# Lecture #5

Introduction to Nonlinear Effects

## Summary of lecture #4

1.  We have used the dispersion relation to describe linear wave
    propagation in plasmas. In a magnetized plasma, the wide variety of
    wave behavior possible is classified in $\omega_{\rho} - \omega_{e}$
    space with the help of the CMA diagram.

2.  We have discussed kinetic corrections to the cold-plasma dielectric
    function that are derived from the Vlasov theory. In particular we
    showed that Landau damping results from the interaction of particles
    traveling near the phase velocity of the wave.

3.  We showed that conservation laws exist for the conservation of
    energy among linear waves, provided the microscopic energy density
    and energy flux of the plasma particles is included.

Overview of Lecture #5

Hierarchy of nonlinear effects.

Three-wave interactions

-   coasting beam experiments.

Quasi-linear diffusion

-   application to beams

Nonlinear Hierarchy of Collective Effects

We have taken a look at the linear oscillations about equilibrium in a
plasma and found a dispersion relation which governs the relationship
between frequency and wavelength in the linear medium.

As the strength of the perturbations is increased, true nonlinear
effects begin to appear. We will consider a hierarchy6 of nonlinear
effects of increasing amplitude.

1.  Wave-wave interactions

> Resonant linear modes satisfying $D(\omega,k) = 0$ can couple to each
> other at sufficiently large amplitude. The wave amplitudes still obey
> conservation laws because the equilibrium distribution function
> remains unchanged.
>
> The primary quantities conserved are the wave energy and wave momentum
> which can be related to the following conditions:
>
> $\begin{array}{r}
> \omega_{0} = \omega_{1} + \omega_{2} \\
> k_{0} = k_{1} + k_{2}
> \end{array}$

2.  Wave-particle interactions

> In this case, sufficient numbers of particles experience acceleration
> through Landau damping of the waves so that the background
> distribution function is altered. This can be the saturation mechanism
> for many unstable waves.
>
> If the waves that produce the particle acceleration are [sufficiently
> random]{.underline}, diffusion in velocity space occurs and the
> phenomenon is known as [quasi-linear diffusion]{.underline}.
>
> ![](media/image1.jpeg){width="3.75in" height="2.96875in"}
>
> Another form of nonlinear wave-particle interaction can occur in which
> two waves produce a beat which can resonate with particles moving with
> velocity v.
>
> $\omega_{1} - \omega_{2} = \underline{k} \cdot \underline{v}$
>
> This effect is called [induced scattering]{.underline}.

3.  Large-amplitude waves

> If the unstable waves are not random in phase space, then well-defined
> stationary waves can sometimes develop, know as
> [solitons]{.underline}.
>
> However, other conditions can exist in which the waves are
> sufficiently large that they overturn (like waves approaching a
> beach), and the much altered evolution of the distribution function
> toward equilibrium is known as [nonlinear Landau damping]{.underline}.
>
> ![](media/image2.jpeg){width="3.75in" height="2.9618055555555554in"}
>
> Still more violent interactions can occur through the interaction of
> solitons, or the crash of a wave, which produces a chaotic spectrum of
> fluctuations known as [turbulence]{.underline}, much like the foam
> that follows the crash of a breaker on the shore.

Equilibrium is finally reached by the dissipation of the fluctuation
power over a broad spectral range.

Spectral scaling laws can be developed which yield the form of the
fluctuation spectrum given a driving mechanism for the instability and a
nonlinear coupling mechanism among waves.

> ![](media/image3.jpeg){width="4.499305555555556in"
> height="3.5541666666666667in"}

Three-wave Interactions

Three-wave coupling is the lowest-order nonlinear coupling among waves
that can occur. In fact, if one of the waves can be considered to be
stationary (constant amplitude), then the argument can be made that it
is not truly nonlinear at all (but is a form of induced scattering).

Three-wave coupling is also known as [parametric coupling]{.underline}
as two waves can be coupled by the periodic variation of some parameter
of the medium, by a third wave.

Any system that exhibits coupling of one frequency to another is
nonlinear to some degree.

![](media/image4.jpeg){width="3.1243055555555554in"
height="2.4680555555555554in"}

We consider the following example to illustrate the basic mechanism of
parametric coupling.

Example: Coupled Pendula

Consider a system of two pendula coupled by a spring with a variable
spring constant.

> ![](media/image5.jpeg){width="3.6243055555555554in"
> height="2.863888888888889in"}
>
> $\Omega_{0}$ is the pump frequency.

The Lagrangian for the system is the following

$L = \frac{m}{2}\left( v_{1}^{2} + v_{2}^{2} \right) - \frac{m}{2}\left( \frac{g}{l_{1}}x_{1}^{2} + \frac{g}{l_{2}}x_{2}^{2} \right) - \frac{k_{0}}{2}\text{cos}\Omega_{0}t\left( x_{1} - x_{2} \right)^{2}$

The frequency of the first pendulum is

$\omega_{1}^{2} \simeq \frac{g}{l_{1}}$

where the change in frequency due to the spring constant is assumed
small and will be neglected. The equation for $\omega_{2}$ is similar.

The system of equations for the motion are then given by

$\begin{array}{r}
{\overset{¨}{x}}_{1} + \omega_{1}^{2}x_{1} = \frac{k_{0}}{m}\text{cos}\Omega_{0}\text{tx}_{2} \\
{\overset{¨}{x}}_{2} + \omega_{2}^{2}x_{2} = \frac{k_{0}}{m}\text{cos}\Omega_{0}\text{tx}_{1}
\end{array}$

Since the parametric coupling between the pendula is assumed to be
small, each pendulum oscillates near its uncoupled frequency, but the
amplitude of the oscillations may vary slowly due to the spring.

So we shall look for solutions of the form

$x_{j} = A_{j}(t)e^{{i\omega}_{j}t} + A_{j}^{}(t)e^{- {i\omega}_{j}t}$

where the coefficients $A_{j}$ are assumed to be slowly varying relative
to the fast time scale of $\omega_{j}$ (note: The division of the
dynamics into [two time scales]{.underline}).

Substituting into the equations of motion, we find

$$\begin{array}{r}
{\overset{¨}{A}}_{} + 2{i\omega}_{}{\overset{˙}{A}}_{} = - \left( {\overset{¨}{A}}_{}^{} - 2{i\omega}_{}{\overset{˙}{A}}_{}^{} \right)e^{- 2{i\omega}_{}t} + \frac{k_{o}}{2m}A_{2}e^{i\left( \Omega_{0} + \omega_{2} - \omega_{} \right)t} + \frac{k_{0}}{2m}A_{2}^{}e^{i\left( \Omega_{0} - \omega_{2} - \omega_{} \right)t} \\
 + \frac{k_{0}}{2m}A_{2}e^{i\left( - \Omega_{0} + \omega_{2} - \omega_{} \right)t} + \frac{k_{0}}{2m}A_{2}^{}e^{i\left( - \Omega_{0} - \omega_{2} - \omega_{} \right)t}
\end{array}$$

To determine the slowly varying coefficients $A_{j}$ we average the
above equations over a period of the fast time scale so that only the
envelope remains.

The exponential terms on the RHS will average to zero unless particular
frequency conditions are met. Suppose we choose

$\Omega_{0} = \omega_{} - \omega_{2}$

and the averaged equations of motion then become

$\begin{array}{r}
{\overset{¨}{A}}_{} + 2{i\omega}_{}{\overset{˙}{A}}_{} = \frac{k_{0}}{2m}A_{2} \\
{\overset{¨}{A}}_{2} + 2{i\omega}_{2}{\overset{˙}{A}}_{2} = \frac{k_{0}}{2m}A_{}
\end{array}$

To find the eigenvalues of these equations, we let

> $A_{} = \text{ae}^{\upsilon t}$ ; $A_{2} = \text{be}^{\upsilon t}$

which leads to the dispersion relation

$\left| \begin{matrix}
\upsilon^{2} + 2{i\omega}_{}\upsilon & \frac{k_{0}}{2m} \\
\frac{k_{0}}{2m} & \upsilon^{2} + 2{i\omega}_{2}\upsilon
\end{matrix} \right| = 0$

Keeping only the lowest order of $\frac{\upsilon}{\omega_{j}}$, (which
is consistent with our two time scale approach), we can find

$\upsilon \simeq i\frac{\frac{k_{0}}{4m}}{\sqrt{\omega_{}\omega_{2}}}$

Then A~/~ and A~2~ are only oscillatory functions. No parametric
instability.

However, if we now assume $\Omega_{0} = \omega_{} + \omega_{2}$ and
follow the same procedure, then we are led to the result

$\upsilon \simeq \frac{\frac{k_{0}}{4m}}{\sqrt{\omega_{}\omega_{2}}}$

which indicates exponential growth of the parametrically-coupled modes.

Since all the frequencies are defined to be positive, this means that
instability can occur for a pump frequency that is higher than either of
the modes that are involved. Such a situation is common (though not
exclusive) and is known as a [decay instability]{.underline} because the
incident photon (at $\Omega_{0}$) decays via the coupling into two
photons of lesser energy. We note that energy of the process is
conserved since

$\overline{h}\Omega_{0} = \overline{h}\omega_{} + \overline{h}\omega_{2}$

In actuality, the coupled-mode amplitudes cannot grow indefinitely at an
exponential rate. All we have done in this example so far is to find the
secular term of the two-time-scale expansion we have derived previously.
A second step would be to find the long time behavior of the
slowly-varying coefficients. We will carry this out for the longitudinal
modes of a coasting beam in the following.

Parametric Coupling of Longitudinal Modes in a Coasting Beam

We note from our previous discussion of the two-time-scale expansion
method, that we found, to lowest order in the expansion, the linear mode
dispersion relation

$1 = \frac{i\left( {e\omega}_{0} \right)^{2}}{(2\pi)^{2}}z_{m}N\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon d\varepsilon}}{\omega - m\left( \omega_{0} + k_{0}\varepsilon \right)}$

and an equation which governed the second-order distribution function

$$\begin{array}{r}
\frac{\partial f_{m}^{(2)}}{\partial\tau_{0}} + \text{im}\omega(\varepsilon)f_{m}^{(2)} + \frac{{e\omega}_{0}}{2\pi}U_{m}^{(2)}\frac{\partial f_{0}}{\partial\varepsilon} = \frac{\text{ie}\omega_{0}}{2\pi}\frac{\frac{\partial f_{0}}{\partial\varepsilon}e^{- i\left( \omega - {m\omega}_{0} \right)\tau_{0}}}{\omega - m\left( \omega_{0} + k_{0}\varepsilon \right)}\frac{\partial U_{m}^{()}}{\partial\tau_{}} + \frac{i\left( {e\omega}_{0} \right)^{2}}{(2\pi)^{2}} \\
\sum_{n + k = m}^{}{U_{n}^{()}U_{k}^{()}}e^{- i\left( \omega_{n} + \omega_{k} \right)\tau_{0}}\frac{\partial}{\partial\varepsilon}\left( \frac{\frac{\partial f_{0}}{\partial\varepsilon}}{\omega_{k} - k\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack} \right)
\end{array}$$

We note the presence of the product terms at second order which produce
sum and difference frequencies.

Using the notation

$D_{m}(\omega) = 1 + i\left( \frac{Nzm}{k_{0}\varepsilon_{0}} \right)^{2}\int_{- \infty}^{\infty}{d\varepsilon}\frac{\frac{\partial f_{0}}{\partial\varepsilon}}{\varepsilon - \frac{1}{k_{0}}\left( \frac{\omega}{m} - \omega_{0} \right)}$

we can write

$\begin{array}{r}
U_{m}^{(2)}D_{m}(s) = - z_{m}{e\omega}_{0}\int_{- \infty}^{\infty}\frac{d \in \frac{\text{ie}\omega_{0}}{2\pi}\frac{\partial f_{0}}{\partial\varepsilon}\frac{\partial U_{m}^{()}}{\partial\tau_{}}}{s + \text{im}\left( \omega_{0} + k_{0}\varepsilon \right)\left( s + {i\omega}_{m} \right)\left( \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)} \\
 + i\left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}\sum_{n + k = m}^{}U_{n}^{()}U_{k}^{()}\frac{\frac{\partial}{\partial\varepsilon}\left( \frac{\frac{\partial f_{0}}{\partial\varepsilon}}{\omega_{k} - k\left( \omega_{0} + k_{0}\varepsilon \right)} \right)}{s + i\left( \omega_{n} + \omega_{k} \right)}
\end{array}$

where $s = - {i\omega}_{m}$

We find that if we [choose]{.underline} the following conditions on the
frequency and the phase in the above equations

$\begin{array}{r}
\omega_{n} + \omega_{k} = \omega_{m} \\
n + k = m
\end{array}$

then certain terms will become secular, and it is these terms that will
give us a condition on the slowly-varying amplitudes.

After a couple of integrations by parts, we find

$$\begin{array}{r}
\frac{{e\omega}_{0}}{2\pi}\frac{\partial U_{m}^{()}}{\partial\tau_{}}\int_{}^{}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\left( \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)} = \\
\left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}\text{mk}_{0}\sum_{n}^{}U_{n}^{()}U_{m - n}^{()}\int_{}^{}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\left( \omega_{m - n}\lbrack m - n\rbrack\left\lbrack \omega_{0} \right\rbrack + k_{0}\varepsilon \right)\left( \omega_{n} + \omega_{m - n} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}}
\end{array}$$

Now as in the pendulum case, we impose an external voltage at mode n of
the form

$U = v_{0}e^{i\left( n\theta - \Omega_{0}t \right)} + v_{0}^{}e^{- i\left( n\theta - \Omega_{0}t \right)}$

Since we are not permitting (yet) the voltage $U_{n}$ to vary, this
situation is essentially a linear scattering of one mode into another by
a periodic medium, i.e. the beam as modulated at frequency $\Omega_{0}$,
but the results of the coupling are found by considering the
second-order distributions.

Keeping only the resonant modes, the expression for the amplitude
becomes

$\begin{array}{r}
\frac{{e\omega}_{0}}{2\pi}\frac{\partial U_{m}^{()}}{\partial\tau_{}}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\left( \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}} = \\
\left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}\text{mk}_{0}V_{0}U_{m - n}^{()}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}}{\left( \omega_{m - n} - \lbrack m - n\rbrack\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)\left( \Omega_{0} + \omega_{m - n} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}}
\end{array}$

And finally we follow the same normal mode analysis as in the pendulum
example by letting

$U_{m}^{()}\left( \tau_{} \right) = \text{ae}^{\text{υτ}_{}}$ ;
$U_{m - n}^{()} = \text{be}^{\text{υτ}_{}}$

This results in the following nonlinear dispersion relation for the
parametric coupling growth rate

$$\begin{array}{r}
\upsilon^{2}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\left( \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\left( \omega_{m - n} - \lbrack m - n\rbrack\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}} = \\
\left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}\left| V_{0} \right|^{2}m(m - n)k_{0}^{2}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\eta}{\left( \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}\left( \omega_{m - n} - \lbrack m - n\rbrack\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)} \\
\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial \in}d\varepsilon}{\left( \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)\left( \omega_{m - n} - \lbrack m - n\rbrack\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right)^{2}}
\end{array}$$

The above expression differs in an important way from the simple
pendulum case: the presence of dissipation through Landau damping and
through the real part of the impedance. Since the RHS of the above
equation involves $V_{0}^{2}$, a threshold amplitude can be found
(numerically) where $\text{Im}\left( \omega_{m} \right) = 0$, and this
is the amplitude required for the parametric oscillation to go unstable.

Solving the above equation numerically yields a solution that indicates
the importance of the frequency (and phase) matching condition found
above.

![](media/image6.jpeg){width="3.0in" height="2.370138888888889in"}

An aside to consider beam transfer function measurements. The purpose of
such a measurement is to determine the machine impedance, i.e. the
proximity of the beam to instability. We will consider a longitudinal
beam transfer function in a coasting beam (consistent with our approach
thus far).

![](media/image7.jpeg){width="3.8743055555555554in"
height="1.3784722222222223in"}

Neglecting second-order effects, the first-order distribution function
is influenced by both an external voltage and a voltage produced by
longitudinal wakes.

$f_{}(\omega) = \frac{\frac{{e\omega}_{0}}{2\pi}\frac{\partial f_{0}}{\partial \in}}{- i\omega + \text{in}\left( \omega_{0} + k_{0}\varepsilon \right)}\left( \frac{{e\omega}_{0}}{2\pi}\text{NZ}(\omega)\int_{- \infty}^{\infty}{f_{}d\varepsilon + V^{\text{ext}}} \right)$

Noting that

$I = \frac{\text{Ne}\omega_{0}}{2\pi}\int_{- \infty}^{\infty}f_{}d\varepsilon$

We can solve for the beam response function given by

$R = \frac{V_{\text{ext}}}{I} = Z + \frac{1}{- i\left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}N\int_{}^{}\frac{\frac{\partial f_{0}}{\partial \in}d\varepsilon}{- \omega + n\left( \omega_{0} + k_{0}\varepsilon \right)}}$

Carrying out the measurement across a revolution harmonic produces a
characteristic resonance curve, shifted by an impedance Z.

![](media/image8.jpeg){width="3.8722222222222222in"
height="3.4069444444444446in"}

Ref: Spentzouris Thesis

![](media/image9.jpeg){width="5.997222222222222in"
height="4.303472222222222in"}

Ref: Spentzouris Thesis

Conservation Law for Three-wave Coupling

We can find an energy (power) conservation law for this coupling process
by rearranging the equation that removed the secular terms

$\frac{\partial U_{m}}{\partial t} = \frac{- U_{n}U_{k}\left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}\int_{- \infty}^{\infty}\frac{\frac{\partial}{\partial\varepsilon}\frac{\frac{\partial f_{0}}{\partial \in}}{\omega_{k} - k\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack}}{\omega_{n} + \omega_{k} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack}}{\frac{{e\omega}_{0}}{2\pi}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack}}$

We have dropped the multiple-time-scale notation. Noting that

$\frac{\partial D}{\partial\omega_{m}} = i\frac{\left( {e\omega}_{0} \right)^{2}}{2\pi}Z_{m}\int_{- \infty}^{\infty}\frac{\frac{\partial f_{0}}{\partial\varepsilon}d\varepsilon}{\left\lbrack \omega_{m} - m\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right\rbrack^{2}}$

we can write the above equation as

$\frac{\partial A_{m}}{\partial t} = s_{m}e^{i\varphi}V_{\text{mnk}}A_{n}A_{k}$

where the interaction term $V_{\text{mnk}}$ is given by

$V_{\text{mnk}} = \frac{- i\frac{2\left( {e\omega}_{0} \right)^{2}}{\sqrt{\in}(2\pi)^{2}}z\int_{- \infty}^{\infty}{\frac{\partial}{\partial\varepsilon}\left( \frac{\frac{\partial f_{0}}{\partial\varepsilon}}{\omega_{k} - k\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack} \right)\frac{d\varepsilon}{\omega_{m} - m\left( \omega_{0} + k_{0}\varepsilon \right)}}}{\sqrt{\left| \frac{\partial D_{m}}{\partial\omega_{m}} \right|\left| \frac{\partial D_{n}}{\partial\omega_{n}} \right|\left| \frac{\partial D_{k}}{\partial\omega_{k}} \right|}}$

Noting that the energy density of one of the modes is given by

$\frac{W_{m}}{m} = \frac{1}{4m}\varepsilon_{0}\omega_{m}\frac{\partial D_{m}}{\partial\omega_{m}}\left| U_{m} \right|^{2}$

we can construct the total energy of the waves (ignoring any significant
dissipation) as

$$\frac{d}{\text{dt}}\left( \sum_{m}^{}\frac{W_{m}}{m} \right) = \frac{2}{3}\text{Re}\left\{ \sum_{m}^{}{\omega_{m}\frac{V_{\text{mnk}}}{m}A_{m}^{}A_{n}A_{k} + \sum_{n}^{}\omega_{- n}\frac{V_{- n - m + k}}{( - n)}A_{- n}^{}A_{- m}A_{k} + \sum_{k}^{}\omega_{- k}\frac{V_{- \text{kn} - m}}{( - k)}A_{- k}^{}A_{n}A_{- m}} \right\}$$

Summing over the modes and using the symmetry relations

$\omega_{- m} = - \omega_{m}$; $A_{- m} = A_{m}^{}$

$\frac{V_{\text{mnk}}}{m} = \frac{V_{- n - \text{mk}}}{- n} = \frac{V_{- \text{kn} - m}}{- k}$

we can finally arrive at the following conservation rule

$\frac{d}{\text{dt}}\left\{ \sum_{m}^{}\frac{W_{m}}{m} \right\} = \frac{2}{3}\text{Re}\sum_{m}^{}\left( \omega_{m} - \omega_{n} - \omega_{k} \right)\frac{V_{\text{mnk}}}{m}A_{m}^{}A_{n}A_{k}$

This says that the total change in energy of the system is conserved. As
the daughter waves grow, the pump must decline, and the process is, in
principle, reversible. In fact [parametric coupling is periodic for zero
dissipation.]{.underline}

We have explored the comparatively simple case of parametric coupling
between longitudinal modes in a coasting beam. However, such coupling
can be expected to occur between any harmonically related modes. Since
synchrotrons are inherently periodic, the possibility of such coupling
is likely.

For instance, within a bunch, the various synchrotron oscillations can
couple

![](media/image10.jpeg){width="3.75in" height="2.9618055555555554in"}

Two colliding beams may well exhibit coupling of betatron modes.

![](media/image11.jpeg){width="3.6243055555555554in"
height="2.863888888888889in"}

Wave-particle interactions

In the above analysis, we have not allowed the distribution function to
change appreciably, however the essence of the wave-particle interaction
in Landau damping suggest that the resonant part of phase space is
greatly altered.

![](media/image12.jpeg){width="3.0in" height="2.738888888888889in"}

In this example, the non-Maxwell-Boltzmann distribution is unstable due
to the bump-on-the-tail. As waves grow, the faster particles slow down
on the average, and the slower particles speed up, filling in the
distribution until the slope=0.

In some cases, namely when there is a spectrum of waves [with random
phases]{.underline}, we can cast the effect on the wave-particle
interaction into a form of diffusion operator, where each particle
experience a series of short, resonant accelerations by the waves,
leading to a random walk in phase space.

To gain some insight on this process, we consider the simple case of
particles with straight trajectories interacting with a wave. The
zero-order motion (constant velocity) is given by

> $z = \text{vt} + z_{0}$

and the perturbation on the velocity due to a sinusoidal wave of the
form

$\underline{E} = \hat{z}E_{0}\text{cos}\left( \text{kz} - \omega t \right)$

is given by

$\begin{array}{r}
m\frac{d}{\text{dt}}\Delta v = \text{qE}_{0}\text{cos}\left( \text{kz}_{0} + \text{kvt} - \omega t \right) \\
 \Rightarrow \Delta v = \frac{\text{qE}_{0}}{m\left( \omega - \text{kv} \right)}\left( \text{sin}\left\lbrack \text{kz}_{0} - \left( \omega - \text{kv} \right)t \right\rbrack - \text{sin}\text{kz}_{0} \right)
\end{array}$

If we now assume the initial starting locations are [uniformly
distributed across a wavelength,]{.underline} we can find the average of
the perturbed velocity as

$\left\langle {\Delta v}^{2} \right\rangle = 2\left( \frac{\text{qE}_{0}}{m\left( \omega - \text{kv} \right)} \right)^{2}\text{sin}^{2}\left( \frac{\left\lbrack \omega - \text{kv} \right\rbrack t}{2} \right)$

Now in the limit of large t, this expression is approximately

$\left\langle {\Delta v}^{2} \right\rangle\underset{t \rightarrow \infty}{}\pi\left( \frac{\text{qE}_{0}}{m} \right)^{2}t\delta\left( \omega - \text{kv} \right)$

and we may define a diffusion coefficient as

$D_{v} \equiv \frac{\left\langle {\Delta v}^{2} \right\rangle}{2t} = \frac{\pi}{2}\left( \frac{\text{qE}}{m} \right)^{2}\delta\left( - \text{kv} \right)$

What does this mean? It means there is effectively a strongly peaked
diffusion operator in the region of resonance in phase space.

![](media/image13.jpeg){width="3.0in" height="2.370138888888889in"}

We can repeat this derivation more rigorously in terms of the Vlasov
Equation. We assume that we can find $f_{}$, the perturbed distribution
function for the linearized Vlasov Equation, for a given electric field
E

$f_{}\left( \omega_{}k \right) = - \frac{\text{iq}}{m}\frac{E\left( \omega_{}k \right)}{\omega - \text{kv}}\frac{\partial f_{0}}{\partial v} + \frac{\text{ig}\left( v_{}k \right)}{\omega - \text{kv}}$

where $\omega = \omega(k)$ is the solution of the dispersion relation
assuming a self-consistent solution for E, and $g\left( v_{}z \right)$
is the initial condition for
$f_{}\left( v_{}z_{}t = 0 \right) = g\left( v_{}z \right)$

If we now average the Vlasov Equation over the fast time scale, [and
over space,]{.underline} the $\frac{\partial f}{\partial z}$ term and
the electric field term (assumed to be sinusoidal) average to zero. This
give

$\frac{\partial f_{0}}{\partial t} = - \frac{q}{m}\left\langle E\frac{\partial f_{}}{\partial v} \right\rangle$

Using the above solution for $f_{}$ we find

$\frac{\partial f_{0}}{\partial t} = \frac{2\pi\text{iq}^{2}}{m^{2}L^{2}}\sum_{k}^{}\left| E_{k} \right|^{2}\frac{\partial}{\partial v}\frac{\frac{\partial f_{0}}{\partial v}}{\omega - \text{kv}}$

Finally for definiteness we assume that the electric field grows in
accordance with the linear growth rates from the dispersion relation so
that

$\frac{\partial}{\partial t}\left| E_{k} \right|^{2} = 2\gamma_{k}\left| E_{k} \right|^{2}$

These are the so-called [quasi-linear diffusion]{.underline} equations
of plasma physics. Their validity rests on two important assumptions:

(1) the phase of the waves is sufficiently random such that the
    averaging process leads to a random walk in phase space and
    $E\left( \omega_{}k \right)\text{>=}0$

(2) the distribution function evolves on a slow time scale so that the
    linear dispersion relation remains valid

In general, the evolution of the distribution function under
quasi-linear diffusion needs to be found in conjunction with the
collision terms of the Fokker-Planck equation discussed previously.

Let us now consider the case of quasi-linear diffusion applied to a
coating beam in a synchrotron. This model was used to explain the
phenomenon of [overshoot]{.underline}, namely the tendency for an
unstable beam to grow larger than the minimum emittance needed to
stabilize a given instability.

Recall that the linear dispersion relation for a coasting beam is given
by

$1 = \left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}\frac{\text{NZ}}{\text{in}}\int_{- \infty}^{\infty}{d\varepsilon\frac{\frac{\partial f_{0}}{\partial\varepsilon}}{k_{0}\varepsilon - \frac{\omega}{n} - \omega_{0}}}$

Following the analysis presented above, we can find the following
diffusion operator for momentum diffusion

$\frac{\partial f_{0}}{\partial t} = - \frac{2\pi}{N}\left( {e\omega}_{0} \right)^{2}\text{Im}\left\{ |Z|^{2}|I|^{2}\frac{\frac{\partial^{2}f_{0}}{\partial\varepsilon^{2}}}{n_{\omega} - \omega_{n}} \right\}$

Now this operator can be expanded in a series of coupled moment
equations of the form

$\begin{array}{r}
F_{k} \equiv \int_{- \infty}^{\infty}{d\varepsilon f}_{0}\varepsilon^{k} \\
G_{k} \equiv \int_{- \infty}^{\infty}{d\varepsilon f}_{}\varepsilon^{k}
\end{array}$

To solve this system we truncate at a given order, which specifies the
maximum phase space asymmetry that can be permitted, and solve the
resulting closed form of the coupled equations for the moment amplitudes

$\begin{array}{r}
F_{0} = 1 \\
\frac{\partial F_{}}{\partial t} = - 2{\pi e\omega}_{0}\frac{\text{Ne}\omega_{0}}{2\pi}\left| \frac{G_{0}}{N} \right|^{2}2\text{Re}(z) \\
\frac{\partial F_{2}}{\partial t} = - 4{\pi e\omega}_{0}\frac{\text{Ne}\omega_{0}}{2\pi}\left| \frac{G_{0}}{N} \right|^{2}2\text{Re}\left\{ \frac{\text{nz}_{n}k_{0}}{\omega_{n} - {n\omega}_{0}} \right\} \\
\text{etc}\text{.}
\end{array}$

The reason the permitted asymmetry is an important issue is because the
quasi-linear operator contains a near delta function, and contains many
phase space harmonics. However, we [assume]{.underline} that the
strength of the operator is sufficiently weak such that collisions can
maintain a near symmetric distribution function. The solution is
typically carried out numerically

![](media/image14.jpeg){width="3.5in" height="2.765277777777778in"}

Ref: Bogacz and Ng

It is also instructive to carry out the same analysis in a bunched beam,
in particular for longitudinal motion. In this case the dynamical
equations are given by the synchrotron motion of particles in the rf
potential

$\begin{array}{r}
H = - \omega_{s}\frac{1}{2}\left( \frac{\upsilon_{s}}{n}\varphi^{2} + \frac{\eta}{\upsilon_{s}}\varepsilon^{2} \right) + U \\
U^{(1)} = \frac{{e\omega}_{0}I_{0}}{E_{0}}\sum_{\rho}^{}\frac{z\left( \rho + \frac{\Omega}{\omega_{0}} \right)}{\rho}\Lambda_{p}e^{i\text{ρφ} - i\Omega t}
\end{array}$

The dispersion relation for single-bunch modes has been derived in the
literature, and has been found to be of the form

$\Lambda_{\rho} = \frac{{e\omega}_{0}I_{0}}{E_{0}}\sum_{k}^{}{\int_{- \infty}^{\infty}\text{dJ}}\frac{2\omega_{s}}{\Omega^{2} - \omega_{s}^{2}}\frac{\partial f_{0}}{\partial J}J_{}\left( p\left\lbrack \frac{2\eta J}{\upsilon_{s}} \right\rbrack^{12} \right) \cdot \sum_{\rho}^{}\frac{z_{\text{//}}}{p}\Lambda_{p}^{(k)}J_{}\left( p\left\lbrack \frac{2\eta J}{\upsilon_{s}} \right\rbrack^{12} \right)$

If we now follow the same procedure as in the plasma case, a
quasi-linear diffusion operator can be derived of the form

$\begin{array}{r}
\frac{\partial f_{0}}{\partial t} = 2i\frac{\partial}{\partial J}\left( \left| U^{()} \right|^{2}\frac{\Omega}{\Omega^{2} - \omega_{s}^{2}}\frac{\partial f_{0}}{\partial J} \right) \\
\sigma = \int_{0}^{\infty}\text{dJJf}_{0}
\end{array}$

This is further simplified by inserting the linear dispersion roots
symbolically in the expression

$\frac{\partial\sigma}{\partial t} = - 2i\frac{{e\omega}_{0}I_{0}z_{\text{//}}}{E_{0}p}\frac{\Omega}{\omega_{s}}$

Of particular note is the fact that the work done on the particles by
the waves through Landau damping is exactly what shows up in the
additional thermal energy of the distribution. This is a universal
property of quasi-linear diffusion wherever it may occur.

By comparison to both simulation and experiment, however, it is seen
that unless decorrelation effects are sufficiently strong, the random
phase approximation does not always apply to a beam, and the
quasi-linear aspect of the dynamics breaks down. In the next lecture we
will explore what happens when the wave motion becomes strongly
nonlinear, but remains coherent.

![](media/image15.jpeg){width="3.8743055555555554in"
height="3.060416666666667in"}

Summary of Lecture #5

1.  We have outlined the hierarchy of nonlinear effects and showed that
    the behavior is ordered according to the strength of the
    perturbation.

2.  We derived the resonant wave-wave interaction known as parametric
    coupling and examined its occurrence in a coasting beam.

3.  We derived a quasi-linear diffusion operator which gives rise to
    slow changes in the distribution function as a result of a spectrum
    of waves. Its applicability depends on the validity of the random
    phase approximation.

References:\
\
1. R.C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press,
1972

2.  D. Nicholson, Introduction to Plasma Theory, Wiley, 1983

3.  L.K. Spentzouris, Coherent Nonlinear Longitudinal Phenomena in
    Unbunched Synchrotron Beams, Ph.D Thesis, Northwestern University,
    1996

4.  N.Krall. And A. Trivilpiece, Principles of Plasma Physics,
    McGraw-Hill, 1973