}\\ -i\omega {m}_{i}{u}_{iy}& =& \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-e{u}_{ix}{B}_{0}& \text{(4.103)}\text{}\text{}\end{array}$

Starting with the $x$ equation, we can write that

 $\left(\omega {m}_{i}-ie{B}_{0}\frac{e{B}_{0}}{i\omega {m}_{i}}\right){u}_{ix}=ke{\varphi }_{1},$ (4.104)

 ${u}_{ix}=\frac{ke}{\omega {m}_{i}}{\varphi }_{1}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}.$ (4.105)

We can immediately use this result to get the form for $y$:

 ${u}_{iy}=-i\frac{e{B}_{0}}{\omega {m}_{i}}{u}_{ix}=-i\frac{{\omega }_{ci}}{\omega }{u}_{ix},$ (4.106)

 ${u}_{iy}=-i\frac{{\omega }_{ci}}{\omega }\frac{ke}{\omega {m}_{i}}{\varphi }_{1}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}.$ (4.107)

In summary, then, the ion motion in the $xy$ plane is given by

$\begin{array}{rcll}{u}_{ix}& =& \frac{ke}{\omega {m}_{i}}{\varphi }_{1}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}& \text{(4.108)}\text{}\text{}\\ {u}_{iy}& =& -i\frac{{\omega }_{ci}}{\omega }\frac{ke}{\omega {m}_{i}}{\varphi }_{1}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}& \text{(4.109)}\text{}\text{}\end{array}$

We can immediately obtain the electron equations of motion by substituting ${m}_{i}\to {m}_{e}$, ${\omega }_{ci}\to {\omega }_{ce}$, and taking the limit ${\omega }_{ce}∕\omega \gg 1$.

$\begin{array}{rcll}{u}_{ex}& =& -\frac{ke}{\omega {m}_{i}}{\varphi }_{1}\left(\frac{{\omega }^{2}}{{\omega }_{ce}^{2}}\right)\approx 0& \text{(4.110)}\text{}\text{}\\ {u}_{ey}& =& -i\frac{{\omega }_{ce}}{\omega }\frac{ke}{\omega {m}_{i}}{\varphi }_{1}\left(\frac{{\omega }^{2}}{{\omega }_{ce}^{2}}\right)\approx -\frac{{E}_{1}}{{B}_{0}}& \text{(4.111)}\text{}\text{}\end{array}$

There is no electron motion in the $x$ direction because the cyclotron motion overpowers it; however, the electrons have the usual $E×B$ drift which is $\parallel ŷ$ in this geometry. Since the current of interest is in the $\stackrel{̂}{x}$ direction, it must be from the ion motion only.

Going back to the dispersion relation,

 ${𝜖}_{0}\left({\omega }^{2}-{c}^{2}{k}^{2}\right){E}_{1}=-i\omega {n}_{0}e{u}_{ix},$ (4.112)

 ${𝜖}_{0}\left({\omega }^{2}-{c}^{2}{k}^{2}\right){E}_{1}=-i\omega {n}_{0}e\frac{ke}{\omega {m}_{i}}{\varphi }_{1}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}.$ (4.113)

Since ${E}_{1}=ik{\varphi }_{1}$, we can eliminate them from the dispersion relation and simplify to:

 $\left({\omega }^{2}-{c}^{2}{k}^{2}\right){E}_{1}=\frac{{n}_{0}{e}^{2}}{{𝜖}_{0}{m}_{i}}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}.$ (4.114)

We recognize the plasma frequency, and simplify further to:

 $\left({\omega }^{2}-{c}^{2}{k}^{2}\right)={\omega }_{pi}^{2}{\left(1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}\right)}^{-1}.$ (4.115)

In the limit $\omega \ll {\omega }_{ci}$, then:

 ${\omega }^{2}-{c}^{2}{k}^{2}=\frac{{\omega }_{pi}^{2}{\omega }^{2}}{{\omega }_{ci}^{2}}={\omega }^{2}\frac{\rho }{{𝜖}_{0}{B}_{0}^{2}}$ (4.116)

Rearranging, after some algebra one obtains the dispersion relation for Alfvén waves:

 $\frac{{\omega }^{2}}{{k}^{2}}=\frac{{c}^{2}}{1+\left(\rho {\mu }_{0}∕{B}_{0}^{2}\right){c}^{2}}$ (4.117)

We recognize the Alfvén velocity:

 ${v}_{A}=\sqrt{\frac{{B}_{0}^{2}}{\rho {\mu }_{0}}}.$ (4.118)

In the limit ${v}_{A}\ll c$ then:

 $\omega ={v}_{A}k$ (4.119)

Basically, this wave is one where the lines of force (magnetic ﬁeld lines) and the plasma move together in the plane perpendicular to the initial magnetic ﬁeld.

4.8.2 Magnetosonic Wave

In the last section we considered waves along the initial magnetic ﬁeld; we now consider low-frequency electromagnetic waves that propagate across ${\stackrel{\to }{B}}_{0}$. Again we take ${\stackrel{\to }{B}}_{0}={B}_{0}ẑ$, ${\stackrel{\to }{E}}_{1}={E}_{1}\stackrel{̂}{x}$, but now $\stackrel{\to }{k}=kŷ$ to satisfy the above. At the beginning of this derivation, we note that ${\stackrel{\to }{E}}_{1}×{\stackrel{\to }{B}}_{0}\parallel \stackrel{\to }{k}$, implying that the $E×B$ drifts will compress and expand the plasma as the wave propagates. This means that we have no choice but to keep the pressure gradient term in the ion momentum equation:

 ${m}_{i}{n}_{i}\left[\frac{\partial {\stackrel{\to }{u}}_{i}}{\partial t}+\left({\stackrel{\to }{u}}_{i}\cdot \nabla \right){\stackrel{\to }{u}}_{i}\right]={n}_{i}e\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{i}×\stackrel{\to }{B}\right)-\nabla p$ (4.120)

However we let ${\stackrel{\to }{E}}_{0}={\stackrel{\to }{u}}_{0}=0$, keep only ﬁrst-order terms, and use $\nabla p=\gamma {T}_{i}\nabla {n}_{i}$,

 ${m}_{i}{n}_{i}\frac{\partial {\stackrel{\to }{u}}_{i1}}{\partial t}={n}_{i}e{\stackrel{\to }{E}}_{1}+{n}_{i}e{\stackrel{\to }{u}}_{i1}×{\stackrel{\to }{B}}_{0}-\gamma {T}_{i}\nabla {n}_{i}$ (4.121)

employing Fourier analysis gets us to:

 $-i\omega {m}_{i}{n}_{0}{\stackrel{\to }{u}}_{i1}={n}_{0}e{\stackrel{\to }{E}}_{1}+{n}_{0}e{\stackrel{\to }{u}}_{i1}×{\stackrel{\to }{B}}_{0}-ik\gamma {T}_{i}{n}_{1}$ (4.122)

 ${\stackrel{\to }{u}}_{i1}=\frac{ie}{\omega {m}_{i}}\left({\stackrel{\to }{E}}_{1}+{\stackrel{\to }{u}}_{i1}×{\stackrel{\to }{B}}_{0}\right)+\frac{k\gamma {T}_{i}}{{m}_{i}\omega }\frac{{n}_{1}}{{n}_{0}}ŷ$ (4.123)

Next we need to split this into $x$ and $y$ components, which will make it tractable:

$\begin{array}{rcll}{u}_{ix}& =& \frac{ie}{\omega {m}_{i}}\left({E}_{1}+{u}_{iy}{B}_{0}\right)& \text{(4.124)}\text{}\text{}\\ {u}_{iy}& =& -\frac{ie}{\omega {m}_{i}}{u}_{ix}{B}_{0}+\frac{k\gamma {T}_{i}}{{m}_{i}\omega }\frac{{n}_{1}}{{n}_{0}}& \text{(4.125)}\text{}\text{}\end{array}$

We now need to use the continuity equation:

 $\frac{\partial n}{\partial t}+\nabla \cdot \left(n\stackrel{\to }{u}\right)=0$ (4.126)

 $-i\omega {n}_{1}+ik{n}_{0}{u}_{iy}=0$ (4.127)

 $\frac{{n}_{1}}{{n}_{0}}=\frac{k}{\omega }{u}_{iy}$ (4.128)

We use this with the $y$ equation of motion to get

 ${u}_{iy}=-\frac{ie{B}_{0}}{\omega {m}_{i}}{\left(1-\frac{{k}^{2}\gamma {T}_{i}}{{\omega }^{2}{m}_{i}}\right)}^{-1}{u}_{ix}$ (4.129)

We can now plug this into the equation of motion for $x$:

 ${u}_{ix}=\frac{ie}{\omega {m}_{i}}\left[{E}_{1}-\frac{ie{B}_{0}^{2}}{\omega {m}_{i}}{\left(1-\frac{{k}^{2}\gamma {T}_{i}}{{\omega }^{2}{m}_{i}}\right)}^{-1}{u}_{ix}\right]$ (4.130)

 $\left[1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}{\left(1-\frac{{k}^{2}\gamma {T}_{i}}{{\omega }^{2}{m}_{i}}\right)}^{-1}\right]{u}_{ix}=\frac{ie}{\omega {m}_{i}}{E}_{1}$ (4.131)

 ${u}_{ix}=\frac{ie}{\omega {m}_{i}}{E}_{1}{\left[1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}{\left(1-\frac{{k}^{2}{\gamma }_{i}{T}_{i}}{{\omega }^{2}{m}_{i}}\right)}^{-1}\right]}^{-1}$ (4.132)

For the ﬁnal current, we also need to know the electron velocity, which can be obtained by ${m}_{i}\to {m}_{e}$ and ${\omega }_{ci}\to {\omega }_{ce}$. We also take the limit $\omega \ll {\omega }_{ce}$ in which case the term in square brackets can be simpliﬁed by the binomial approximation, and we get:

 ${u}_{ex}=\frac{ie}{\omega {m}_{e}}{E}_{1}\frac{{\omega }^{2}}{{\omega }_{ce}^{2}}\left[1-\frac{{k}^{2}{\gamma }_{e}{T}_{e}}{{\omega }^{2}{m}_{e}}\right]\approx \frac{ie}{\omega {m}_{e}}\frac{{\omega }^{2}}{{\omega }_{ce}^{2}}\frac{{k}^{2}{\gamma }_{e}{T}_{e}}{{\omega }^{2}{m}_{e}}{E}_{1}$ (4.133)

Similarly to the last section, we know that the dispersion relation here will be:

 ${𝜖}_{0}\left({\omega }^{2}-{c}^{2}{k}^{2}\right){E}_{x}=-i\omega {n}_{0}e\left({u}_{ix}-{u}_{ex}\right)$ (4.134)

using the ﬂuid velocities we just derived,

 ${𝜖}_{0}\left({\omega }^{2}-{c}^{2}{k}^{2}\right)=\frac{{n}_{0}{e}^{2}}{{m}_{e}}\left(\frac{{m}_{i}}{{m}_{e}}{\left[1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}{\left(1-\frac{{k}^{2}{\gamma }_{i}{T}_{i}}{{\omega }^{2}{m}_{i}}\right)}^{-1}\right]}^{-1}-\frac{{k}^{2}}{{\omega }_{ce}^{2}}\frac{{\gamma }_{e}{T}_{e}}{{m}_{e}}\right)$ (4.135)

the plasma frequency,

 ${\omega }^{2}-{c}^{2}{k}^{2}={\omega }_{pe}^{2}\left(\frac{{m}_{i}}{{m}_{e}}{\left[1-\frac{{\omega }_{ci}^{2}}{{\omega }^{2}}{\left(1-\frac{{k}^{2}{\gamma }_{i}{T}_{i}}{{\omega }^{2}{m}_{i}}\right)}^{-1}\right]}^{-1}-\frac{{k}^{2}}{{\omega }_{ce}^{2}}\frac{{\gamma }_{e}{T}_{e}}{{m}_{e}}\right)$ (4.136)

taking the limit $\omega \ll {\omega }_{ci}$,

 ${\omega }^{2}-{c}^{2}{k}^{2}={\omega }_{pe}^{2}\left[\frac{{m}_{i}}{{m}_{e}}\frac{{\omega }^{2}}{{\omega }_{ci}^{2}}\left(1-\frac{{k}^{2}{\gamma }_{i}{T}_{i}}{{\omega }^{2}{m}_{i}}\right)-\frac{{k}^{2}}{{\omega }_{ce}^{2}}\frac{{\gamma }_{e}{T}_{e}}{{m}_{e}}\right]$ (4.137)

After a bit of algebra, we can rearrange this to be

 ${\omega }^{2}-{c}^{2}{k}^{2}\left(1+\frac{{\gamma }_{e}{T}_{e}}{{m}_{i}{v}_{A}^{2}}\right)+\frac{{\omega }_{pi}^{2}}{{\omega }_{ci}^{2}}\left({\omega }^{2}-{k}^{2}\frac{{\gamma }_{i}{T}_{i}}{{m}_{i}}\right)=0$ (4.138)

where we have used the deﬁnition of the Alfvén speed, ${v}_{A}\equiv {B}_{0}∕\sqrt{{\mu }_{0}\rho }=c{\omega }_{ci}∕{\omega }_{pi}$. From the derivation of the ion acoustic wave, recall the sound speed is:

 ${c}_{s}=\sqrt{\frac{{\gamma }_{e}{T}_{e}+{\gamma }_{i}{T}_{i}}{{m}_{i}}}$ (4.139)

so the dispersion relation becomes, after some algebraic hammering,

 ${\omega }^{2}\left(1+\frac{{c}^{2}}{{v}_{A}^{2}}\right)={c}^{2}{k}^{2}\left(1+\frac{{c}_{s}^{2}}{{v}_{A}^{2}}\right)$ (4.140)

or after rearranging, the magnetosonic wave dispersion relation:

 $\frac{{\omega }^{2}}{{k}^{2}}={c}^{2}\frac{{v}_{A}^{2}+{c}_{s}^{2}}{{v}_{A}^{2}+{c}^{2}}.$ (4.141)

This wave is essentially an acoustic (sound) wave but the compression/expansion is created by $E×B$ drifts. In the limit ${B}_{0}\to 0$, this wave becomes the ion acoustic wave. In the limit $T\to 0$ the pressure gradient forces drop out, ${c}_{s}\to 0$, and this wave becomes a modiﬁed Alfvén wave.

Because the phase velocity of this wave is almost always higher than ${v}_{A}$, it is sometimes called the ‘fast’ hydromagnetic wave.

4.9 Brief Summary of More Waves

We give a brief summary of the rest of the menagerie of plasma waves here, with dispersion relations given from general plasma physics references (e.g. Chen).

4.9.1 Upper Hybrid Waves

Next we consider electron plasma waves. It turns out that the electron plasma wave along ${\stackrel{\to }{B}}_{0}$ is unaﬀected by the magnetic ﬁeld. The perpendicular case, however, is the upper hybrid wave. The easiest situation to consider is the zero temperature limit. If we consider longitudinal waves, ${\stackrel{\to }{E}}_{1}\parallel \stackrel{\to }{k}$ and $\stackrel{\to }{k}\perp {\stackrel{\to }{B}}_{0}$ we can immediately see that there will be an $E×B$ drift for the electron motion in the wave. This aﬀects the electron equation of motion. At the end of the derivation, we would arrive at the upper hybrid frequency:

 ${\omega }_{uh}^{2}={\omega }_{pe}^{2}+{\omega }_{ce}^{2}$ (4.142)

Once again, we immediately see that the ${B}_{0}\to 0$ limit corresponds to the usual electron plasma oscillation.

4.9.2 Ion Cyclotron Waves

In the derivation of the ion acoustic wave we assumed unmagnetized plasma. Instead we consider the ion acoustic wave in a magnetized plasma. It is tempting to set $\stackrel{\to }{k}\cdot {\stackrel{\to }{B}}_{0}=0$, but the problem for this situation is that the electrons are unable to move between wave fronts because of their small gyroradii. Instead, if $\stackrel{\to }{k}$ and ${\stackrel{\to }{B}}_{0}$ are almost perpendicular but not quite, then this wave can propagate. It turns out that the critical angle is $\chi \approx \sqrt{{m}_{e}∕{m}_{i}}$ which is indeed small.

When one goes through the derivation, analogously to the ion acoustic wave, one arrives at the electrostatic ion cyclotron wave dispersion relation:

 ${\omega }^{2}={\omega }_{ci}^{2}+{k}^{2}{c}_{s}^{2}$ (4.143)

we can see that this reduces to the ion acoustic wave in the limit ${B}_{0}\to 0$ (equivalently ${\omega }_{ci}\to 0$).

4.9.3 Lower Hybrid Waves

So what happens to the ion acoustic wave when the propagation angle is exactly $\pi ∕2$, i.e. the wave is exactly perpendicular to the magnetic ﬁeld? In this case, keeping ﬁnite electron mass, it turns out that the compression/expansion of the normal ion acoustic wave is unimportant because the electron motion is constrained. Starting with the electron and ion equations of motion, it turns out that there is an oscillation at the lower hybrid frequency:

 ${\omega }_{lh}=\sqrt{{\omega }_{ci}{\omega }_{ce}}$ (4.144)

4.9.4 O/X Waves

We now move on to considering the electromagnetic wave in magnetized plasmas, ﬁrst for the waves with $\stackrel{\to }{k}\perp {\stackrel{\to }{B}}_{0}$. First, if ${\stackrel{\to }{E}}_{1}\parallel {\stackrel{\to }{B}}_{0}$ then there is no change to the plasma response, and we have the Ordinary (‘O’) Wave:

 ${\omega }^{2}={\omega }_{pe}^{2}+{c}^{2}{k}^{2}$ (4.145)

which is the same as the unmagnetized result.

On the other hand, if ${\stackrel{\to }{E}}_{1}\perp {\stackrel{\to }{B}}_{0}$ then there is a drift which alters the plasma response. Generally the wave’s electric ﬁeld is taken as elliptically polarized in the plane perpendicular to ${\stackrel{\to }{B}}_{0}$. Working through the math is somewhat involved, but at the end of the day the dispersion relation is

 $\frac{{c}^{2}{k}^{2}}{{\omega }^{2}}=1-\frac{{\omega }_{pe}^{2}}{{\omega }^{2}}\frac{{\omega }^{2}-{\omega }_{pe}^{2}}{{\omega }^{2}-{\omega }_{uh}^{2}}$ (4.146)

which includes the previously-encountered upper-hybrid frequency.

4.9.5 R/L Waves

The last set of waves are the electromagnetic waves with $\stackrel{\to }{k}\parallel {\stackrel{\to }{B}}_{0}$. In general the wave’s electric ﬁeld can lie anywhere in the plane perpendicular to the initial magnetic ﬁeld. This results in two solutions, which are elliptically polarized waves with right-hand or left-hand polarization, and which are respectively the R wave

 ${n}^{2}=\frac{{c}^{2}{k}^{2}}{\omega }=1-\frac{{\omega }_{pe}^{2}∕{\omega }^{2}}{1-{\omega }_{ce}∕\omega }$ (4.147)

and the L wave

 ${n}^{2}=\frac{{c}^{2}{k}^{2}}{\omega }=1-\frac{{\omega }_{pe}^{2}∕{\omega }^{2}}{1+{\omega }_{ce}∕\omega }$ (4.148)

Chapter 5Transport

5.1 Diﬀusion in weakly-ionized gases

If we consider a weakly-ionized gas, then the diﬀusion of plasma species is dominated by collisions with the neutral atoms. This is the simplest diﬀusion case, so we start with it. We begin with the momentum equation (for an arbitrary plasma species):

 $mn\left[\frac{\partial \stackrel{\to }{u}}{\partial t}+\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{u}\right]=nq\left(\stackrel{\to }{E}+\stackrel{\to }{u}×\stackrel{\to }{B}\right)-\nabla p-mn\nu \left(\stackrel{\to }{u}-{\stackrel{\to }{u}}_{n}\right)$ (5.1)

if we consider a steady-state equilibrium system, then the term $\partial \stackrel{\to }{u}∕\partial t$ is zero. We take ${\stackrel{\to }{u}}_{0}={\stackrel{\to }{B}}_{0}=0$. If $\stackrel{\to }{u}$ is small or $\nu$ is large then we can likewise neglect the convective derivative term as a small term, in which case,

 $\stackrel{\to }{v}=\stackrel{\to }{u}-{\stackrel{\to }{u}}_{n}=\frac{q}{m\nu }\stackrel{\to }{E}-\frac{T}{m\nu }\frac{\nabla n}{n}$ (5.2)

where we have also used $p=nT$. We can simplify this expression by deﬁning the mobility:

 $\mu \equiv \frac{|q|}{m\nu }$ (5.3)

and diﬀusion coeﬃcients:

 $D\equiv \frac{T}{m\nu }$ (5.4)

so that the total ﬂux of a plasma species $j$ is a generalized Fick’s law:

 ${\stackrel{\to }{\Gamma }}_{j}=±{\mu }_{j}{n}_{j}\stackrel{\to }{E}-{D}_{j}\nabla {n}_{j}$ (5.5)

of course if $q=0$ or $\stackrel{\to }{E}=0$ then this expression reduces to the typical Fick’s law from classical physics.

5.2 Ambipolar Diﬀusion

To start oﬀ with, we rewrite the continuity equation in terms of the particle ﬂux:

 $\frac{\partial n}{\partial t}+\nabla \cdot \Gamma =0$ (5.6)

It is clear that the particle ﬂux can cause time-varying concentrations of plasma species. If we consider a hydrogen ion - electron plasma, with just two species, the immediate question is if the diﬀusion process violates quasi-neutrality. We note that both the mobility and diﬀusion coeﬃcients depend inversely on the mass, and so naïvely one might ask what stops the electrons from all diﬀusing away and leaving an ion-only plasma. The answer is that an imposed positive potential on the high-density regions of plasma creates an electric ﬁeld which enhances ion diﬀusion and suppresses electron diﬀusion. The plasma will quickly charge to essentially force ${\Gamma }_{e}={\Gamma }_{i}$. This process is known as ambipolar diﬀusion.

We can derive the spontaneously-generated ambipolar electric ﬁeld by setting the diﬀusion rates for electrons and ions equal to each other:

 ${\mu }_{i}n{\stackrel{\to }{E}}_{a}-{D}_{i}\nabla n=-{\mu }_{e}n{\stackrel{\to }{E}}_{a}-{D}_{e}\nabla n$ (5.7)

where we have taken ${n}_{e}={n}_{i}=n$. We can solve this equation for the ambipolar electric ﬁeld:

 ${\stackrel{\to }{E}}_{a}=\frac{{D}_{i}-{D}_{e}}{{\mu }_{i}+{\mu }_{e}}\frac{\nabla n}{n}$ (5.8)

with deﬁnitions for the (electron and ion) mobility and diﬀusion coeﬃcients given in the previous section.

We can also derive the common resulting diﬀusion as:

 $\Gamma ={\mu }_{i}n{\stackrel{\to }{E}}_{a}-{D}_{i}\nabla n$ (5.9)

 $\Gamma =\left(\frac{{D}_{i}-{D}_{e}}{{\mu }_{i}+{\mu }_{e}}-{\mu }_{i}{D}_{i}\right)\nabla n$ (5.10)

 $\Gamma =-\left(\frac{{\mu }_{i}{D}_{e}+{\mu }_{e}{D}_{i}}{{\mu }_{i}+{\mu }_{e}}\right)\nabla n$ (5.11)

which is simply Fick’s Law with a new ambipolar diﬀusion coeﬃcient:

 ${D}_{a}=\frac{{\mu }_{i}{D}_{e}+{\mu }_{e}{D}_{i}}{{\mu }_{i}+{\mu }_{e}}$ (5.12)

5.3 Diﬀusion in a Slab

We consider the simple application to diﬀusion in a slab geometry. If the ambipolar diﬀusion coeﬃcient is constant, then the continuity equation becomes:

 $\frac{\partial n}{\partial t}={D}_{a}{\nabla }^{2}n.$ (5.13)

We can solve this via a separation of variables technique. Take solutions of the form:

 $n\left(\stackrel{\to }{r},t\right)=T\left(t\right)X\left(x\right)$ (5.14)

plugging this into the continuity equation, we get that:

 $X\frac{dT}{dt}=T{D}_{a}\frac{{d}^{2}X}{d{x}^{2}}$ (5.15)

which we can rearrange to get:

 $\frac{1}{T}\frac{dT}{dt}=\frac{{D}_{a}}{X}\frac{{d}^{2}X}{d{x}^{2}}.$ (5.16)

We note that the left-hand side depends only on $t$ and the right-hand side depends only on $x$, so by the separation of variables technique they must both be equal to a common constant. We can therefore solve them individually. Starting with the time,

 $\frac{1}{T}\frac{dT}{dt}=\mathrm{\text{const}}=-\frac{1}{\tau }$ (5.17)

where we have set the constant equal to $-1∕\tau$ for reasons that will be apparent later. This equation can be simply solved:

 $\frac{dT}{dt}=-\frac{T}{\tau }$ (5.18)

 $T\left(t\right)={T}_{0}{e}^{-t∕\tau }$ (5.19)

The plasma decays exponentially in time, with time constant $\tau$.

Next, we consider the solution in the spatial ($x$) direction. The equation to solve is:

 $\frac{{D}_{a}}{X}\frac{{d}^{2}X}{d{x}^{2}}=-\frac{1}{\tau }$ (5.20)

 $\frac{{d}^{2}X}{d{x}^{2}}=-\frac{1}{\tau {D}_{a}}X$ (5.21)

which has sinusoidal solutions:

 $X\left(x\right)=Asin\left(x∕\sqrt{\tau {D}_{a}}\right)+Bcos\left(x∕\sqrt{\tau {D}_{a}}\right)$ (5.22)

We require that, for a plasma extending to $x=±L$, the solution must satisfy $X\left(±L\right)=0$ so we can immediately set $A=0$ and $B={X}_{0}$. We also now have the requirement that:

 $\frac{L}{\sqrt{\tau {D}_{a}}}=\frac{\pi }{2}$ (5.23)

or:

 $\tau ={\left(\frac{2L}{\pi }\right)}^{2}\frac{1}{{D}_{a}}$ (5.24)

is the timescale for the plasma to decay by ambipolar diﬀusion. Combining these results, the solution for the plasma density as a function of time and space is:

 $n\left(x,t\right)={n}_{0}{e}^{-t∕\tau }cos\left(\frac{\pi x}{2L}\right)$ (5.25)

5.4 Diﬀusion in Magnetic Fields

We revisit the problem of diﬀusion in weakly-ionized plasmas but now introduce a magnetic ﬁeld, which will have the eﬀect of reducing the diﬀusion. Of course, this is the main goal of magnetic conﬁnement fusion!

In the direction parallel to $\stackrel{\to }{B}$, there is no eﬀect on the diﬀusion rates. If we take $\stackrel{\to }{B}\parallel ẑ$, then:

 ${\Gamma }_{z}=±\mu n{E}_{z}-D\frac{\partial n}{\partial z}$ (5.26)

For the perpendicular direction, we have to return to the ﬂuid momentum equation:

 $mn\frac{d{\stackrel{\to }{v}}_{\perp }}{dt}=±en\left(\stackrel{\to }{E}+{\stackrel{\to }{v}}_{\perp }×\stackrel{\to }{B}\right)-{\nabla }_{\perp }p-mn\nu \stackrel{\to }{v}$ (5.27)

Assuming that the collisions are fast enough that the velocity derivative term (left hand side) can be neglected, and also assuming an isothermal plasma, the two components are:

$\begin{array}{rcll}mn\nu {v}_{x}& =& ±en{E}_{x}±en{v}_{y}{B}_{0}-T\frac{dn}{dx}& \text{(5.28)}\text{}\text{}\\ mn\nu {v}_{y}& =& ±en{E}_{y}\mp en{v}_{x}{B}_{0}-T\frac{dn}{dy}& \text{(5.29)}\text{}\text{}\end{array}$

which can be simpliﬁed immediately to:

$\begin{array}{rcll}{v}_{x}& =& ±\mu {E}_{x}±\frac{{\omega }_{c}}{\nu }{v}_{y}-\frac{D}{n}\frac{dn}{dx}& \text{(5.30)}\text{}\text{}\\ {v}_{y}& =& ±\mu {E}_{y}\mp \frac{{\omega }_{c}}{\nu }{v}_{x}-\frac{D}{n}\frac{dn}{dy}& \text{(5.31)}\text{}\text{}\end{array}$

We can solve these for ${v}_{y}$ via substitution of ${v}_{x}$. This gives us:

 $\left(1+{\omega }_{c}^{2}{\tau }^{2}\right){v}_{y}=±\mu {E}_{y}-\frac{D}{n}\frac{dn}{dy}-{\omega }_{c}^{2}{\tau }^{2}\frac{{E}_{x}}{{B}_{0}}±{\omega }_{c}^{2}{\tau }^{2}\frac{T}{eBn}\frac{dn}{dx}$ (5.32)

and similarly for ${v}_{x}$:

 $\left(1+{\omega }_{c}^{2}{\tau }^{2}\right){v}_{x}=±\mu {E}_{x}-\frac{D}{n}\frac{dn}{dx}+{\omega }_{c}^{2}{\tau }^{2}\frac{{E}_{y}}{{B}_{0}}\mp {\omega }_{c}^{2}{\tau }^{2}\frac{T}{eBn}\frac{dn}{dy}$ (5.33)

We notice that the last two terms are the $E×B$ drift and the diamagnetic drift, respectively, which are important generally but not for this problem. The ﬁrst two terms are the standard mobility and diﬀusion terms, but because of the factor $\left(1+{\omega }_{c}^{2}{\tau }^{2}\right)$ on the left-hand side the coeﬃcients are reduced from the no-ﬁeld case:

 ${\mu }_{\perp }=\frac{\mu }{1+{\omega }_{c}^{2}{\tau }^{2}}$ (5.34)

 ${D}_{\perp }=\frac{D}{1+{\omega }_{c}^{2}{\tau }^{2}}$ (5.35)

5.5 Diﬀusion in Fully Ionized Plasmas

We give a brief qualitative discussion of the diﬀerence between diﬀusion in partially- and fully-ionized plasmas. The previous sections dealt with diﬀusion in partially-ionized plasmas, in which both electrons and ions diﬀused via collisions with the background neutral particles, and we did not care about the concentrations of neutrals.

In the case of a fully-ionized plasma, the collisions which matter are ion-ion, electron-electron, or ion-electron / electron-ion. These can be lumped into like-particle and unlike-particle collisions.

In the case of a Coulomb collision between two identical (like) particles, it turns out that the individual particle velocities can be changed quite signiﬁcantly in the collision - i.e. they are reversed in a $18{0}^{\circ }$ collision, and a $9{0}^{\circ }$ collision changes the velocity vector direction by $9{0}^{\circ }$. However, in a magnetized plasma, the important quantity is actually the particle’s guiding center. It turns out that the ‘center of mass of the guiding centers’ does not change in like-particle collisions. This means that like-particle collisions cannot lead to diﬀusion.

All diﬀusion in fully-ionized plasmas therefore comes from unlike-particle collisions. In an electron colliding with an ion, for instance, there can be net momentum exchange between the species which leads to diﬀusion. The ion itself is much less aﬀected because of its large mass, of course, but at the end of the day momentum conservation requires that both species will diﬀuse based on these unlike-particle collisions.

5.6 Plasma Resistivity

We present a simple and intuitive derivation of the plasma resistivity, which is of course an important quantity. From the momentum equations, we know that the rate of momentum transfer between electrons and ions is:

 ${P}_{ie}=-{P}_{ei}=mn{\nu }_{ei}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)$ (5.36)

We can also write this another way as follows. The plasma resistivity for two interpenetrating ﬂuids with relative ﬂow will be proportional to the relative velocity, $\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)$, the scattering species density (${n}_{i}$), the electron density ${n}_{e}$, and the scattering force which is proportional to the two charges (${e}^{2}$). There is also going to be a constant of proportionality, so in total:

 ${P}_{ie}=\eta {e}^{2}{n}^{2}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)$ (5.37)

setting these two expressions for the momentum exchange equal to each other,

 $mn{\nu }_{ei}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right)={\eta }^{2}{e}^{2}{n}^{2}\left({\stackrel{\to }{u}}_{i}-{\stackrel{\to }{u}}_{e}\right),$ (5.38)

which reduces to:

 $\eta =\frac{m}{n{e}^{2}}{\nu }_{ei}.$ (5.39)

The constant of proportionality $\eta$ is the plasma resistivity. We now need to come up with an expression for the electron-ion collision rate.

The simple derivation is as follows. Consider a $9{0}^{\circ }$ collision between electron and ion. We can deﬁne this as the case where the initial kinetic energy is equal to half the potential energy at the initial impact parameter:

 $\frac{1}{2}{m}_{e}{v}^{2}=\frac{1}{2}\frac{{e}^{2}}{4\pi {𝜖}_{0}{b}_{90}}$ (5.40)

There is some degree of arbitrariness in this deﬁnition by about a factor of 2. More rigorous derivations are done but are much lengthier. Anyways, we can write the $9{0}^{\circ }$ impact parameter as:

 ${b}_{90}=\frac{{e}^{2}}{4\pi {m}_{e}{𝜖}_{0}{v}^{2}}.$ (5.41)

A simple expression for the scattering cross section is thus:

 $\sigma =\pi {b}_{90}^{2}=\frac{{e}^{4}}{16{\pi }^{2}{m}_{e}^{2}{𝜖}_{0}^{2}{v}^{4}}$ (5.42)

and the electron-ion collision rate can be written:

 ${\nu }_{ei}=n\sigma v=\frac{n{e}^{4}}{16{\pi }^{2}{m}_{e}^{2}{𝜖}_{0}^{2}{v}^{3}}.$ (5.43)

It turns out that in many real plasmas, small-angle collisions actually dominate Coulomb interactions, or are at the very least important and cannot be neglected. We will see this more rigorously later. The relative importance of small- and large-angle scattering is characterized by the Coulomb logarithm, and the electron-ion collision rate is enhanced by $ln\Lambda$:

 ${\nu }_{ei}=n\sigma v=\frac{n{e}^{4}ln\Lambda }{16{\pi }^{2}{m}_{e}^{2}{𝜖}_{0}^{2}{v}^{3}}.$ (5.44)

We can now write down an expression for the plasma resistivity:

 $\eta =\frac{m}{n{e}^{2}}{\nu }_{ei}=\frac{{e}^{2}ln\Lambda }{16{\pi }^{2}{m}_{e}{𝜖}_{0}^{2}{v}^{3}}$ (5.45)

Further, if the electrons have a Maxwellian distribution then $v=\sqrt{{T}_{e}∕{m}_{e}}$ and this becomes:

 $\eta =\frac{\sqrt{{m}_{e}}{e}^{2}ln\Lambda }{16{\pi }^{2}{𝜖}_{0}^{2}{T}_{e}^{3∕2}}\propto \frac{ln\Lambda }{{T}_{e}^{3∕2}}.$ (5.46)

Since the Coulomb logarithm is very insensitive to plasma conditions, we can see that the plasma resistivity is mostly dependent on the electron temperature. We also note that the resistivity decreases (conductivity increases) at high temperature, which places limits on processes like Ohmic heating.

5.7 Coulomb Collisions

We must treat the collision of two charged particles; the classical Coulomb collision, in which the two particles interact only through the electrostatic potential:

 $\stackrel{\to }{E}=-\nabla \varphi ,$ (5.47)

 $\varphi =\frac{{q}_{1}{q}_{2}}{4\pi {𝜖}_{0}r}.$ (5.48)

In the lab frame we can deﬁne the kinematics by the positions and velocities of the two individual particles:

$\begin{array}{rcll}{m}_{1}\frac{d{\stackrel{\to }{v}}_{1}}{dt}& =& -\frac{{q}_{1}{q}_{2}}{4\pi {𝜖}_{0}}\frac{{\stackrel{\to }{r}}_{2}-{\stackrel{\to }{r}}_{1}}{|{\stackrel{\to }{r}}_{2}-{\stackrel{\to }{r}}_{1}{|}^{3}}& \text{(5.49)}\text{}\text{}\\ {m}_{2}\frac{d{\stackrel{\to }{v}}_{2}}{dt}& =& -\frac{{q}_{1}{q}_{2}}{4\pi {𝜖}_{0}}\frac{{\stackrel{\to }{r}}_{1}-{\stackrel{\to }{r}}_{2}}{|{\stackrel{\to }{r}}_{1}-{\stackrel{\to }{r}}_{2}{|}^{3}}& \text{(5.50)}\text{}\text{}\\ \frac{d{\stackrel{\to }{r}}_{1}}{dt}& =& {\stackrel{\to }{v}}_{1}& \text{(5.51)}\text{}\text{}\\ \frac{d{\stackrel{\to }{r}}_{2}}{dt}& =& {\stackrel{\to }{v}}_{2}& \text{(5.52)}\text{}\text{}\end{array}$

Of course each of the previous four equations is in three dimensions, generally, and we must simplify from 12 equations before this is a tractable problem.

The ﬁrst is to notice that conservation of momentum:

 $\frac{d}{dt}\left({m}_{1}{\stackrel{\to }{v}}_{1}+{m}_{2}{\stackrel{\to }{v}}_{2}\right)=0,$ (5.53)

suggests a new coordinate system moving with the center of mass velocity:

 $\stackrel{\to }{V}=\frac{{m}_{1}{\stackrel{\to }{v}}_{1}+{m}_{2}{\stackrel{\to }{v}}_{2}}{{m}_{1}+{m}_{2}}$ (5.54)

which is a constant of the motion, plus the relative velocity:

 $\stackrel{\to }{v}={\stackrel{\to }{v}}_{1}-{\stackrel{\to }{v}}_{2}.$ (5.55)

Of course we can transform back to the original lab frame coordinates via:

$\begin{array}{rcll}{\stackrel{\to }{v}}_{1}& =& \stackrel{\to }{V}+\frac{{m}_{2}}{{m}_{1}+{m}_{2}}\stackrel{\to }{v},& \text{(5.56)}\text{}\text{}\\ {\stackrel{\to }{v}}_{2}& =& \stackrel{\to }{V}+\frac{{m}_{1}}{{m}_{1}+{m}_{2}}\stackrel{\to }{v}.& \text{(5.57)}\text{}\text{}\end{array}$

The motion in the center-of-mass frame is simply the relative motion:

 $\frac{d\stackrel{\to }{v}}{dt}=\frac{{q}_{1}{q}_{2}}{4\pi {𝜖}_{0}{m}_{r}}\frac{\stackrel{\to }{r}}{{r}^{3}}$ (5.58)

where we have introduced the reduced mass:

 ${m}_{r}=\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}.$ (5.59)

The problem is now equivalent to a single particle of mass ${m}_{r}$ moving around a central potential at the center of mass. At this point we will need to introduce additional constraints via conserved quantities, in particular energy and angular momentum. One can derive these rigorously, but for simplicity:

 $\frac{1}{2}{m}_{r}{v}^{2}+\frac{{q}_{1}{q}_{2}}{4\pi {𝜖}_{0}}\frac{1}{r}={E}_{0}=\mathrm{\text{const}}$ (5.60)

 ${m}_{r}\stackrel{\to }{r}×\stackrel{\to }{v}={\stackrel{\to }{L}}_{0}=\mathrm{\text{const}}$ (5.61)

One nice result of conservation of angular momentum is that we can see that the motion is entirely in a plane, therefore we can simplify from three to two dimensions for the remainder of the derivation.

We can write the conserved quantities in terms of the original velocity ${v}_{0}$ and impact parameter $b$ (relative to the center of mass).

 ${E}_{0}=\frac{1}{2}{m}_{r}{v}_{0}^{2}$ (5.62)

 ${L}_{0}=-{m}_{r}b{v}_{0}$ (5.63)

Finally, we use cylindrical coordinates for this problem:

 $x=rcos𝜃\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}y=rsin𝜃$ (5.64)

and we remember that:

 $\stackrel{\to }{v}=ṙ\stackrel{̂}{r}+r\stackrel{̇}{𝜃}\stackrel{̂}{𝜃}$ (5.65)

so that the conservation of energy and momentum, respectively, give us that:

 ${ṙ}^{2}+{r}^{2}{\stackrel{̇}{𝜃}}^{2}+\frac{{q}_{1}{q}_{2}}{2\pi {𝜖}_{0}{m}_{r}r}={v}_{0}^{2}$ (5.66)

and

 ${r}^{2}\stackrel{̇}{𝜃}=-b{v}_{0}$ (5.67)

the second equation can be used to eliminate $𝜃$ in the ﬁrst to get:

 $\frac{1}{{v}_{0}^{2}}{ṙ}^{2}=1-\frac{{b}^{2}}{{r}^{2}}-2\frac{{b}_{90}}{r}$ (5.68)

where we have deﬁned the $9{0}^{\circ }$ impact parameter as:

 ${b}_{90}=\frac{{q}_{1}{q}_{2}}{4\pi {𝜖}_{0}{m}_{r}{v}_{0}^{2}}$ (5.69)

The next task is to solve for the scattering angle $\chi$ as a function of the impact parameter and relative velocity. If we deﬁne the point of closest approach by an angle ${𝜃}_{min}$ and distance ${r}_{min}$ we can see via geometric arguments that

 $\chi =2{𝜃}_{min}-\pi$ (5.70)

We will solve for ${𝜃}_{min}$ using our previously-derived equation of motion, starting with:

 ${𝜃}_{min}\equiv \pi -{\int }_{{𝜃}_{min}}^{\pi }d𝜃=\pi -{\int }_{{r}_{min}}^{\infty }\frac{d𝜃}{dr}dr$ (5.71)

with a change of variables from $𝜃$ to $r$ in the last step. Next we need to use:

 $\frac{d𝜃}{dr}=\frac{\stackrel{̇}{𝜃}}{ṙ}=-\frac{b{v}_{0}}{{r}^{2}ṙ}$ (5.72)

we can plug our previous relation for $ṙ$ into this equation and then into the expression for ${𝜃}_{min}$ to get that:

 ${𝜃}_{min}=\pi -b{\int }_{{r}_{min}}^{\infty }\frac{dr}{r\sqrt{{r}^{2}-{b}^{2}-2{b}_{90}r}}$ (5.73)

Actually evaluating this integral requires a non-obvious trick substitution (see, e.g. Friedberg):

 $\frac{b}{r}=-\frac{{b}_{90}}{b}+\sqrt{1+{b}_{90}^{2}∕{b}^{2}}sin\alpha$ (5.74)

 ${𝜃}_{min}=\pi -{\int }_{{\alpha }_{1}}^{{\alpha }_{2}}d\alpha =\pi -\left({\alpha }_{2}-{\alpha }_{1}\right)$ (5.75)

with ${\alpha }_{2}=\pi ∕2$ and

 $sin{\alpha }_{1}=\frac{{b}_{90}}{\sqrt{{b}^{2}+{b}_{90}^{2}}}$ (5.76)

What we end up with, omitting some algebra,

 $cot\frac{\chi }{2}=\frac{b}{{b}_{90}}=\frac{4\pi {𝜖}_{0}{m}_{r}}{{q}_{1}{q}_{2}}{v}_{0}^{2}b$ (5.77)

5.8 Coulomb Collision Frequencies

Now that we have the general result in the previous section for Coulomb collisions, the immediate question is how often does a particle in a plasma, a ‘test particle’, collide with other particles. In this derivation, which follows along with Friedberg Section 9.3, we explicitly consider test electrons but the results can be applied generally.

First, we can write simply the change in the test electron’s momentum as:

 $\frac{d}{dt}\left({m}_{e}{v}_{e}\right)=-\left(\Delta {m}_{e}{v}_{e}\right){n}_{i}\sigma {v}_{e}=-{\nu }_{ei}\left({m}_{e}{v}_{e}\right)$ (5.78)

where in the last step we have essentially deﬁned the electron-ion collision frequency ${\nu }_{ei}$ as a rate of momentum loss. In general the cross section is not a constant and we in fact must rewrite this as:

 ${\nu }_{ei}=\frac{1}{{m}_{e}{v}_{e}}\int \left(\Delta {m}_{e}{v}_{e}\right){f}_{i}\left(\stackrel{\to }{{v}_{i}}\right)|{\stackrel{\to }{v}}_{e}-{\stackrel{\to }{v}}_{i}|b\phantom{\rule{1em}{0ex}}db\phantom{\rule{1em}{0ex}}d\alpha \phantom{\rule{1em}{0ex}}d{\stackrel{\to }{v}}_{i}$ (5.79)

where the momentum loss must be integrated over all possible impact parameters $b$, scattering angles $\alpha$ in the plane perpendicular to the scattering, and also over the scattering body distribution, in this case the ions.

The ﬁrst task is to evaluate $\Delta {m}_{e}{v}_{e}$ and then we must tackle the integral. If the initial velocity is in the $x$ direction in the center-of-mass frame, then we can write the change in momentum as:

 $\Delta {m}_{e}{v}_{e}={m}_{r}\left({\stackrel{\to }{v}}_{i}-{\stackrel{\to }{v}}_{f}\right)\cdot \stackrel{̂}{x}$ (5.80)

In terms of the scattering angle $\chi$ used previously we can see by intuition that the change in $x$-directed momentum will be

 $\Delta {m}_{e}{v}_{e}={m}_{r}\left({v}_{e}-{v}_{x}^{\prime }\right)\left(1-cos\chi \right)$ (5.81)

and based on what we previously derived for Coulomb scattering:

 $\Delta {m}_{e}{v}_{e}={m}_{r}\left({v}_{e}-{v}_{x}^{\prime }\right)×\frac{{b}_{90}^{2}}{{b}^{2}+{b}_{90}^{2}}$ (5.82)

Now we can tackle the integrations over impact parameter and $\alpha$:

 ${\int }_{0}^{\alpha }d\alpha {\int }_{0}^{{b}_{max}}\frac{{b}_{90}^{2}}{{b}^{2}+{b}_{90}^{2}}\phantom{\rule{1em}{0ex}}b\phantom{\rule{1em}{0ex}}db=\pi {b}_{90}^{2}ln\left(1+\frac{{b}_{max}^{2}}{{b}_{90}^{2}}\right)$ (5.83)

There is an important point to be made here. In a naïve sense one might expect to take the integral over the impact parameter $b$ to inﬁnity since the Coulomb interaction has indeﬁnite range, and we have previously argued that long-range interactions are fundamental aspects of a plasma. However, in this integration taking the $b$ integration to inﬁnity would lead to a logarithmic divergence, which cannot be allowed.

We can make a simple physical argument that as the impact parameter is taken to the large limit, it cannot exceed the Debye length because at that point the plasma will screen the Coulomb interaction. We therefore typically take ${b}_{max}\to {\lambda }_{De}$. We typically use the deﬁnition $\Lambda ={b}_{max}∕{b}_{90}$ such that

 $\pi {b}_{90}^{2}ln\left(1+\frac{{b}_{max}^{2}}{{b}_{90}^{2}}\right)\approx 2\pi {b}_{90}^{2}ln\Lambda$ (5.84)

In magnetically-conﬁned plasmas we typically have $ln\Lambda \sim 15-20$, and this quantity is called the Coulomb logarithm. It is essentially a metric of the relative importance of small- and large-angle collisions. In the regime $ln\Lambda >10$ the behavior is dominated by small-angle collisions, the classical plasma regime. If $ln\Lambda <10$ then we are in a moderately- or strongly-coupled plasma, which are more applicable to ICF plasmas.

We now use this in the expression for the collision frequency:

 ${\nu }_{ei}=\frac{4\pi {m}_{r}}{{m}_{e}{v}_{e}}ln\Lambda \int \left({v}_{e}-{v}_{x}^{\prime }\right){b}_{90}^{2}{f}_{i}\left({\stackrel{\to }{v}}_{i}\right)|{\stackrel{\to }{v}}_{e}-{\stackrel{\to }{v}}_{i}|\phantom{\rule{1em}{0ex}}d{\stackrel{\to }{v}}_{i}$ (5.85)

we now have to tackle the integration over target velocities, which is best done in spherical coordinates:

$\begin{array}{rcll}{v}_{x}^{\prime }& =& {v}_{e}+vcos𝜃& \text{(5.86)}\text{}\text{}\\ {v}_{y}^{\prime }& =& vsin𝜃sin\varphi & \text{(5.87)}\text{}\text{}\\ {v}_{z}^{\prime }& =& vsin𝜃cos\varphi & \text{(5.88)}\text{}\text{}\end{array}$

in which case we can write that:

 $d{\stackrel{\to }{v}}_{i}={v}^{2}sin𝜃\phantom{\rule{1em}{0ex}}dv\phantom{\rule{1em}{0ex}}d𝜃\phantom{\rule{1em}{0ex}}d\varphi$ (5.89)

 $|{\stackrel{\to }{v}}_{e}-{\stackrel{\to }{v}}_{i}|=v$ (5.90)

 ${b}_{90}^{2}={\left(\frac{{e}^{2}}{4\pi {𝜖}_{0}{m}_{r}}\right)}^{2}\frac{1}{{v}^{4}}$ (5.91)

and returning to the expression for the collision frequency, some algebra reveals that it reduces to

 ${\nu }_{ei}=\left(\frac{\sqrt{2}{e}^{4}}{8{\pi }^{3∕2}{𝜖}_{0}^{2}}\frac{{n}_{i}}{{T}_{i}^{3∕2}}\frac{{m}_{i}^{3∕2}}{{m}_{e}{m}_{r}}ln\Lambda \right)I\left({w}_{e}\right)$ (5.92)

with velocities normalized to the ion thermal value: ${w}_{e}={v}_{e}∕{v}_{Ti}$ and ${w}_{i}=v∕{v}_{Ti}$. We have also introduced a dimensionless integral:

 $I\left({w}_{e}\right)=-\frac{{e}^{-{w}_{e}^{2}}}{{w}_{e}}{\int }_{0}^{\infty }dw{\int }_{0}^{\pi }d𝜃sin𝜃cos𝜃{e}^{-{w}^{2}-2{w}_{e}wcos𝜃}$ (5.93)

This can be evaluated in terms of the error function, and then taking appropriate limits. We skip the details and go straight to the result:

 $I\left({w}_{e}\right)=\frac{1}{{w}_{e}^{2}}\left(\frac{\sqrt{\pi }}{2{w}_{e}}\mathrm{\text{erf}}\left({w}_{e}\right)-{e}^{-{w}_{e}^{2}}\right)\approx \frac{\sqrt{\pi }}{2}\frac{1}{{w}_{e}^{3}+3\sqrt{\pi }∕4}$ (5.94)

Using this to complete the expression for the collision frequency, we get:

 ${\nu }_{ei}=\left(\frac{{e}^{4}}{4\pi {𝜖}_{0}^{2}}\frac{{n}_{i}}{{T}_{i}^{3∕2}}\frac{1}{{m}_{e}{m}_{r}}ln\Lambda \right)\frac{1}{{v}_{e}^{3}+1.3{v}_{Ti}^{3}}$ (5.95)

For electron-ion collisions in particular we can take the limits ${m}_{r}\approx {m}_{e}$ and ${v}_{e}\sim {v}_{Te}\gg {v}_{Ti}$, in which case:

 ${\nu }_{ei}=\left(\frac{{e}^{4}}{4\pi {𝜖}_{0}^{2}{m}_{e}^{2}}\frac{{n}_{i}}{{T}_{i}^{3∕2}}ln\Lambda \right)\frac{1}{{v}_{e}^{3}}$ (5.96)

One can also examine the expression for the collision frequency and come up with a hierarchy of various species colliding with each other, in terms of the mass ratio $\mu ={m}_{e}∕{m}_{i}$:

$\begin{array}{rcll}{\nu }_{ee}& \sim & {\nu }_{ei}& \text{(5.97)}\text{}\text{}\\ {\nu }_{ii}& \sim & \sqrt{\mu }{\nu }_{ei}& \text{(5.98)}\text{}\text{}\\ {\nu }_{ie}& \sim & \mu {\nu }_{ei}& \text{(5.99)}\text{}\text{}\end{array}$

5.9 Distribution Function

In the previous sections we have often explicitly or implicitly assumed that the particles obey a Maxwellian distribution, but in many plasmas of interest this is not the case and leads to full-ﬂedged kinetic theory. First we discuss some properties of the distribution function.

In general the particle distribution can be a function of full phase space: position $\stackrel{\to }{r}$, velocity $\stackrel{\to }{v}$, and time $t$:

 $f=f\left(\stackrel{\to }{r},\stackrel{\to }{v},t\right)$ (5.100)

One can obtain functions such as the density via integration over part of phase space:

 $n\left(\stackrel{\to }{r},t\right)=\int f\left(\stackrel{\to }{r},\stackrel{\to }{v},t\right)\phantom{\rule{1em}{0ex}}d\stackrel{\to }{v}$ (5.101)

In this particular case we sometimes use a normalized distribution function:

 $\stackrel{̂}{f}=f∕n$ (5.102)

Of course the most famous and commonly-used distribution function is the Maxwellian:

 $\stackrel{̂}{{f}_{m}}={\left(\frac{m}{2\pi {k}_{B}T}\right)}^{3∕2}{e}^{-{v}^{2}∕{v}_{th}^{2}}$ (5.103)

with ${v}_{th}=\sqrt{2{k}_{B}T∕m}$ and $v=|\stackrel{\to }{v}|$.

5.10 Overview of Kinetic Theories

The basic problem of kinetic theory is to deﬁne how the distribution function of a plasma changes with time. One can simply derive this by taking the chain rule total time derivative of $f$:

 $\frac{df}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt}+\frac{\partial f}{\partial {v}_{x}}\frac{d{v}_{x}}{dt}+\frac{\partial f}{\partial {v}_{y}}\frac{d{v}_{y}}{dt}+\frac{\partial f}{\partial {v}_{z}}\frac{d{v}_{z}}{dz}$ (5.104)

The ﬁrst term is simply the explicit time dependence of $f$. The second through fourth terms are observed to be:

 $\stackrel{\to }{v}\cdot \nabla f$ (5.105)

and the ﬁfth through seventh terms are, using $\stackrel{\to }{F}=m\stackrel{̇}{\stackrel{\to }{v}}$, simply

 $\frac{\stackrel{\to }{F}}{m}\cdot \frac{\partial f}{\partial \stackrel{\to }{v}}$ (5.106)

Overall the total change in time of the distribution function is due to external interactions, which we call ‘collisions’ and now we can write the Boltzmann Equation:

 $\frac{\partial f}{\partial t}+\stackrel{\to }{v}\cdot \nabla f+\frac{\stackrel{\to }{F}}{m}\cdot \frac{\partial f}{\partial \stackrel{\to }{v}}={\left(\frac{\partial f}{\partial t}\right)}_{c}.$ (5.107)

Of course in plasmas one often has the situation that the forces present are simply due to the particle’s Coulomb interaction, and furthermore, collisions can often be neglected in hot plasmas. In this limit we obtain the Vlasov Equation:

 $\frac{\partial f}{\partial t}+\stackrel{\to }{v}\cdot \nabla f+\frac{q}{m}\left(\stackrel{\to }{E}+\stackrel{\to }{v}×\stackrel{\to }{B}\right)\cdot \frac{\partial f}{\partial \stackrel{\to }{v}}=0.$ (5.108)

If there are collisions with neutral atoms, then one can use the Krook collision term:

 ${\left(\frac{\partial f}{\partial t}\right)}_{c}=\frac{{f}_{n}-f}{\tau }$ (5.109)

And another familiar limit is with Coulomb collisions, we get the Fokker-Planck Equation:

 $\frac{df}{dt}=-\frac{\partial }{\partial \stackrel{\to }{v}}\cdot \left(f⟨\Delta \stackrel{\to }{v}⟩\right)\frac{1}{2}\frac{{\partial }^{2}}{\partial \stackrel{\to }{v}\partial \stackrel{\to }{v}}:\left(f⟨\Delta \stackrel{\to }{v}\Delta \stackrel{\to }{v}⟩\right)$ (5.110)

which is too complex to delve into at this level.

5.11 Landau Damping

One of the most important and famous results of kinetic theory is Landau damping, or more generally wave-plasma collisionless coupling. In this case we consider electron plasma oscillations in an initially uniform plasma with ${\stackrel{\to }{E}}_{0}={\stackrel{\to }{B}}_{0}=0$. We also want to treat the wave perturbatively, i.e. let:

 $f\left(\stackrel{\to }{r},\stackrel{\to }{v},t\right)={f}_{0}\left(\stackrel{\to }{v}\right)+{f}_{1}\left(\stackrel{\to }{r},\stackrel{\to }{v},t\right)$ (5.111)

the ﬁrst-order Vlasov equation for electrons is:

 $\frac{\partial {f}_{1}}{\partial t}+\stackrel{\to }{v}\cdot \nabla {f}_{1}-\frac{e}{m}{\stackrel{\to }{E}}_{1}\cdot \frac{\partial {f}_{0}}{\partial \stackrel{\to }{v}}=0$ (5.112)

we will assume inﬁnite inertia ions and use plane wave perturbations in the $x$ direction, so that:

 ${f}_{1}\propto {e}^{i\left(kx-\omega t\right)}$ (5.113)

so the Vlasov equation becomes, to ﬁrst order,

 $-i\omega {f}_{1}+ik{v}_{x}{f}_{1}=\frac{e}{m}{E}_{x}\frac{\partial f}{\partial {v}_{x}}$ (5.114)

 ${f}_{1}=\frac{ie{E}_{x}}{m}\frac{\partial {f}_{0}∕\partial {v}_{x}}{\omega -k{v}_{x}}$ (5.115)

Now we have to use Poisson’s Equation:

 ${𝜖}_{0}\nabla \cdot {\stackrel{\to }{E}}_{1}=ik{𝜖}_{0}{E}_{x}=-i{n}_{1}=-e\int {f}_{1}{d}^{3}v$ (5.116)

Using these two results we can eliminate ${E}_{x}$ to get:

 $1=-\frac{{e}^{2}}{km{𝜖}_{0}}\int \frac{\partial {f}_{0}∕\partial {v}_{x}}{\omega -k{v}_{x}}{d}^{3}v$ (5.117)

If $f$ is a Maxwellian, then we can easily separate the three coordinates and reduce this to a one-dimensional integral:

 $1=\frac{{w}_{pe}^{2}}{{k}^{2}}\int \frac{\partial {\stackrel{̂}{f}}_{0}\left({v}_{x}\right)∕\partial {v}_{x}}{{v}_{x}-\left(\omega ∕k\right)}d{v}_{x}$ (5.118)

We notice immediately the pole at ${v}_{x}=\omega ∕k$. While the pole is not necessarily on the contour of integration if $\omega ∕k$ is complex, it matters nevertheless. In the simplest limit of large phase velocity and weak damping, the pole lies near the real axis and we can take the contour of integration as along the axis except for a small semi-circle around the pole. In this case, by the residue theorem,

 $1=\frac{{\omega }_{pe}^{2}}{{k}^{2}}\left[P{\int }_{-\infty }^{\infty }\frac{\partial {\stackrel{̂}{f}}_{0}∕\partial v}{v-\left(\omega ∕k\right)}\phantom{\rule{1em}{0ex}}dv+i\pi {\frac{\partial {\stackrel{̂}{f}}_{0}}{\partial v}|}_{v=\omega ∕k}\right]$ (5.119)

The ﬁrst term in square brackets ends up being the electron plasma wave dispersion relation:

 ${\omega }^{2}={\omega }_{pe}^{2}+\frac{3T}{2m}{k}^{2}$ (5.120)

see Chen section 7.4 for a derivation of this. The interesting part is the second term, which serves as a small and imaginary correction to the dispersion relation in the limit of small k:

 ${\omega }^{2}\left(1-i\pi \frac{{\omega }_{pe}^{2}}{{k}^{2}}{\left[\frac{\partial {\stackrel{̂}{f}}_{0}}{\partial v}\right]}_{v={v}_{\varphi }}\right)={\omega }_{pe}^{2}$ (5.121)

Or rearranging, and using again the smallness of the second term in parentheses,

 $\omega ={\omega }_{pe}\left(1+i\frac{\pi }{2}\frac{{\omega }_{pe}^{2}}{{k}^{2}}{\left[\frac{\partial {\stackrel{̂}{f}}_{0}}{\partial v}\right]}_{v={v}_{\varphi }}\right)$ (5.122)

Essentially, we can see that this derivation has introduced damping to the electron plasma wave. The damping depends on the wavenumber, and most critically on the shape of the distribution function where the particle velocity matches the wave’s phase velocity. At this point there is coupling predicted between the wave and the plasma particles. If $\left(\partial {f}_{0}∕\partial v\right)<0$ then we can see that the wave is damped and transfers energy to the particles. On the other hand, if $\left(\partial {f}_{0}∕\partial v\right)>0$ the the wave is actually unstable and grows in amplitude.

Chapter 6Diagnostics

6.1 Refractive measurements of density

One can use the refractive index of a plasma, or dispersion, as a density measurement via interferometry. Consider the following scheme, shown in Fig. 6.1, where we stick an unmagnetized plasma in one leg of an interferometer (Mach-Zender scheme shown). In this case, the detectors will measure the phase shift between the two legs, so we must calculate the phase shift due to plasma.

Consider the source $S$ as a monochromatic source of electromagnetic radiation that obeys:

 $\omega \gg {\omega }_{pe}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}f\gg {f}_{pe}$ (6.1)

Remember the back-of-the-envelope rule ${f}_{pe}=9\sqrt{n}$ Hz, with $\left[n\right]={m}^{-3}$.

The dispersion relation for electromagnetic waves in a plasma is

 ${\omega }^{2}={\omega }_{pe}^{2}+{c}^{2}{k}^{2},$ (6.2)

 $ck=\omega \sqrt{1-{\omega }_{pe}^{2}∕{\omega }^{2}}$ (6.3)

which leads to a refractive index

 $n\equiv \frac{ck}{\omega }={\left(1+{\omega }_{pe}^{2}∕{\omega }^{2}\right)}^{1∕2}$ (6.4)

We know that we can write the phase shift for propagating waves as

 $\varphi =\int kd\ell =\int n\frac{\omega }{c}d\ell$ (6.5)

So the diﬀerence between plasma and vacuum (with $n=1$) is:

 $\Delta \varphi ={\int }_{0}^{L}\left(n-1\right)\frac{\omega }{c}d\ell ,$ (6.6)

 $\Delta \varphi =-{\int }_{0}^{L}\left[{\left(1-{\omega }_{pe}^{2}∕{\omega }^{2}\right)}^{1∕2}-1\right]\frac{\omega }{c}d\ell$ (6.7)

Based on our initial assumption that $\omega \gg {\omega }_{pe}$ we can use the binomial expansion:

 $\Delta \varphi ={\int }_{0}^{L}\frac{1}{2}\frac{{\omega }_{pe}^{2}}{{\omega }^{2}}\frac{\omega }{c}d\ell =-\frac{1}{2\omega c}{\int }_{0}^{L}{\omega }_{pe}^{2}d\ell$ (6.8)

Using the deﬁnition of the plasma frequency:

 ${\omega }_{pe}^{2}=\frac{{n}_{e}{e}^{2}}{{𝜖}_{0}{m}_{e}}$ (6.9)

we can write the phase shift as:

 $\Delta \varphi =-\frac{1}{2\omega c}\frac{{e}^{2}}{{𝜖}_{0}{m}_{e}}{\int }_{0}^{L}{n}_{e}d\ell$ (6.10)

The phase shift is directly proportional to the line-integrated electron number density.

6.2 Refractive Measurements of Astrophysical Length

Now consider a somewhat diﬀerent system, where we have a broadband source of radiation $S$ that traverses a plasma of scale length $L$ and known density ${n}_{e}$ to a detector. Given some information, for example the diﬀerence in arrival time between radiation of frequencies ${f}_{1}$ and ${f}_{2}$, what is the length $L$?

We start oﬀ by stating:

 $L={v}_{g}t$ (6.11)

so for two diﬀerent frequencies:

 $\Delta t=L\left({v}_{g1}^{-1}-{v}_{g2}^{-1}\right)$ (6.12)

The electromagnetic radiation propagates at the group velocity:

 ${v}_{g}\equiv \frac{\partial \omega }{\partial k}$ (6.13)

so we need the dispersion relation. We know that for electromagnetic radiation in an unmagnetized plasma:

 ${\omega }^{2}={c}^{2}{k}^{2}+{\omega }_{pe}^{2}$ (6.14)

taking the derivative $\partial ∕\partial k$ gives us:

 $2\omega {v}_{g}=2{c}^{2}k,$ (6.15)

 ${v}_{g}=\frac{{c}^{2}k}{\omega }.$ (6.16)

Using the original dispersion relation to substitute for $\omega$,

 ${v}_{g}=\frac{{c}^{2}k}{ck}{\left(1+\frac{{\omega }_{pe}^{2}}{{c}^{2}{k}^{2}}\right)}^{-1∕2}$ (6.17)

We typically take the limit ${\omega }_{pe}\ll \omega$, which can be checked via the back-of-the-envelope relation ${f}_{pe}=9\sqrt{n}$ with $f$ in Hz and $\left[n\right]={m}^{-3}$. In that limit we can binomial approximate the last term,

 ${v}_{g}=c\left(1-\frac{{\omega }_{pe}^{2}}{2{c}^{2}{k}^{2}}\right)\approx c\left(1-\frac{1}{2}\frac{{f}_{pe}^{2}}{{f}^{2}}\right)$ (6.18)

returning to the original expression between $L$ and $\Delta t$, we can rewrite as:

 $L=\Delta t{\left({v}_{g1}^{-1}-{v}_{g2}^{-1}\right)}^{-1}$ (6.19)

since the terms ${f}_{pe}∕f$ are small, when we substitute in for the group velocities,

 $L=c\Delta t{\left(\frac{{f}_{pe}^{2}}{2{f}_{2}^{2}}-\frac{{f}_{pe}^{2}}{2{f}_{1}^{2}}\right)}^{-1}$ (6.20)

at which point one must take some values for ${f}_{1}$, ${f}_{2}$, $\Delta t$, and ${n}_{e}$ and plug them into this expression.

Next we consider using electromagnetic radiation as a probe of magnetic ﬁeld conditions. This topic is similar to the ﬁrst considered in Diagnostics, in that we will calculate a phase shift due to the plasma and consider that it is being compared to the vacuum value in an interferometer.

We consider the electromagnetic source of radiation as producing monochromatic plane-polarized waves, which propagate in the plasma such that $\stackrel{\to }{k}\parallel {\stackrel{\to }{B}}_{0}\parallel ẑ$. In this case the two fundamental solutions are the R and L waves, or right- and left-hand circularly polarized waves. The initial wave can be written as a superposition of these two, following Hutchinson’s notation:

 $\stackrel{\to }{E}\left(z=0\right)=\frac{{E}_{0}}{2}\left[\left(1,i\right)+\left(1,-i\right)\right]$ (6.21)

where we are denoting the polarizations by $±i$ from the basic:

 $\frac{{E}_{x}}{{E}_{y}}=±i$ (6.22)

for circular polarization. Also note that we have taken $\stackrel{\to }{E}\left(z=0\right)\parallel \stackrel{̂}{x}$ without loss of generality. In general, some distance later, the electric ﬁeld will be given by:

 $\stackrel{\to }{E}=\frac{{E}_{0}}{2}\left[{e}^{i{k}_{+}z}+{e}^{i{k}_{-}z}\right]$ (6.23)

where the two circular polarizations can have diﬀerent wave numbers. Using $k=n\omega ∕c$,

 $\stackrel{\to }{E}=\frac{{E}_{0}}{2}\left[{e}^{i{n}_{+}\omega z∕c}+{e}^{i{n}_{-}\omega z∕c}\right]$ (6.24)

which we can rewrite using our $\left(x,y\right)$ notation as:

 $\stackrel{\to }{E}\left(z\right)={E}_{0}exp\left[i\left({n}_{+}+{n}_{-}\right)\omega z∕2c\right]\left(cos\frac{\Delta \varphi }{2},sin\frac{\Delta \varphi }{2}\right)$ (6.25)

where

 $\Delta \varphi =\left({n}_{+}-{n}_{-}\right)\frac{\omega z}{c}$ (6.26)

there is a phase diﬀerence between the RHCP and LHCP components, which leads to a rotation in the plane polarization as the wave propagates along $z$.

In general, for a wave propagating in an plasma with small $𝜃$ between $\stackrel{\to }{k}$ and ${\stackrel{\to }{B}}_{0}$, the dispersion relation can be written as:

 ${n}^{2}=1-X±XYcos𝜃$ (6.27)

with

 $X=\frac{{\omega }_{pe}^{2}}{{\omega }^{2}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}Y=\frac{{\omega }_{ce}}{\omega }$ (6.28)

the trick here is to write

 $\frac{{n}_{±}^{2}}{1-X}=1±\frac{XYcos𝜃}{1-X}$ (6.29)

taking the square root, and approximating the second term as small ($X$ is small):

 $\frac{{n}_{±}}{\sqrt{1-X}}=1±\frac{XYcos𝜃}{2\left(1-X\right)}$ (6.30)

taking the diﬀerence between the two polarizations:

 $\left({n}_{+}-{n}_{-}\right)=\frac{XYcos𝜃}{\sqrt{1-X}}.$ (6.31)

We now plug in for $X$ and $Y$:

 $\Delta \varphi =\frac{\omega z}{c}\frac{{\omega }_{pe}^{2}{\omega }_{ce}cos𝜃}{{\omega }^{3}\sqrt{1-{\omega }_{pe}^{2}∕{\omega }^{2}}}$ (6.32)

in the limit of small $X$, i.e. $\omega \gg {\omega }_{pe}$:

 $\Delta \varphi =\frac{\omega z}{c}\frac{{\omega }_{pe}^{2}{\omega }_{ce}cos𝜃}{{\omega }^{3}}$ (6.33)

we note that the polarization plane rotation is $\propto z$. The Faraday rotation angle is typically deﬁned as

 $\alpha \equiv \frac{\Delta \varphi }{2}=\frac{\omega z}{2c}\frac{{\omega }_{pe}^{2}{\omega }_{ce}}{{\omega }^{3}}cos𝜃$ (6.34)

The Faraday rotation angle is linearly proportional to the propagation distance $z$, the plasma density (through plasma frequency), and the initial magnetic ﬁeld strength through the cyclotron frequency.

In the common event that the plasma is not perfectly uniform, a WKB-style analysis can be adopted:

 $\alpha ={\int }_{0}^{L}\frac{\partial \varphi }{\partial z}\frac{\omega }{c}d\ell$ (6.35)

some algebraic manipulation and our previous deﬁnition of $\Delta \varphi$ leads to:

 $\alpha =\frac{e}{2mc}{\int }_{0}^{L}\frac{{n}_{e}}{{n}_{c}}\stackrel{\to }{B}\cdot \stackrel{\to }{d\ell }$ (6.36)

the Faraday rotation depends on both density and magnetic ﬁeld proﬁles. To measure one, the other must be known ab initio.

6.4 Magnetic Field Probes

6.4.1 $Ḃ$ Probes

One of the most common and simplest probes for magnetic ﬁeld is just a coil. The schematic is shown in Fig. 6.3. The magnetic ﬁeld is sampled via ﬂux through a coil, which induces an EMF and voltage. We know from Faraday’s law that

 $\oint \stackrel{\to }{E}\cdot \stackrel{\to }{d\ell }={\int }_{S}\stackrel{̇}{\stackrel{\to }{B}}\cdot \stackrel{\to }{dA}$ (6.37)

so,

 $V=NA\stackrel{̇}{{B}_{\perp }}$ (6.38)

where $V$ is the induced voltage, $N$ is the number of coils, and $A$ is the area of one coil. Typically these probes are used with an circuit integrator, in which case the post-integrator voltage is:

 $V=\frac{NA{B}_{\perp }}{RC}$ (6.39)

6.4.2 Hall Probes

The Hall probe is shown schematically in Fig. 6.4. The concept is that a current, $\stackrel{\to }{j}$, is passed through the probe. There is a $\stackrel{\to }{j}×\stackrel{\to }{B}$ force on the current carriers. Since this is typically electrons, there is an induced charge separation and electric ﬁeld as shown in the ﬁgure. A Hall probe utilizes this eﬀect by measuring the potential induced across the probe, which can be related to the magnetic ﬁeld and current. In most plasma experiments, however, there is too much pickup for this to be a useful technique.

6.4.3 Rogowski Coils

The last simple diagnostic considered in this section is actually a current diagnostic, but the concept is very similar to the $Ḃ$ probe. Consider the schematic Fig. 6.5. In this case the coil surfaces are perpendicular to the magnetic ﬁeld induced by the current $I$. By

 $\oint \stackrel{\to }{B}\cdot \stackrel{\to }{d\ell }={\mu }_{0}{I}_{enc},$ (6.40)

the induced ﬂux on the probe is

 $\Phi =NA{\mu }_{0}I$ (6.41)

where $N$ is the number of coils and $A$ is the area of each coil. We can thus write the voltage induced as

 $V=\stackrel{̇}{\Phi }=NA{\mu }_{0}İ.$ (6.42)

Once again we typically would pair a Rogowski coil with an integrating circuit, in which case the ﬁnal signal is:

 $V=\frac{NA{\mu }_{0}I}{RC}$ (6.43)

6.5 Langmuir Probes

The last diagnostic considered is the important and basic Langmuir probe. Consider an electrically conducting probe placed into the plasma at arbitrary potential. Fig. 6.6 shows a schematic of the probe potential. A sheath develops around the probe; over the sheath the probe’s potential is Debye screened. In this example the probe potential $V$ is less than the plasma potential.

The ﬁrst important case is when the probe potential is much less than the plasma. This could arise, for example, if the probe is grounded and the plasma is positively charged due to ambipolar diﬀusion. If the diﬀerence in potentials is greater than a few times $k{T}_{e}∕e$, then the thermal electrons do not have enough energy to overcome the potential and the probe collects only ions. This is called the ion saturation current. We can estimate this as:

 ${I}_{SI}={n}_{s}evA$ (6.44)

where ${n}_{s}$ is the sheath density, $v$ is the ion drift velocity, and $A$ is the probe area. If we take the drift velocity as

 $v\approx \sqrt{{k}_{B}T∕{m}_{i}}$ (6.45)

and the sheath density as given by Debye screening:

 ${n}_{s}={n}_{0}{e}^{e\varphi ∕{k}_{B}{T}_{e}}\approx {n}_{0}{e}^{-1∕2}=0.6{n}_{0}$ (6.46)

where we have approximated the sheath boundary as $e\varphi \sim -{k}_{B}{T}_{e}∕2$. In this case the ion saturation current is given by:

 ${I}_{SI}=0.6{n}_{0}eA\sqrt{{k}_{B}{T}_{e}∕{m}_{i}}$ (6.47)

By measuring the ion saturation current, one can determine the plasma density if the electron temperature is known.

Of course one must also have a way to measure the electron temperature. This can actually be done with the same probe! Consider qualitatively what happens as the probe potential is increased. When it reaches the plasma potential there is no potential diﬀerence to drive a current, so the current must go to zero. As the potential is further increased, the probe potential exceeds the plasma potential. Now ions are repulsed by the probe potential and electrons are preferentially collected instead. The rate at which this transition occurs depends on the temperature, since the probe is essentially sampling an increasing fraction of the electron distribution as the voltage is increased. The derivation is omitted here, but the qualitative behavior of the $I-V$ curve is shown in Fig. 6.7.

Chapter 7Tokamaks

7.1 Introduction

In this section we attempt to give a brief motivation of the basic design of a tokamak and its general properties. In previous sections we have considered only linear machines. If one wishes to build a fusion reactor, then end losses in a linear pinch (Z or $𝜃$) are too great to overcome. The magnetic mirror was proposed as a way around this limit, yet it turns out that the mirror scheme is not enough to overcome the end losses and the absolute theoretical maximum $Q$ for a mirror machine is $1.1$. For magnetic conﬁnement, we must design a scheme with no end losses.

In the 1950s Soviet physicists Sakharov and Tamm proposed a scheme in which a linear pinch plasma is bent into a torus, thus eliminating the end losses. The basic geometry is shown in Fig. 7.1; here we call $R$ the major radius of the torus and $a$ the minor radius. If one imagines using coils around the torus to create a toroidal magnetic ﬁeld, the conﬁnement scheme will be very similar to the $𝜃$ pinch MHD equilibrium.

The basic ﬁeld geometry thus far is shown in Fig. 7.2. In the top half of the ﬁgure we show the toroidal current-carrying coils which create a toroidal ﬁeld $B$. This ﬁeld conﬁguration is equivalent to a single inﬁnite current-carrying wire on the origin for this simple analysis. We know that the magnetic ﬁeld due to an inﬁnite wire can be derived from Ampère’s Law:

 $\oint \stackrel{\to }{B}\cdot \stackrel{\to }{d\ell }={\mu }_{0}{I}_{enc}$ (7.1)

 $|B|=\frac{{\mu }_{0}I}{2\pi r}$ (7.2)

The important point to take away from this is that $|B|\propto 1∕r$.

So there is a magnetic ﬁeld gradient. We know from Section 2.3 that a gradient in the magnetic ﬁeld causes particle drifts:

 ${v}_{\nabla B}=\mp \frac{{v}_{\perp }^{2}}{2{\omega }_{c}}\frac{\nabla B×\stackrel{\to }{B}}{{B}^{2}}$ (7.3)

In our simple torus geometry, the result of the $\nabla B$ drift is shown in Fig. 7.3. Because the drift velocity is $\propto 1∕q$ the electrons and ions drift in opposite directions. Working through the cross product with the right hand rule, the reader can verify that ions drift up and electrons drift down in this geometry.

After some time has passed, the electron and ion drifts will necessarily induce charge separation. Positive charge will gather at the top of the machine and negative charge at the bottom. This is shown in Fig. 7.4. Naïvely one might expect that this will simply continue until the induced electric ﬁeld cancels out the $\nabla B$ drift and thus quiescence can be achieved.

There is, however, a major problem for this design. As shown in Fig. 7.4, this induced electric ﬁeld is perpendicular to the toroidal ﬁeld. There is therefore a $E×B$ drift, which is outwards as shown in the ﬁgure. As we know from Section 2.2, both electrons and ions will drift in the same direction at the same velocity due to this drift. The entire plasma is therefore shoved radially outwards. Numerical estimates show that this process will happen very fast (sub-ms) relative to necessary energy conﬁnement times in tokamaks (s).

One must therefore prevent the $\nabla B$ drift from causing charge separation. The solution is shown in Fig. 7.5. If we add a toroidal current within the plasma itself, it will create a poloidal ﬁeld and the overall ﬁeld lines will wrap around the torus like stripes on a barber’s pole (or screw threads - indeed this is the toroidal analog of the screw pinch). In magnetic conﬁnement machines we are typically limited to $\beta \ll 1$, for instance $\beta \sim 0.1$, in which case we know that the transport properties along the ﬁeld lines are much faster than perpendicular to them. In this case then, we know that the electron conductivity along the lines of force is very high. With the ﬁeld lines wrapped around the plasma, the top and bottom of the machine are essentially electrically shorted and no potential diﬀerence can be generated between them. The $\nabla B$ drift can therefore generate some currents within the plasma but it cannot cause a charge separation, and we therefore avoid the disastrous $E×B$ drift.

7.2 Lawson Criterion

The immediate question for any conﬁnement scheme is how well it can perform relative to the demands set by fusion ignition and energy gain. We therefore follow a simple derivation for tokamaks by J. Lawson [Proc. Phys. Soc. B70, 6 (1957)].

We obviously consider a DT plasma conﬁned in a tokamak. The fusion power which can be used for self-heating is generated by the alpha particles from the DT fusion reaction:

 $D\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}T\phantom{\rule{1em}{0ex}}\to \phantom{\rule{1em}{0ex}}n\phantom{\rule{1em}{0ex}}\left(14.1\phantom{\rule{1em}{0ex}}\mathrm{\text{MeV}}\right)+\alpha \phantom{\rule{1em}{0ex}}\left(3.5\phantom{\rule{1em}{0ex}}\mathrm{\text{MeV}}\right)$ (7.4)

The total fusion power is:

 ${S}_{\alpha }=\frac{1}{4}{E}_{\alpha }{n}_{i}^{2}⟨\sigma v⟩$ (7.5)

where the factor of $1∕4$ comes from ${n}_{D}={n}_{T}={n}_{i}∕2$, ${E}_{\alpha }=3.5$ MeV, and $⟨\sigma v⟩$ is the fusion reactivity. Using that the total density $n=\left({n}_{e}+{n}_{i}\right)\approx 2{n}_{i}$ and that the total pressure is $p=nT$,

 ${S}_{\alpha }=\frac{1}{16}{E}_{\alpha }\frac{{p}^{2}}{{T}^{2}}⟨\sigma v⟩$ (7.6)

The fusion power must overpower loss mechanisms. In this simple analysis, we consider only Bremsstrahlung and heat conduction losses. The bremsstrahlung power derivation won’t be treated until later, so we simply give the result:

 ${S}_{B}={C}_{B}{Z}_{eff}{n}_{e}^{2}\sqrt{T}\approx \frac{1}{4}{C}_{B}{Z}_{eff}\frac{{p}^{2}}{{T}^{3∕2}}$ (7.7)

where ${C}_{B}$ is a constant of proportionality.

The last term to derive is a heat conduction loss term. In general, this would be

 $\frac{1}{V}\int \stackrel{\to }{q}\cdot \stackrel{\to }{dA}=-2\frac{\kappa }{r}{\frac{\partial T}{\partial r}|}_{r=a}$ (7.8)

Unfortunately the thermal conductivity $\kappa$ is not generally well-known, and neither is the temperature gradient at the edge of the plasma. It is therefore convenient to introduce / deﬁne a general ‘energy conﬁnement time’ such that this loss term is

 $\frac{3}{2}\frac{p}{{\tau }_{E}}$ (7.9)

where ${\tau }_{E}$ characterizes the loss timescale for thermal energy due to thermal conduction, or other loss mechanisms not considered.

So, in general the power balance equation is:

 ${S}_{\alpha }+{S}_{h}\ge {S}_{B}+{S}_{\kappa }$ (7.10)

where we also include an arbitrary heating term ${S}_{h}$, which is generally taken to be $0$ in a steady-state ignited plasma. Substituting in the terms we have derived above:

 $\frac{{E}_{\alpha }⟨\sigma v⟩}{16}\frac{{p}^{2}}{{T}^{2}}\ge \frac{{C}_{B}}{4}\frac{{p}^{2}}{{T}^{3∕2}}+\frac{3}{2}\frac{p}{{\tau }_{E}}$ (7.11)

We can further simplify the constant coeﬃcients:

 ${K}_{\alpha }\frac{⟨\sigma v⟩{p}^{2}}{{T}^{2}}\ge {K}_{B}\frac{{p}^{2}}{{T}^{3∕2}}+{K}_{\kappa }\frac{p}{{\tau }_{E}}$ (7.12)

this can be simpliﬁed algebraically to obtain a requirement on the pressure-conﬁnement product:

 $p{\tau }_{E}\ge \frac{{K}_{\kappa }{T}^{2}}{{K}_{\alpha }⟨\sigma v⟩-{K}_{B}\sqrt{T}}$ (7.13)

Since the fusion reactivity is just a function of the temperature, the right-hand side is a function of temperature only. This is the Lawson Criterion. For a given temperature that we can achieve in a fusion tokamak, the Lawson Criterion tells us what pressure-conﬁnement product $p{\tau }_{E}$ we must achieve. For example, a detailed calculation reveals that if one can operate a tokamak at 15 keV then one must have $p{\tau }_{E}\ge 8.3$ atm-s. Typical energy conﬁnement times are of order one second, which means that typical pressures in reactors will have to be of order ten atmospheres.

A plot of the minimum $p{\tau }_{E}$ is shown in Fig. 7.6

7.3 MHD Equilibrium in Tokamaks

Recall the work done previously in section 3.3. Now that we have added the toroidal current, and thus a poloidal ﬁeld, we can see that the ﬁeld conﬁguration in the tokamak is at a most basic level the toroidal analog of the screw pinch:

 $\frac{d}{dr}\left[p+\frac{{B}_{z}^{2}}{2{\mu }_{0}}+\frac{{B}_{𝜃}^{2}}{2{\mu }_{0}}\right]+\frac{{B}_{𝜃}^{2}\left(r\right)}{{\mu }_{0}r}=0$ (7.14)

In general, with two of the three unknowns speciﬁed (e.g. ${B}_{z}$ and ${B}_{𝜃}$) then MHD determines the third (e.g. $p$). However, because of the toroidal geometry there are several complications that arise, and are qualitatively discussed in the following sections.

7.3.2 Toroidal Force Balance - Hoop

First we consider the hoop force on a tokamak plasma. The situation is shown schematically in Fig. 7.7. The ﬁeld of interest in this case is poloidal, and this is therefore applicable to Z-pinch or Screw-pinch plasmas. The plasma is shown in red in the ﬁgure, the current ﬂows in or out of the page, and the poloidal ﬁeld thus wraps around the plasma as shown.

Because of the toroidal geometry, the magnetic ﬁeld induced has a $1∕r$ dependence and is therefore stronger on the inside of the plasma relative to the outside. On the other hand, the surface area on the inner half, ${S}_{1}$, is smaller than the outer surface area since they are going like $r$. But the total pressure on one half surface of the plasma due to the magnetic pressure will be $\propto {B}^{2}S$ and the quadratic dependence on the magnetic ﬁeld wins. The force on the inside of the plasma is thus greater than on the outside, and this eﬀect creates a net ‘Hoop’ force directed outwards. The name comes from the fact that this is analogous to the tension in a circular current-carrying wire loop.

7.3.3 Toroidal Force Balance - Tire Tube

The next situation to consider is analogous to the pressure exerted on the walls of a tire. Consider that the plasma surface is an isobar. In that case the force exerted on the surface of the plasma can be thought of as $pS$.Since ${S}_{2}>{S}_{1}$ and ${p}_{1}={p}_{2}$ by assumption this creates a net force, which is also directed outwards. This is shown schematically in Fig. 7.8. This force does not depend on what the magnetic ﬁeld is doing, and thus will occur in toroidal plasmas of all types (Z-pinch, $𝜃$-pinch, Screw-pinch).

7.3.4 Toroidal Force Balance - 1/R

The next toroidal force balance problem comes from the $1∕R$ dependence of the toroidal ﬁeld. Because this requires a toroidal ﬁeld, it occurs in the $𝜃$-pinch or screw-pinch problems only. This situation is shown in Fig. 7.9. The coils carry a current ${I}_{c}$ and the plasma has an induced opposite current ${I}_{p}$ since it is diamagnetic. We assume that the plasma current is carried in an inﬁnitesimal sheath on the outer surface.

We remember Ampère’s Law in integral form, which we can apply over a toroidal loop:

 $\oint \stackrel{\to }{B}\cdot \stackrel{\to }{d\ell }=2\pi r{B}_{\varphi }={\mu }_{0}{I}_{enc}.$ (7.15)

Just outside the plasma, the toroidal ﬁeld is:

 ${B}_{\varphi a}=\frac{{\mu }_{0}{I}_{c}}{2\pi r}$ (7.16)

and just inside the inner plasma surface:

 ${B}_{\varphi i}=\frac{{\mu }_{0}\left({I}_{c}-{I}_{p}\right)}{2\pi r}$ (7.17)

The net force on a plasma surface is given by:

 $F=\frac{\left({B}_{\varphi a}-{B}_{\varphi i}\right){S}_{2}}{2{\mu }_{0}}$ (7.18)

depending on the jump in ﬁeld and the surface area.

We now need to consider the force on the inside of the plasma relative to the outside. Inside, $1∕R$ is smaller and the surface area $S$ is also smaller, but because of the quadratic dependence on $B$ and the fact that $B\propto 1∕R$ the total force on the inside of the plasma is actually greater than on the outside. This is analogous to the hoop force. The net result is that the $1∕R$ eﬀect described here results in an outward directed net force on the plasma.

7.3.5 Toroidal Force Balance - Wall Stabilization

No matter which type of pinch we choose, the above eﬀects all act to push the plasma radially outwards in a toroidal machine or tokamak. Obviously this needs to be avoided in a conﬁnement or fusion machine, since the plasma must remain isolated from the ﬁrst wall. The problem of stabilization thus arises.

One way to do this is to consider the eﬀect of the wall. This is shown schematically in Fig. 7.10. In particular, consider a perfectly conducting (superconductor) wall. In this case we know that no magnetic ﬂux can penetrate the wall. The initial situation is shown in the top half of the ﬁgure. The poloidal ﬁeld is greater on the inside edge of the plasma, and due to the eﬀects described in the previous sections we know that there is a net force on the plasma directed radially outwards. After some time, the plasma will be moved closer to the wall (bottom half of the ﬁgure). Since magnetic ﬂux cannot penetrate a superconductor, the ﬂux piles up on the outside of the plasma, increasing the strength of the poloidal magnetic ﬁeld until equilibrium is achieved since this counteracts the outwards force.

This seems like an excellent stabilization property. Of course, the problem is that a superconducting ﬁrst wall is technologically infeasible. In a real machine the ﬁrst wall will be resistive. In this case the magnetic ﬂux can penetrate the wall and resistively dissipate. The wall eﬀect might slow down the outwards drift of the plasma, but it will not indeﬁnitely stop it and provide stable conﬁnement.

7.3.6 Toroidal Force Balance - Vertical $B$ Stabilization

We note that many of the above eﬀects depend on the fact that the poloidal ﬁeld is stronger on the inside edge of the plasma. This is shown schematically in Fig. 7.11. A simple way to counteract this is to add a constant vertical magnetic ﬁeld, shown below the schematic. This adds to the poloidal ﬁeld on the plane and tends to increase the ﬁeld strength on the outer part of the plasma while decreasing it on the inner edge. This counteracts the toroidal force eﬀects and can negate the radial force. If we need to counteract eﬀects like the tire tube force, which do not depend on a poloidal ﬁeld, the vertical ﬁeld could be further increased until the ﬁeld on the outer edge is actually greater than the ﬁeld on the inner edge, thus providing conﬁnement.

This technique is used in many real machines and is part of the design for ITER.

7.4 Heating

This section gives a brief overview of potential heating mechanisms in tokamak systems. We recall that the Lawson criterion requires we heat the plasma to make keV before fusion $\alpha$ self-heating can take over.

7.4.1 Ohmic Heating

We have already seen that for several reasons we will need to have a toroidal current in the plasma. Since the plasma has ﬁnite resistivity, the toroidal current will provide some Ohmic heating:

 $P=IV={I}^{2}R$ (7.19)

the power is proportional to the square of the toroidal current and the plasma resistivity. This looks great but we recall that the plasma resistivity is $\propto {T}^{-3∕2}$. This means that Ohmic heating becomes less eﬃcient as the plasma is heated. A detailed analysis reveals that Ohmic heating is great for initial heating of the plasma but cannot get a plasma to fusion-relevant temperatures on its own.

For initial start-up, most machines use a transformer scheme (with the plasma as the secondary winding) to drive current inductively for Ohmic heating.

7.4.2 Neutral Beams

Another method for heating a plasma is to use neutral beams of energetic ions. For example, one could use an accelerator to generate beams of energetic D which impinge on the plasma. As the atoms hit the plasma they are ionized, and then slow quickly via Coulomb collisions, transferring their energy to both plasma species. The required ion energy is set by the temperature $E\gg {k}_{B}T$ and also by the areal density of the tokamak plasma - ideally the ions stop near the center of the plasma.

Neutral beams are required so that the particles do not undergo cyclotron motion until they are collisionally ionized within the plasma itself. The scheme to create a neutral beam is a typical electrostatic ion accelerator plus a beam neutralizer.

In current machines, beam energies are typically of order 100 keV. A reactor-scale machine would require energies of order 1 MeV. Neutral beam heating is expected to play a part in any magnetic fusion machine.

7.4.3 RF Heating

Another very common scheme is to use high-frequency electromagnetic waves to heat the plasma. Generally the wave frequency is chosen to match either the electron or ion cyclotron resonance. We know that the electron cyclotron frequency rule-of-thumb is ${f}_{ce}\sim 28$ GHz/T. So for a toroidal ﬁeld in the machine of $10$ T, we must produce radiation at $280$ GHz which is technically challenging to produce that radiation. Ion cyclotron heating, on the other hand, is at much lower frequency ${f}_{ci}\sim 14∕A$ MHz/T. So in a 10 T ﬁeld with $A=2$ (D resonance), the ICH heating system must produce $70$ MHz radiation. The absorption is collisionless so these processes are still eﬃcient at high temperature. This is simple, but it turns out that the launching antenna must be very close to the plasma surface which is technically challenging.

ECH and/or ICH is expected to play a role in any fusion machine.

7.4.4 Lower Hybrid Current Drive

A related question is how we drive current in a tokamak. In the Ohmic heating section we discussed the use of a pulsed transformer to induce large toroidal currents which can Ohmically heat the plasma early in the discharge. However, we must have a scheme to drive steady-state toroidal currents in a fusion machine. Neutral beams and ECH/ICH can be used to do this but it turns out the eﬃciency is low. The problem of ‘current drive’ is an area of investigation. One way to do this is to launch waves at the lower-hybrid resonance such that the $\stackrel{\to }{k}$ of the radiation is toroidally directed. It turns out that these waves tend to collisionlessly Landau damp on the electrons, which drives a toroidal current.

Detailed analysis shows that this is a relatively eﬃcient way to induce a steady-state toroidal current but it cannot provide quite enough. A real fusion machine will depend on a large amount of ‘bootstrap’ current for steady-state operation.

7.5 Banana Orbits

We ﬁnish this section with a brief qualitative description of an eﬀect of toroidal geometry on single-particle motion.

We recall that the ﬁeld strength exhibits a $1∕R$ dependence, which is shown schematically at the top of Fig. 7.12. Consider a particle which is conﬁned by cyclotron motion to one line of force. Because a tokamak is a screw-pinch style conﬁnement scheme, as we have argued previously, the lines of force wrap around the plasma. If we project a single line of force into the poloidal plane, though, it will be circular. This is shown by the dashed line in Fig. 7.12. Naïvely, one might expect that the particles will simply gyrate around the line of force all the way around. However, as the particle moves from the outside of the plasma towards the inside of the machine the magnetic ﬁeld strength is increasing. If the particle has non-zero transverse momentum, then it will exhibit a mirror machine style conﬁnement and reﬂection. Recall Section 2.8, where we derived that:

 ${sin}^{2}{𝜃}_{c}=\frac{{B}_{min}}{{B}_{max}}$ (7.20)

any particles at the location of ${B}_{min}$ with pitch angle greater than ${𝜃}_{c}$ will be conﬁned to a banana-style orbit as shown in green.

Of course particles have a distribution (typically Maxwellian) and thus there is a wide range of banana orbits. This is shown very schematically in Fig. 7.13. The green orbit is a highly conﬁned particle, i.e. with high pitch angle. The blue orbit is moderately conﬁned, and the red orbit is barely conﬁned. Of course there are also unconﬁned orbits, which transit the entire machine.

In a real machine, the banana orbits will aﬀect transport properties in the plasma. We also note that, depending on the initial direction of the velocity, the banana orbits are qualitatively diﬀerent. See Fig. 7.14. This can also aﬀect transport properties.

Chapter 8ICF

In this chapter we derive a few scattered plasma physics topics related to inertial conﬁnement fusion.

8.1 Transport

For simplicity we begin with the cross section for emission of a photon with energy $h\nu$ by an electron with velocity ${v}_{e}$ interacting with an ion via the Coulomb force. This is the Kramers cross section:

 $\frac{d\sigma }{d\nu }=\frac{32{\pi }^{2}}{3\sqrt{3}}\frac{{Z}^{2}{e}^{6}}{{m}_{e}^{2}{c}^{3}{v}_{e}^{2}}\frac{1}{h\nu }.$ (8.1)

The next task is to calculate the spectral emissivity per unit mass. To do this we have to integrate the Kramers cross section over the electron distribution:

 ${\eta }_{\nu }=\frac{h\nu }{4\pi A{m}_{p}}{\int }_{{v}_{min}}^{\infty }{d}^{3}vf\left(v\right)v\frac{d\sigma }{d\nu },$ (8.2)

where the lower limit of integration is the minimum electron velocity needed to create a photon of energy $h\nu$, given by $h\nu ={m}_{e}{v}_{min}^{2}∕2$. Using the Kramers cross section together with the Maxwellian distribution:

 $f\left(v\right)=\sqrt{\frac{2}{\pi }}{\left(\frac{m}{kT}\right)}^{3∕2}{e}^{-{m}_{e}{v}^{2}∕2{k}_{B}T},$ (8.3)

we can write the spectral emissivity as

 ${\eta }_{\nu }=\frac{h\nu }{4\pi A{m}_{p}}\frac{32{\pi }^{2}}{3\sqrt{3}}\frac{{Z}^{2}{e}^{6}}{{m}_{e}^{2}{c}^{3}}\frac{1}{h\nu }\sqrt{\frac{2}{\pi }}{\left(\frac{m}{kT}\right)}^{3∕2}{\int }_{{v}_{min}}^{\infty }dv\phantom{\rule{1em}{0ex}}vexp\left[-{m}_{e}{v}^{2}∕2{k}_{B}T\right]$ (8.4)

after doing the integral, only the lower limit remains and we substitute $h\nu ={m}_{e}{v}_{min}^{2}∕2$. Also with some algebraic cleanup, the emissivity expression becomes:

 ${\eta }_{\nu }=\frac{16\sqrt{\pi }}{3\sqrt{6}}\frac{{e}^{6}}{{m}_{e}^{2}{c}^{3}}\frac{{Z}^{2}{n}_{e}}{A{m}_{p}\sqrt{{k}_{B}{T}_{e}∕{m}_{e}}}exp\left[-\frac{h\nu }{{k}_{B}{T}_{e}}\right]$ (8.5)

The total power radiated into $4\pi$ is obtained by integrating $\eta$ over all frequencies:

 ${P}_{br}=4\pi {\int }_{0}^{\infty }{\eta }_{\nu }d\nu ,$ (8.6)

 ${P}_{br}=\frac{32\sqrt{\pi }}{3\sqrt{6}}\frac{{e}^{6}}{{m}_{e}\hslash {c}^{2}}\sqrt{\frac{{k}_{B}{T}_{e}}{{m}_{e}{c}^{2}}}\frac{{Z}^{3}\rho }{{\left(A{m}_{p}\right)}^{2}}$ (8.7)

We note that ignoring the constants gives a proportionality:

 ${P}_{br}\propto \sqrt{{T}_{e}}{Z}^{3}\rho ∕{A}^{2}$ (8.8)

8.1.2 Electron Heat Conduction

In many ICF-relevant situations the electron heat conduction is critically important for the implosion dynamics. The classical heat conductivity is derived from the Fokker-Planck equation:

 $\frac{\partial f}{\partial t}+\stackrel{\to }{v}\cdot \nabla f-\frac{e\stackrel{\to }{E}}{m}\cdot {\nabla }_{v}f=A{\nabla }_{v}\cdot \left(\frac{{\nabla }_{v}f}{v}-\frac{\stackrel{\to }{v}\left(\stackrel{\to }{v}\cdot {\nabla }_{v}f\right)}{{v}^{3}}\right)+{C}_{ee}\left(f\right)$ (8.9)

where we have introduced $A=\left(2\pi nZ{e}^{4}∕{m}^{2}\right)ln{\Lambda }_{e}$, with $ln{\Lambda }_{e}$ the Coulomb log for electron-ion collisions. ${C}_{ee}$ represents a formal treatment of electron-electron collisions, which are neglected in this derivation.

Here will will assuming a temperature gradient in the $\stackrel{̂}{x}$ direction, and consider a perturbative approach in which the distribution function becomes:

 $f\left(v,\mu \right)={f}_{0}\left(v\right)+\mu {f}_{1}\left(v\right)$ (8.10)

where ${f}_{0}$ is the isotropic equilibrium Maxwellian. $\mu =cos𝜃$ where $𝜃$ is the pitch angle between the electron velocity and the temperature gradient. We are also implicitly assuming that the electron mean free path is much less than the gradient length scale:

 $L\equiv {\left|dlnT∕dx\right|}^{-1}\gg {\lambda }_{e}.$ (8.11)

The 0-th order distribution function is written:

 ${f}_{0}\left(v\right)=\frac{n}{{\left(2\pi \right)}^{3∕2}{v}_{th}^{3}}exp\left(-{v}^{2}∕2{v}_{th}^{2}\right)$ (8.12)

If we insert the perturbed distribution function $f\left(v,\mu \right)$ into the Fokker-Planck equation and keep only terms that are ﬁrst order in $\mu$, we get:

 $\frac{\partial {f}_{1}}{\partial t}+v\frac{\partial {f}_{0}}{\partial x}-\frac{eE}{m}\frac{\partial {f}_{0}}{\partial v}=-\frac{2A}{{v}^{3}}{f}_{1}$ (8.13)

rearranging gives

 ${f}_{1}=-\frac{{v}^{4}}{2A}\left[\frac{\partial {f}_{0}}{\partial x}-\frac{eE}{mv}\frac{\partial {f}_{0}}{\partial v}\right]$ (8.14)

We need another equation at this point. We use the plasma property of quasi-neutrality to require that the current due to ${f}_{1}$ vanishes in steady state, or:

 ${j}_{x}=0=-e\stackrel{̂}{x}\cdot \int \stackrel{\to }{dv}\mu \stackrel{\to }{v}\mu {f}_{1}$ (8.15)

using the previous expression for ${f}_{0}$ and ${f}_{1}$ , glossing over some algebra, yields:

 $eE=-\frac{5}{2}{k}_{B}\frac{dT}{dx}$ (8.16)

It is an interesting aside that we have shown here that a gradient in temperature generates an electric ﬁeld. Anyways, this can be replaced in the equation for ${f}_{1}$ which depends only on derivatives of ${f}_{0}$ and can be shown to be:

 ${f}_{1}\left(v\right)=-{f}_{0}\left(v\right)\frac{{v}_{th}^{4}}{4A}\frac{dlnT}{dx}{\left(\frac{v}{{v}_{th}}\right)}^{4}\left[\frac{{v}^{2}}{{v}_{th}^{2}}-8\right]$ (8.17)

It follows from this that the heat ﬂux is:

 $q=\int {d}^{3}v\left(m{v}^{2}∕2\right)\mu v\mu {f}_{1}\left(v\right)=-\chi \frac{dT}{dx}$ (8.18)

where we have deﬁned the thermal conductivity:

 ${\chi }_{S}={\left(\frac{8}{\pi }\right)}^{3∕2}G\left(Z\right)\frac{{\left({k}_{B}T\right)}^{5∕2}{k}_{B}}{Z{e}^{4}\sqrt{m}ln\Lambda }$ (8.19)

where the function $G\left(Z\right)$ takes into account eﬀects of electron collisions, which are more important in low-Z plasmas. We note:

 ${\chi }_{S}\propto \frac{G\left(Z\right)}{Z}\frac{{T}^{5∕2}}{ln\Lambda }$ (8.20)

The conductivity is primarily a function of temperature.

A detailed look at the velocity integral reveals that electrons with ${v}_{e}\sim 3.7{v}_{th}$ end up doing the brunt of the work in conducting thermal energy. It is therefore important to revisit our assumption that ${f}_{1}$ was a small perturbation to ${f}_{0}$. If we write

 $\frac{|{f}_{1}|}{{f}_{0}}\approx \frac{10q}{{v}_{th}n{k}_{B}T}\le 1$ (8.21)

the real heat ﬂux must satisfy this inequality. We can deﬁne a free-streaming ﬂux:

 ${q}_{F}\equiv {v}_{th}n{k}_{B}T$ (8.22)

physically, this represents the entire thermal energy of the plasma $\sim n{k}_{B}T$ moving with the thermal velocity, and represents an upper limit on ﬂux. The real conductivity thus must satisfy:

 $\frac{q}{{q}_{F}}\le 0.1$ (8.23)

The heat ﬂux cannot exceed $10%$ of the free-streaming limit. Generally in hydrodynamics simulations one takes:

 $q=\mathrm{\text{min}}\left({q}_{S},{f}_{L}{q}_{F}\right)$ (8.24)

which is the minimum of the Spitzer conductivity and the free-streaming ﬂux limit multiplied by a ﬂux limiter which is generally taken in the range $0.03-0.1$.

Detailed Fokker-Planck simulations reveal that the Spitzer conductivity is accurate when $L∕{\lambda }_{e}\ge 1{0}^{2}$, i.e. in the long gradient length limit. As the gradient length scale is decreased, ﬂux-limited conduction must be used. This is particularly important in calculations of direct-drive ICF targets, where the thermal conduction between the critical surface and ablation front is fundamental to the entire dynamics of the problem, and the temperature gradient is very steep between the hot plasma at the critical surface and cold plasma at the ablation front.

8.2 Laser-Plasma Interactions

8.2.1 Critical Density

In Section 4.2 we derived the dispersion relation for electromagnetic waves in unmagnetized plasmas, which we reproduce here:

 ${\omega }^{2}={\omega }_{pe}^{2}+{c}^{2}{k}^{2}$ (8.25)

We can immediately see that there is a cutoﬀ ($k=0$, no propagation) when the wave frequency is equal to the electron plasma frequency. Since the latter depends on the plasma density, this presents a limit on density:

 ${\omega }^{2}={\omega }_{pe}^{2}=\frac{{n}_{c}{e}^{2}}{{𝜖}_{0}{m}_{e}}$ (8.26)

rearranging:

 ${n}_{c}=\frac{{\omega }^{2}{𝜖}_{0}{m}_{e}}{{e}^{2}}$ (8.27)

where ${n}_{c}$ is the critical density. Electromagnetic waves can only propagate for densities $n<{n}_{c}$.

8.2.2 Inverse Bremsstrahlung

If we consider a laser beam propagating from low-density plasma towards a higher density region ($n>{n}_{c}$) at some point the laser energy must be either scattered or absorbed in the plasma. The best situation for ICF is when the laser wave is absorbed collisionally, or more colloquially via the inverse bremsstrahlung mechanism.

If we treat the laser electric ﬁeld as:

 $\stackrel{\to }{E}\left(\stackrel{\to }{r},t\right)=\stackrel{\to }{E}\left(\stackrel{\to }{r}\right){e}^{-i\omega t}$ (8.28)

and write the electron momentum equation as:

 ${m}_{e}{n}_{e}\left[\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}+{\stackrel{\to }{u}}_{e}\cdot \left(\nabla \cdot \stackrel{\to }{u}\right)\right]={n}_{e}{q}_{e}\left(\stackrel{\to }{E}+{\stackrel{\to }{u}}_{e}×\stackrel{\to }{B}\right)-{m}_{e}{n}_{e}{\nu }_{ei}{\stackrel{\to }{u}}_{e}$ (8.29)

In previous derivations of EM waves in plasmas we neglected collisions, i.e. we had assumed ${\nu }_{ei}=0$. In this problem we keep ﬁnite collision frequency, since that is the eﬀect we are looking for. If we assume that ${\stackrel{\to }{u}}_{0}={\stackrel{\to }{E}}_{0}={\stackrel{\to }{B}}_{0}=0$ and keep only ﬁrst-order terms:

 ${m}_{e}{n}_{e}\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}=-{n}_{e}e\stackrel{\to }{E}-{m}_{e}{n}_{e}{\nu }_{ei}{\stackrel{\to }{u}}_{e},$ (8.30)

 $\frac{\partial {\stackrel{\to }{u}}_{e}}{\partial t}=-\frac{e}{{m}_{e}}\stackrel{\to }{E}-{\nu }_{ei}{\stackrel{\to }{u}}_{e}.$ (8.31)

Where we have implicitly assumed that the ion inertia is inﬁnite over the timescales of this process. Since the ﬁeld is harmonic, we can use Fourier analysis on this equation:

 $-i\omega {\stackrel{\to }{u}}_{e}=-\frac{e}{{m}_{e}}\stackrel{\to }{E}-{\nu }_{ei}{\stackrel{\to }{u}}_{e}$ (8.32)

we can rearrange this equation to acquire the plasma response:

 ${\stackrel{\to }{u}}_{e}=\frac{-ie\stackrel{\to }{E}}{{m}_{e}\left(\omega +i{\nu }_{ei}\right)}$ (8.33)

 $\stackrel{\to }{j}=-{n}_{e}e{\stackrel{\to }{u}}_{e}=\frac{i{𝜖}_{0}{\omega }_{pe}^{2}}{\omega +i{\nu }_{ei}}\stackrel{\to }{E}$ (8.34)

We now go back to Maxwell’s equations for a moment:

$\begin{array}{rcll}\nabla ×\stackrel{\to }{E}& =& -\frac{\partial \stackrel{\to }{B}}{\partial t}& \text{(8.35)}\text{}\text{}\\ \nabla ×\stackrel{\to }{B}& =& {\mu }_{0}\stackrel{\to }{j}+\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{E}}{\partial t}& \text{(8.36)}\text{}\text{}\end{array}$

Combining these and using Fourier analysis, we get that:

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)={\mu }_{0}\frac{\partial }{\partial t}\stackrel{\to }{j}+\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{E}}{\partial {t}^{2}}$ (8.37)

Using the vector identity for the left hand side:

 $\nabla ×\left(\nabla ×\stackrel{\to }{E}\right)={\nabla }^{2}\stackrel{\to }{E}-\nabla \left(\nabla \cdot \stackrel{\to }{E}\right),$ (8.38)

and using Fourier analysis we get that

 $-{k}^{2}\stackrel{\to }{E}=-\frac{{\omega }^{2}}{{c}^{2}}-i\omega {\mu }_{0}\stackrel{\to }{j}$ (8.39)

Now we need to substitute into this equation for the current to get:

 ${\omega }^{2}={c}^{2}{k}^{2}+{\omega }_{pe}^{2}{\left(1+i{\nu }_{ei}∕\omega \right)}^{-1}$ (8.40)

If the damping is weak, we can use the binomial approximation:

 ${\omega }^{2}={c}^{2}{k}^{2}+{\omega }_{pe}^{2}\left(1-\frac{i{\nu }_{ei}}{\omega }\right)$ (8.41)

Clearly we now have wave damping via the imaginary part of the dispersion relation. If we take $k$ as imaginary with $k={k}_{r}+i{k}_{i}∕2$ with weak damping (i.e. small ${k}_{i}$), we can isolate the imaginary part of the above:

 $0={c}^{2}{k}_{r}{k}_{i}-{\omega }_{pe}^{2}{\nu }_{ei}∕\omega$ (8.42)

From the undamped dispersion relation we can approximate:

 ${k}_{r}=\frac{1}{c}\sqrt{{\omega }^{2}-{\omega }_{pe}^{2}}$ (8.43)

we follow Kruer and write the damping coeﬃcient as:

 ${k}_{i}=\frac{{\omega }_{pe}^{2}}{{\omega }^{2}}\frac{{\nu }_{ei}}{{v}_{g}}$ (8.44)

the energy damping length for inverse bremsstrahlung is simply ${k}_{i}^{-1}$.

8.2.3 Resonance Absorption

We start oﬀ a series of qualitative descriptions of non-collisional absorption processes with resonance absorption. For p-polarized light, the schematic is shown in Fig. 8.1. At the top we show a schematic density proﬁle where the plasma has a gradient from 0 past the critical density, with $\nabla n\parallel ẑ$. Next we show the p-polarized wave propagating in the plasma in green. The cross-hatches represent the electric ﬁeld’s direction. The wave is propagating at an angle $\alpha$ relative to the density gradient. Due to refraction, the wave bends, and reaches a maximum in density at $z=-Lsin\alpha$ where $L$ is the gradient length scale. However, since the electric ﬁeld has a component in the $ẑ$ direction there will be an evanescent component in the $z$ direction. This reaches the surface of critical density, and can resonantly couple to the plasma there. The bottom part of Fig. 8.1. The resonant ﬁeld coupling to the plasma at the critical density can excite electron plasma waves, or ‘plasmons’. For more detail see Kruer.

8.2.4 Parametric Instabilities

Now we discuss a few parametric instabilities. The essential process is three-wave coupling:

 ${\omega }_{L}={\omega }_{1}+{\omega }_{2}$ (8.45)

 ${\stackrel{\to }{k}}_{L}={\stackrel{\to }{k}}_{1}+{\stackrel{\to }{k}}_{2}$ (8.46)

An incident laser photon decays into two daughter waves, 1 and 2. These can be several combinations of scattered photons, plasmons (electron plasma waves), and phonons (ion acoustic waves).

Two-Plasmon Decay

First up is the two-plasmon decay, which is shown schematically in Fig. 8.2. The dispersion relation for light waves is shown in green and the dispersion relation for electron plasma waves, ‘plasmons’, is shown in blue. A single incident photon can decay into two plasmons via the frequency and $\stackrel{\to }{k}$ matching conditions. As shown, one plasmon propagates in the direction of the original photon and the other goes backwards. Because the electron plasma wave dispersion relation is so much ﬂatter than the light wave, by a ratio $\sim {v}_{th}∕c$, we can approximate ${\omega }_{pe}\approx {\omega }_{L}∕2$. Since ${\omega }_{pe}\propto \sqrt{n}$, this implies that the two-plasmon decay can occur occur very close to a plasma density of ${n}_{c}∕4$,the ‘quarter-critical’.

Stimulated Raman Scattering

The next possibility we consider is called ‘Stimulated Raman Scattering’ (SRS). The schematic for this decay is shown in Fig. 8.3. Once again the green curve shows the light wave dispersion relation, and the blue curve shows the plasmon dispersion relation. Once incident photon, labelled ‘inc’, decays into a scattered photon and a plasmon. The situation shown is a backscattered photon, but this is not necessarily the case. SRS-scattered photons can go forwards, sideways, etc. The maximum density at which SRS can occur is quarter-critical. However, it can also occur at much lower densities, and is thus distinct from TPD.

Stimulated Brillouin Scattering

The last parametric decay instability is Stimulated Brillouin Scattering (SBS). Once again, this is shown schematically in Fig. 8.4. Yet again green denotes light waves. SBS, consists of an incident laser photon decaying into a backscattered photon and an Ion Acoustic Wave (IAW), or ‘phonon’. These are shown in red. Because of $\omega \propto k$ for IAWs, and they are low frequency, the wave matching conditions require that the scattered photon is backscattered and the phonon goes nearly in the same direction as the original wave to conserve momentum ${k}_{i}\sim 2{k}_{L}$. The SBS process can occur anywhere in under-dense plasma.

8.2.5 Hot Electrons

It is important to have a brief discussion of hot electrons. In ICF implosions, it is important to keep the fuel adiabat low so that it is easily compressible to very high densities during the implosion. This means that any unintended heating of the fuel early on in the target illumination is undesirable, since it will raise the adiabat. The relevance here is that many of these processes can generate very high-energy electrons. Any time a laser-plasma instability generates plasmons, or electron plasma waves, those waves can Landau damp on the plasma electrons. When the wave is very strong, there is a non-linear wave-breaking process which eﬃciently couples wave energy to particle energy and generates a two-temperature electron distribution. The hot electrons can have a temperature of many tens of keV. These electrons are energetic enough to penetrate the fuel layer in an implosion target and preheat it. It is therefore important to understand and control LPI, in addition to controlling LPI to increase laser coupling to the target.

8.3 Misc

We now discuss a few totally random but important topics of relevance to ICF.

8.3.1 EM Field Generation

Field generation in plasmas is an important topic, particularly for spontaneous generation of ﬁelds in ICF. If we recall Ohm’s law from the discussion of magnetohydrodynamics, there is a pressure gradient term. This work will follow that of Haines. To ﬁrst order we can write from Ohm’s Law:

 $\stackrel{\to }{E}=-\frac{\nabla {p}_{e}}{{n}_{e}e}$ (8.47)

Physically, if there is a pressure gradient in the plasma the electrons quickly leave until there is an electrostatic ﬁeld set up to mitigate this. See also the previous discussion of ambipolar diﬀusion. Other terms in the generalized Ohm’s law are omitted for clarity. This is also ignoring any ﬁelds from incident lasers in an ICF scheme. Using Faraday’s law we can get the magnetic ﬁeld:

 $\frac{\partial \stackrel{\to }{B}}{\partial t}=-\nabla ×\stackrel{\to }{E}$ (8.48)

and using $p=n{k}_{B}T$,

 $\frac{\partial \stackrel{\to }{B}}{\partial t}=\nabla ×\left(\frac{\nabla {p}_{e}}{{n}_{e}e}\right)=\frac{{k}_{B}\nabla {T}_{e}×\nabla {n}_{e}}{{n}_{e}e}$ (8.49)

This is the most basic mechanism for spontaneous magnetic ﬁeld generation. If the temperature and density gradients are not parallel in a plasma, then a magnetic ﬁeld is generated. This is also known as the Biermann Battery.

8.3.2 Rayleigh-Taylor Instability

In a situation where a heavy ﬂuid is supported against a gravitational potential by a lighter ﬂuid, then it is clearly energetically and thus dynamically favorable to exchange material of the heavier ﬂuid for the lighter one. Doing this from ﬁrst principles is non-trivial. However, if we consider two homogenous ﬂuids with an interface at $z=0$ with densities ${\rho }_{1}$ and ${\rho }_{2}$. Stealing a result from Drake, the equation of motion for perturbations of this interface is

 $\frac{\partial }{\partial z}\left[-\rho n\frac{\partial w}{\partial z}\right]={k}^{2}\left[-\rho nw+w\frac{g}{n}\frac{\partial \rho }{\partial z}\right]$ (8.50)

We take the perturbation in each ﬂuid as:

$\begin{array}{rcll}{w}_{1}& =& {w}_{0}{e}^{kz}& \text{(8.51)}\text{}\text{}\\ {w}_{2}& =& {w}_{0}{e}^{-kz}& \text{(8.52)}\text{}\text{}\end{array}$

plugging into the ﬁrst equation, we get that:

 ${w}_{0}\frac{g}{n}\left({\rho }_{2}-{\rho }_{1}\right)=\frac{n}{{k}^{2}}\left({\rho }_{2}+{\rho }_{1}\right)k{w}_{0}$ (8.53)

and thus, the growth rate, which we (Drake) have called ${n}_{0}$, is given by:

 ${n}_{0}=\sqrt{\frac{{\rho }_{2}-{\rho }_{1}}{{\rho }_{2}+{\rho }_{1}}kg}=\sqrt{{A}_{n}kg}$ (8.54)

where $g$ is the gravitational acceleration, $k$ is the wave number, and ${A}_{n}$ is the Atwood Number:

 ${A}_{n}\equiv \frac{{\rho }_{2}-{\rho }_{1}}{{\rho }_{2}+{\rho }_{1}}$ (8.55)

Obvious ${A}_{n}$ is the stable situation, and larger ${A}_{n}$ correspond to larger diﬀerences in density across the interface and thus higher growth rates.

Chapter 9Back of the Envelope

In this chapter we simplify a few of the previously-derived relations to ‘back of the envelope’ or Formulary-style equations which can be used to quickly calculate important plasma properties. The NRL Plasma Formulary (available online) is also an excellent resource for this sort of calculation.

9.1 Plasma Frequency

We start oﬀ with the plasma frequency.

9.1.1 Electron

Way back in Section 1.3 we derived the general result for the electron plasma frequency:

 ${\omega }_{pe}=\sqrt{\frac{n{e}^{2}}{{𝜖}_{0}{m}_{e}}}\phantom{\rule{1em}{0ex}}\left(mks\right)\phantom{\rule{1em}{0ex}}=\sqrt{\frac{4\pi n{e}^{2}}{{m}_{e}}}\phantom{\rule{1em}{0ex}}\left(cgs\right)$ (9.1)

where the ﬁrst expression is in the MKS system of units (generally used in this work) and the second is the CGS system (commonly used in older physics materials, and also in ICF). We also note that often $f$ is a more desired quantity than $\omega$, e.g. if one wishes to compare to a source of electromagnetic radiation. We can therefore generate Table 9.1.

Table 9.1: Electron Plasma Frequencies

 MKS CGS ${\omega }_{pe}$ $56\sqrt{n}$ $5.6E4\sqrt{n}$ ${f}_{pe}$ $9\sqrt{n}$ $9E3\sqrt{n}$

9.1.2 Ion

The ion plasma frequency is not as widely used, however it is useful to note that

 ${\omega }_{p}\propto \frac{Z}{\sqrt{m}}$ (9.2)

and therefore we can write the ion plasma frequency in terms of the electron values:

 ${\omega }_{pi}=\frac{1}{42}\frac{Z}{\sqrt{A}}\phantom{\rule{1em}{0ex}}{\omega }_{pe}$ (9.3)

where the fraction out front comes from the square root of the ion to electron mass ratio.

9.2 Debye Length

Recall the result of Section 1.2:

 ${\lambda }_{De}=\sqrt{\frac{{𝜖}_{0}{k}_{B}T}{n{e}^{2}}}\phantom{\rule{1em}{0ex}}\left(MKS\right)\phantom{\rule{1em}{0ex}}=\sqrt{\frac{{k}_{B}T}{4\pi n{e}^{2}}}\phantom{\rule{1em}{0ex}}\left(CGS\right)\phantom{\rule{1em}{0ex}}$ (9.4)

In addition to the constants, we note that the Debye length depends on both temperature and density. Once again, we can give the result in both systems of units for convenience, though in both equations we absorb Boltzmann’s constant into the temperature and take $\left[T\right]=$eV:

 ${\lambda }_{De}=7430\phantom{\rule{1em}{0ex}}\sqrt{T∕n}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(MKS\right)$ (9.5)

 ${\lambda }_{De}=743\phantom{\rule{1em}{0ex}}\sqrt{T∕n}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(CGS\right)$ (9.6)

9.3 Cyclotron Motion

In this section, refer back to Section 2.1 for derivations. In this section, for convenience, we always take $\left[B\right]=$T.

We recall the result, in MKS units:

 ${r}_{L}=\frac{{v}_{\perp }}{{\omega }_{c}}=\frac{m{v}_{\perp }}{q{B}_{z}}$ (9.7)

 ${r}_{Le}=2.4\sqrt{{T}_{e}}∕B\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm}}$ (9.8)

 ${r}_{Li}=100\sqrt{A{T}_{e}}∕\left(ZB\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm}}$ (9.9)

where the results are given in cm, but the conversion to meters is trivial since we have used ‘convenient’ units for both ${T}_{e}$ and $B$.

9.3.2 Cyclotron Frequencies

From the previous results we can immediately write that:

 ${\omega }_{c}=\frac{qB}{m}$ (9.10)

The cyclotron frequency only depends on the particle type and the magnetic ﬁeld. By far the easiest values to remember are:

 ${f}_{ce}=28\phantom{\rule{1em}{0ex}}\mathrm{\text{GHz}}∕\mathrm{\text{T}}$ (9.11)

and

 ${f}_{ci}=14\left(Z∕A\right)\phantom{\rule{1em}{0ex}}\mathrm{\text{MHz}}∕\mathrm{\text{T}}$ (9.12)

one simply has to multiply by the magnetic ﬁeld strength in Tesla to get the gyro frequencies.

9.4 Collision Frequencies

Here we will use the results of 5.8.

 ${\nu }_{ei}=\left(\frac{{e}^{4}}{4\pi {𝜖}_{0}^{2}{m}_{e}^{2}}\frac{{n}_{i}}{{T}_{i}^{3∕2}}ln\Lambda \right)\frac{1}{{v}_{e}^{3}}$ (9.13)

For a thermal distribution, this expression simpliﬁes to:

 ${\nu }_{ei}=\left(3×1{0}^{-6}\right)\frac{{n}_{e}ln\Lambda }{{T}_{e}^{3∕2}}\phantom{\rule{1em}{0ex}}{\mathrm{\text{s}}}^{-1}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(CGS\right)$ (9.14)

 ${\nu }_{ei}=\left(3×1{0}^{-12}\right)\frac{{n}_{e}ln\Lambda }{{T}_{e}^{3∕2}}\phantom{\rule{1em}{0ex}}{\mathrm{\text{s}}}^{-1}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(MKS\right)$ (9.15)

where of course $\left[\nu \right]={\mathrm{\text{s}}}^{-1}$ and the only diﬀerent between systems of units is $\left[n\right]$.

Also recall the scaling result:

$\begin{array}{rcll}{\nu }_{ee}& \sim & {\nu }_{ei}& \text{(9.16)}\text{}\text{}\\ {\nu }_{ii}& \sim & \sqrt{\mu }{\nu }_{ei}& \text{(9.17)}\text{}\text{}\\ {\nu }_{ie}& \sim & \mu {\nu }_{ei}& \text{(9.18)}\text{}\text{}\end{array}$

where $\mu ={m}_{e}∕{m}_{i}$, so based on the species mass ratio, and there is an assumption buried in here that we are working in a $Z=1$ plasma. For non-hydrogenic ions, the ion collision rate is enhanced by a factor of ${Z}^{4}$.

9.5 Thermal Velocities

For a Maxwellian distribution function, we know from basic physics that the mean thermal velocity is:

 ${v}_{th}=\sqrt{2{k}_{B}T∕m}$ (9.19)

This leads to the simple expressions:

 ${v}_{Te}=\left(4.2×1{0}^{7}\right)\sqrt{{T}_{e}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm/s}}$ (9.20)

 ${v}_{Ti}=\left(9.8×1{0}^{5}\right)\sqrt{{T}_{i}∕A}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm/s}}$ (9.21)

which are obviously in CGS units. Again we have used $\left[T\right]=$eV. The extension to MKS is trivial.

9.6 Mean Free Path

We just derived the collision frequencies and thermal velocities. We can simply write that:

 $\lambda =\frac{{v}_{th}}{\nu }$ (9.22)

which leads to the following for electrons:

 ${\lambda }_{e}=\left(4.2×1{0}^{7}\right)\sqrt{{T}_{e}}×{\left[\left(3×1{0}^{-6}\right)\frac{{n}_{e}ln\Lambda }{{T}_{e}^{3∕2}}\right]}^{-1}$ (9.23)

 ${\lambda }_{e}=\left(1.2×1{0}^{13}\right)\frac{{T}_{e}^{2}}{{n}_{e}ln\Lambda }\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm}}$ (9.24)

and for ions:

 ${\lambda }_{i}=\left(9.8×1{0}^{5}\right)\sqrt{{T}_{i}∕A}×{\left[\left(3×1{0}^{-6}\right)\sqrt{\mu }{Z}^{4}\frac{{n}_{i}ln\Lambda }{{T}_{i}^{3∕2}}\right]}^{-1}$ (9.25)

 ${\lambda }_{i}=\left(1.4×1{0}^{13}\right)\frac{{T}_{i}^{2}}{\sqrt{A}{Z}^{4}{n}_{i}ln\Lambda }\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm}}$ (9.26)

all of these expressions are in CGS units with our usual convention of $\left[T\right]=$eV.

9.7 Ion Sound Speed

We use the derivation presented in Sec. 4.6. In particular:

 ${c}_{s}=\sqrt{\frac{{T}_{e}+\gamma {T}_{i}}{{m}_{i}}}.$ (9.27)

for the ion sound speed. We can simplify this somewhat by writing:

 ${c}_{s}=\sqrt{\frac{\gamma {T}_{e}}{{m}_{i}}}.$ (9.28)

where we have to be somewhat careful about what exactly is meant by $\gamma$. Anyways, plugging in some numbers gets us that

 ${c}_{s}\approx \left(9.8×1{0}^{5}\right)\sqrt{Z∕A}\sqrt{\gamma {T}_{e}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{cm/s}}$ (9.29)

which is obviously in CGS units. Once again, $\left[{T}_{e}\right]=$eV.

9.8 $E×B$ drift speed

The drift velocity is:

 $\stackrel{\to }{v}=\frac{\stackrel{\to }{E}×\stackrel{\to }{B}}{{B}^{2}}=\frac{E}{B}$ (9.30)

where we assume the electric and magnetic ﬁelds are perpendicular. A handy formula, with $\left[E\right]=$ V/m and $\left[B\right]=$T is

 $v=1{0}^{4}\frac{E}{B}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\text{m/s}}$ (9.31)

9.9 Plasma Parameter

The plasma parameter is deﬁned as the number of particles in a Debye sphere:

 ${N}_{D}=\frac{4\pi }{3}n{\lambda }_{De}^{3}$ (9.32)

this is useful to know, since for classical plasma behavior we typically require ${N}_{D}\gg 1$. For simple calculations,

 ${N}_{D}\approx \left(1.7×1{0}^{9}\right)\frac{{T}^{3∕2}}{\sqrt{n}}$ (9.33)

in CGS units.

9.10 Plasma Beta

The plasma beta is deﬁned as the ratio of thermal to magnetic pressure in a plasma;

 $\beta \equiv \frac{n{k}_{B}T}{{B}^{2}∕8\pi },$ (9.34)

and is an important parameter for magnetically-conﬁned plasmas. In the CGS system of units:

 $\beta \approx \left(4×1{0}^{-11}\right)\frac{nT}{{B}^{2}}$ (9.35)

where as usual $\left[T\right]=$eV and $\left[B\right]=$T.

9.11 Fermi Energy

From Statistical Mechanics we know that the Fermi Energy is:

 ${E}_{F}=\frac{{\hslash }^{2}{\left(3{\pi }^{2}\right)}^{\left(}2∕3}{\right)}\phantom{\rule{1em}{0ex}}{n}^{2∕3}$ (9.36)

in CGS units, this reduces to:

 ${E}_{F}\approx \left(3.6×1{0}^{-15}\right)\phantom{\rule{1em}{0ex}}{n}^{2∕3}\phantom{\rule{1em}{0ex}}\mathrm{\text{eV}}$ (9.37)

we can also calculate the Fermi pressure:

 ${p}_{F}\equiv \frac{2}{5}n{E}_{f}\approx \left(2.3×1{0}^{-33}\right)\phantom{\rule{1em}{0ex}}{n}^{5∕3}\phantom{\rule{1em}{0ex}}\mathrm{\text{bar}}$ (9.38)

9.12 Plasma Degeneracy

We just discussed the Fermi energy. In some ICF-relevant plasmas, the thermal energy can be comparable to or less than the Fermi energy, in which case we call it a degenerate plasma. It is useful to characterize these plasmas by:

 (9.39)

so, for quick back of the envelope calculations in CGS units,

 $𝜃\approx \left(2.7×1{0}^{14}\right)\phantom{\rule{1em}{0ex}}\frac{{T}_{e}}{{n}^{2∕3}}$ (9.40)

the result, $𝜃$, is a dimensionless quantity.

9.13 Plasma Coupling

In a similar vein, it is useful to think of the plasma coupling. We want to compare the thermal energy of plasma ions to the strength of the coupling between them. We deﬁne:

 (9.41)

so, once again for CGS units,

 $\Gamma =\left(1.4×1{0}^{-7}\right)\phantom{\rule{1em}{0ex}}\frac{{n}^{1∕3}}{T}$ (9.42)

9.14 Critical Density

We previously discussed the fact that the cutoﬀ or critical density is very important for laser-plasma interactions, and thus for ICF. Recall, from section 8.2.1,

 ${n}_{c}=\frac{{\omega }^{2}{𝜖}_{0}{m}_{e}}{{e}^{2}}\phantom{\rule{1em}{0ex}}\left(MKS\right)\phantom{\rule{1em}{0ex}}=\frac{{\omega }^{2}{m}_{e}}{4\pi {e}^{2}}\phantom{\rule{1em}{0ex}}\left(CGS\right)$ (9.43)

It is most useful to write this in terms of the laser wavelength $\lambda =2\pi c∕\omega$. Then, when getting rid of the constants, we get:

 ${n}_{c}=\frac{1.1×1{0}^{21}}{{\lambda }_{\mu m}^{2}}\phantom{\rule{1em}{0ex}}{\mathrm{\text{cm}}}^{-3}$ (9.44)

the result is in CGS, and we have written the wavelength in units of $\mu m$ for convenience. Thus, for a Nd:glass laser at $1\omega$, the critical density is $\sim 1{0}^{21}$. If we frequency triple it, then the critical density is $\sim 1$