\begin{spacing}{0}
\chapter{Mercury Jet Surface Development in a Magnetic Field}
\label{chap:exp_result_velocity}
\end{spacing}
\
\par

In this chapter, the velocity of filament emanating from jet surface caused by disruption in a magnetic field
is investigated. The energy deposition induced by proton beam generates filaments on the Hg jet surface due to thermal stress. The filament velocity leaving from the jet surface and the effect of magnetic field on it
are discussed. It explains that the joule damping dissipates the kinetic energy on a time scale of joule damping term.
\ \\
\ \\

\section{Filament Model on Jet Surface}

\subsection{Geometry of viewing mercury filament}

It is investigated \cite{McDonald20082} that the observed motion of filament by 2D image has geometric relations with the viewing angle by focal length in optics.
The filaments ejected from Hg jet by the proton beam interaction are viewed via
shadow photography from a focal length $f$ = 9.15~cm from the center of the jet.
The jet is supposed to have elliptical cross section. The schematic geometry of viewing Hg filament
is depicted at Fig.~\ref{fig:schematic_drop}. The measurement describes the projection $y_{m}(t)$ onto the $y$ axis of a ray from the observer to the surface. McDonald \cite{McDonald20082} assumes that the filaments leave
perpendicularly as shown in Fig.~\ref{fig:schematic_drop}. The elliptic expression is given as Eqn.~(\ref{e:eq61}):

\begin{equation}
 \label{e:eq61}
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1  \ .
\end{equation}

Suppose a filament leaves the surface with velocity $v_{o}$ at time $t_{o}$ from point $(x_{o}, y_{o})$,
at time $t > t_{o}$, the travel distance $d$ is then $v_{o}(t - t_{o})$ assuming that the velocity is
constant. The position of the filament is $x_{d} = x_{o} + d \sin(\theta)$ and $y_{d} = y_{o} + d \cos(\theta)$.
Using the trigonometric notation of slope at point $(x_{o}, y_{o})$, the position of the filament, $y_{m}$,
as projected onto the $y$ axis is

\begin{equation}
 \label{e:eq62}
\begin{split}
&y_{m} = y_{d} \frac{f}{f-x_{d}} \approx y_{d} (1 + \frac{x_{d}}{f}) \\
&= b \cos \theta + v_{o} (t-t_{o}) \cos \theta + \frac{[a+v_{o}(t-t_{o})][b+v_{o}(t-t_{o})]}{2 f} \sin 2 \theta  \ .
\end{split}
\end{equation}

Thus, the apparent velocity of the filament along $y$ axis is

\begin{equation}
 \label{e:eq63}
v_{m} = \frac{d y_{m}}{d t} \approx v_{o} [ \cos \theta + \frac{a + b + 2 v_{o} (t-t_{o})}{2 f}
         \sin 2 \theta ] \ .
\end{equation}

The earliest time $t_{om}$ that a filament can be seen via projected shadow photography when
$y_{m} = b$ is given as follows:

\begin{equation}
 \label{e:eq64}
t_{om} \approx t_{o} + \frac{b(1-\frac{v_{m}}{v_{o}})}{v_{m}}
\end{equation}

and

\begin{equation}
 \label{e:eq65}
v_{m} \approx \frac{v_{o}}{1 + v_{o} \frac{(t_{om} - t_{o})}{b}} \ .
\end{equation}

\subsection{Distribution of filaments on jet surface}

McDonald \cite{McDonald20082} suggested three cases of possible distribution of filaments on jet surface,
which can indicate the probable existence of filaments in observation.
First, in case that the filaments are distributed uniformly in angle $\theta$,
the probability of the existence of the filaments is

\begin{equation}
 \label{e:eq66}
P(\theta) d \theta =  \frac{d \theta}{2 \pi} \ .
\end{equation}

Second, in case that the filaments are distributed uniformly in angle $\theta$,
the probability of the existence of the filaments is

\begin{equation}
 \label{e:eq67}
P(\theta) d \theta =  \frac{d \phi}{2 \pi} = \frac{a b}{a^{2} \sin^{2} \theta + b^{2} \cos^{2} \theta } \frac{d \theta}{ 2 \pi } \ .
\end{equation}

Third, in case that the filaments are distributed uniformly in position $s$ around the circumference $C$ of
the ellipse, the probability of the existence of the filaments is

\begin{equation}
 \label{e:eq68}
P(\theta) d \theta =  \frac{d s}{C} \approx \frac{2 \sqrt{a^{2} \cos^{2} \theta + b^{2} \sin^{2} \theta }}{3(a+b) - \sqrt{(3a + b) (a + 3b)}} \frac{d \theta}{2 \pi} \ .
\end{equation}

\subsection{Estimation of filament velocity}

Sievers and Pugnat \cite{Sievers} reported the response of solid and liquid target to rapid heating by the incident
proton beam. The parabolic radial energy deposition density $E_{beam}$ is considered, dropping to 0 at the
outer radius $a$ = 1~cm as follow:

\begin{equation}
 \label{e:eq612}
E_{p} = E_{o} [ 1 - (r/a)^{2} ] \ .
\end{equation}

The increase in temperature causes pressure rise. It is assumed that the rise time for the temperature
is of the same order of magnitude with the beam energy deposition, $10^{-9}$~s, and thermal expansion is initially
prevented by the mass inertial of the material. From the definition of bulk modulus $K$, the resulting
instantaneous thermal pressure for Hg is

\begin{equation}
 \label{e:eq612a}
\Delta p (r) = K \alpha_{v} \Delta T (r) \ .
\end{equation}

If the thermal heating occurs very slowly comparable to the material's dynamic frequency, it would correspond to
quasi-static thermal expansion. It is believed that the energy stored in the material due to the initial thermal
expansion may be converted into kinetic energy bombarding the liquid flow away. Corresponding to the thermal
expansion caused by the pressure rise, strain energy is stored in the liquid flow due to the compression,
which can be expressed as

\begin{equation}
 \label{e:eq612b}
\frac{E_{strain}}{V} = \frac{K ( \alpha_{v} \Delta T (r) )^{2} }{2}   \ ,
\end{equation}

where $E_{strain}$ represents the converted beam energy. Hence, it indicates that the thermal volumetric
expansion is proportional to the jet expansion velocity with the coefficient of
compressibility of jet material. The order of the velocity with which the boundary of the
liquid material is given by the thermal expansion at the boundary divided by the time over which the
sound travels across the radius of the jet, which is in units of $c \alpha_{v} T_{o} $ \cite{Sievers}. The pressure
and the velocity at the boundary are reduced by extending the time of heating, which depends on
the compressibility like

\begin{equation}
 \label{e:eq612c}
p \sim \frac{1}{\kappa}   \ \ \ , \ \ \ v \sim \frac{1}{\sqrt{\kappa}} \ ,
\end{equation}

where $\kappa$ is the compressibility of material.

\section{Observation of Filament Development on Mercury Jet Surface}

\subsection{Image calibration}

\subsubsection{\textit{image calibration with proton beam arrival signal}}

In order to investigate the time response of filament, we need to establish the accuracy and
calibration of the measurement based on the optical diagnostics setup.
Figure~\ref{fig:scope_beam} shows the traced signals on an oscilloscope when the beam and the
beam triggering signal are delivered. The scintillating fiber signal gives the beam arrival time.
Therefore, it is possible to set up the trigger timing for the cameras and
laser drivers, which is $\sim$ 2~$\mu$s after the master electronic triggering signal from
the proton synchrotron.
Therefore, the first image of the SMD camera shows the condition of jet at the time before the beam
arrives since the exposure time of SMD camera is 150~ns. All of electronic delays including the
cable delays are less than 1~$\mu$s. The maximum frame rate of SMD camera is up to 1~MHz.
The accuracy of camera frame rate is checked by using laser pulses. Laser pulses with certain
periods are generated and then monitored at oscilloscope through photodiode. The frame rate of camera
is set at the same values of laser pulse period. The frame rate is checked by monitoring
the variation of intensity of image captured from camera, which is judged as negligibly uniform.
Otherwise, the contrasts of consecutively captured images was varied under a slight time difference by
frame rate and/or pulse period because of short laser pulse time 150~ns.

\subsubsection{\textit{time delay structure of the triggered image to the beam arrival signal}}

Figure~\ref{fig:tof_beam} (b) shows the time structures between freezing image after laser enabling and
proton beam arrival. Figure~\ref{fig:tof_beam} (a) shows the specifications of 25~W laser, where the
response time to reach the peak laser, wavelength of laser, and optical power for various pulse
rates are investigated \cite{Tsang}. Laser emits $ \sim $ 250~ns after receiving the 16 pulse trigger
from the pulse generator. The time of flight of light to the primary vessel is $ \sim $ 60~ns.
Once the light source arrives at the primary vessel, the freezing image of Hg jet flow is
instantaneously generated and it is then transmitted through the optical fiber corresponding to the
light speed $\sim $ 4~ns/m, where $\sim $ 60~ns is taken for the used imaging fiber length.
From the traced signals in Fig.~\ref{fig:scope_beam}, the proton beam arrival time is measured.
Considering the time of flight from primary vessel through scintillating fiber, $\sim $ 60~ns delay
is estimated, so that the time delay between the 1$^{st}$ acquired image and the actual proton beam arrival
is given as $T_{3} - T_{2}$ in Fig.~\ref{fig:tof_beam} (b), which is considered for the velocity analysis of
filaments.

\subsection{Parameter optimization with uncertainty}

\subsubsection{\textit{nonlinear curve fit for estimation of model}}

Selecting a model of the right form to fit a set of data requires the use of empirical evidence in the data,
knowledge of the process, and some trial-and-error experimentation. Much of the need to iterate stems from
the difficulty in initially selecting a function that describes the data well. Some scientific theory
describing the mechanics of a physical system provide a functional form for the process, which type of function
makes an ideal starting point for model development. So, a practical approach is to choose the simplest possible
functions that have properties ascribed to the process.
Fitting models that are more complex than necessary means that random noise in the
data will be modeled as deterministic structure. This will unnecessarily reduce the amount of data available
for estimation of the residual standard deviation, potentially increasing the uncertainties of the
results obtained when the model is used. Numerical methods for model validation, such as $R^{2}$ statistic,
are useful, which is focused on a particular aspect of the relationship between the model and the data and
try to compress that information into a single descriptive number.
The residuals from a fitted model represent the differences between the responses observed and the
corresponding prediction of the response computed using the regression function.

The nonlinear regression model is

\begin{equation}
 \label{e:eq69}
\textbf{Y} = f(\textbf{X},\Theta) + \epsilon \ ,
\end{equation}

where $\textbf{X} = (x_{1}, x_{2}, \ldots , x_{m})$ are independent variables, $\Theta = (\Theta_{1}, \Theta_{2}, \ldots , \Theta_{k})$ are parameters, and $\epsilon $ is the random error assuming the mean
is equal to 0 with normal distribution, where $k$ is the degree.
A measure of the quality of nonlinear fitting parameters is the chi-square value:

\begin{equation}
 \label{e:eq69a}
\chi^{2} = \sum_{i=1}^{n} w_{i} ( y_{i} - \hat{y_{i}} )^{2} \ ,
\end{equation}

where $w_{i}$ is the weighting coefficient, $y_{i}$ are the experimental data points, and
$\hat{y_{i}}$ are the predicted points. To fit the model, the residual is defined as

\begin{equation}
 \label{e:eq69b}
r_{i} = y_{i} -  \hat{y_{i}} \ .
\end{equation}

It conforms to a normal distribution with the mean equal to 0 and the variance equal to
$\sigma_{i}^{2}$. Then the maximum likelihood estimates of the parameters $\Theta_{i}$ can be obtained
by minimizing the chi-square value, defined as

\begin{equation}
 \label{e:eq69c}
\chi^{2} = \sum_{i=1}^{n} \frac{(y_{i} - \hat{y_{i}})^{2}}{\sigma_{i}^{2}} \ .
\end{equation}

If the error is treated as weight, $ w_{i} = \frac{1}{\sigma_{i}^{2}}$ can be defined in the
chi-square minimizing equation, where $\sigma_{i}$ are the measurement errors.
The quality of regression can be measured by the coefficient of determination, $R^{2}$,
which is defined as

\begin{equation}
 \label{e:eq69d}
R^{2} = 1 - \frac{\sum_{i=1}^{n}  w_{i} (y_{i} - \hat{y_{i}})^{2}}{\sum_{i=1}^{n} (y_{i} - \bar{y})^{2}} \ .
\end{equation}

In order to express the relationship between independent variables and dependent variables and
a degree of confidence in regression model, the adjusted $R^{2}$ for validation of goodness of fit
measurement is defined as

\begin{equation}
 \label{e:eq69e}
\bar{R^{2}} = 1 - \frac{ \frac{ \sum_{i=1}^{n} w_{i} (y_{i} - \hat{y_{i}})^{2}}{n^{*} - k}}{\frac{\sum_{i=1}^{n} (y_{i} - \bar{y})^{2}}{n^{*}}} \ .
\end{equation}

Note that if intercept is included in the model, the degree of freedom is $n^{*} = n - 1$.
Otherwise, $n^{*} = n$ \cite{Originlab}.
The adjusted $R^{2}$ avoids the effect of the degrees of freedom by adding variables in the model, which
results in rising of $R^{2}$. Therefore, the adjusted $R^{2}$ overcomes the rise in $R^{2}$ when fitting
a small sample size by multiple predictor model.

The covariance value indicates the correlation between two variables, and the matrices of
covariance in regression show the inter-correlations among all parameters. The correlation
matrix rescales the covariance values. Parameter errors are equal to the square root of diagonal terms
in covariance matrix and the covariance matrix of the regression and correlation between
parameters are defined as

\begin{equation}
 \label{e:eq69f}
Cov \ (\Theta_{i} , \Theta_{j}) = \sigma^{2} (X' \ X)^{-1}
\end{equation}

and

\begin{equation}
 \label{e:eq69g}
Cor  \ (\Theta_{i} , \Theta_{j}) = \frac{Cov \ (\Theta_{i}, \Theta_{j})}{\sqrt{Cov \ (\Theta_{i},\Theta_{i})} \sqrt{Cov \ (\Theta_{j},\Theta_{j})}} \ .
\end{equation}

\subsubsection{\textit{Levenberg-Marquardt minimization}}

To estimate the $\hat{\Theta}$ value with the least square method, we need to solve
the normal equations which are set to be zero for the partial derivatives of $\chi^{2}$ with respect to each
$\hat{\Theta}_{p}$ :

\begin{equation}
 \label{e:eq610}
\frac{\partial \chi^{2}}{\partial \hat{\Theta_{p}}} = 0  \ .
\end{equation}

Employing an iterative strategy to estimate the parameter values, it starts with some initial values $\Theta_{o}$.
With each iteration, $\chi^{2}$ value is computed and then the parameter values are adjusted to reduce the
$\chi^{2}$. When $\chi^{2}$ values computed in two successive iterations are small enough compared with the
tolerance, the fitting is converged. The Levenberg-Marquardt algorithm is employed for an iterative technique
that locates a local minimum of a multivariate function that is expressed as the sum of squares of nonlinear
function. Levenberg-Marquardt is considered as a combination of steepest descent and the Gauss-Newton method \cite{Pujol}.
When the solution is far from a local minimum, the algorithm behaves like a steepest descent method.
When the solution is close to a local minimum, it becomes a Gauss-Newton method and exhibits fast convergence \cite{Pujol}.

Given the residuals $r_{i} \ (i = 1, \  \ldots , \ n )$ of parameters $\Theta = (\Theta_{1}, \Theta_{2},
 \ldots, \Theta_{p})$ , with $n \geq k$,
the Gauss-Newton algorithm finds the minimum of $\chi^{2}$ given in Eqn.~(\ref{e:eq69c}).
Starting with an initial guess $\Theta_{o}$ for the minimum, the method proceeds by the iteration
$\Theta^{s+1} = \Theta^{s} + \delta \Theta $ with an increment
$\delta \Theta$ satisfying the equation given as Eqn.~(\ref{e:eq610a}) using
Eqn.~(\ref{e:eq610}):

\begin{equation}
 \label{e:eq610a}
(\textbf{J}_{r}^{T} \textbf{J}_{r}) \delta \Theta = - \textbf{J}_{r}^{T} \textbf{r}  \ ,
\end{equation}

where $\textbf{r}$ is the vector of $r_{i}$ and $\textbf{J}_{r}$ is the Jacobian of $\textbf{r}$ with respect to
$\Theta$. The residuals $r_{i}$ are defined as $r_{i} (\Theta)= y_{i} - f ( x_{i}, \ \Theta )$. In order to find the parameters $\Theta$ that a given model function
$y=f(x, \Theta)$ fits best data points, the increment $\delta \Theta$ can be expressed in
terms of Jacobian of the function as follow:

\begin{equation}
 \label{e:eq610b}
(\textbf{J}_{f}^{T} \textbf{J}_{f}) \delta \Theta =  \textbf{J}_{f}^{T} \textbf{r} \ .
\end{equation}

The Levenberg-Marquardt iteration is a variation on the Newton iteration. The equation
$ \textbf{N} \delta \Theta = \textbf{J}_{f}^{T} \textbf{r} $
are augmented to $ \textbf{N}' \delta \Theta = \textbf{J}_{f}^{T} \textbf{r} $ where
$N^{'}_{ij} = (1 \ + \ \delta_{ij} \ \lambda) \ N_{ij}$ with $\delta_{ij}$ the Kronecker delta.
The $\lambda$ is initialized to a small value, e.g. 10$^{-3}$. If the value obtained for $\delta \Theta$
reduce the residuals, the increment is accepted and $\lambda$ is divided by 10 before the next iteration.
If the residuals increase then $\lambda$ is multiplied by 10 and the augmented equations are solved
again until an increment is obtained that reduces the residuals. For large $\lambda$, the iteration
approaches a steepest descent \cite{Originlab}.

\subsubsection{\textit{chi-square probability}}

The chi square statistic for an experiment with $n$ possible outcomes, performed $m$ times, in which
$Y_{1}, Y_{2}, \cdots, Y_{n}$ are the number of experiments which resulted in each possible outcome,
with probabilities of each outcome $p_{1}, p_{2}, \cdots, p_{n}$ is:

\begin{equation}
 \label{e:eq610c}
\chi^{2} = \sum_{1 \leq i \leq n}  \frac{(Y_{i} - m p_{i})^{2}}{m p_{i}}  \ .
\end{equation}

Note that $y_{1}, \cdots , y_{n}$ are independently normally distributed with mean $\mu$ and
variance $\sigma^{2}$, then $\bar{y}$ will be precisely normally distributed with mean mean $\mu$ and
variance $\sigma^{2}/n$. By substitution of $d \ S^{2} / \sigma^{2} = t$ into sampling distribution,
the probability density function $P$ that a $\chi^{2}$ value calculated for an experiment with $d$
degrees of freedom is due to chance is:

\begin{equation}
 \label{e:eq610d}
P_{\chi^{2}, \ d} = [ 2^{d/2} \ \Gamma(\frac{d}{2}) ]^{-1} \int_{\chi^{2}}^{\infty}  (t)^{\frac{d}{2} - 1} e^{-\frac{t}{2}} dt \ ,
\end{equation}

where $\Gamma$ is the generalization of the factorial function to real and complex arguments:

\begin{equation}
 \label{e:eq610e}
\Gamma_{x} =  \int_{0}^{\infty}  (t)^{x - 1} e^{-t} dt \ .
\end{equation}

Tables for the chi-square distribution with $d$ degrees of freedom are given in percentiles \cite{Evans}.
The $P$ percentile is given as

\begin{equation}
 \label{e:eq610f}
Pr (\chi^{2} (d) \leq \chi^{2}_{p} (d)) = 1- P \ .
\end{equation}

Noth that the probability calculation from $\chi^{2}$ is an approximation which is valid for
large values of $n$, and is only meaningful when calculated from a large number of independent
experiments.

\subsection{Filament distribution and uncertainty of measurement}

\subsubsection{\textit{observation of filaments on jet surface}}

Figure~\ref{fig:velocity_photo} shows photographs of filament evolution on the Hg jet surface at 25~$\mu$s frame
rate, where the beam is 10~Tp, 24~GeV and the magnetic field is 10~T.
The first collected image among 15 images is brighter than the rest of 14 images.
It indicates that the radiation generated by the interaction of Hg with proton beam affects the
transmittance and/or reflectance of optical components, resulting in the production of darker
images as one sees the rest of collected 14 images.

\subsubsection{\textit{measurement of traveled distance of filament}}

To obtain the vertical filament velocity, the distance traveled by a fixed point
on the jet surface is tracked over a given time period.
The jet volume, where the maximal energy is deposited, results in the initial generation of the filaments.
The higher jet velocity occurs when the filaments is initially protruded out of the jet surface and then
the jet velocity decreases due to the magnetic damping and viscous dissipation. So, the velocity at steady
state is obtained in order to evaluate its effect on the beam intensity and magnetic field.

The quality of optical images varies from shot to shot since the radiation or jet dispersion may make image
quality varies. The most difficulty in measurement is to discern the edge of filaments as it moves somewhat
far away from surface because the initial jet filament edge is dense(clearly black) but it looks like dissipating,
dilute, and disappearing (grey or similar with background) as it moves further. Because measurement is done in
several points, there may be some error in measurement after some steady velocity(constant peak velocity) for
weak filament velocity measurement.

The image size at Viewport 2 is 240 by 240. Using graphic software, pixels on image is picked to locate the edge of
filament. Therefore, the uncertainty, while locate the position $y_{m}$, is reported to be $\pm$ 2 pixels,
which corresponds to the difference of $\sim \pm$ 17~m/s filament velocity. This uncertainty can occur
randomly. The peak strong filament which gives constant velocity within $\pm$ 2 pixels until the end
of 15 frames is assumed to be considered as there is constant uncertainty, $\pm$ 2 pixels. The weak filament
which gives constant velocity within $\pm$ 2 pixels until the filament reaches some frames, for example,
3 $\sim$ 7 frames, is also assumed to be considered as there is a constant uncertainty, $\pm$ 2 pixels, where
the black edge of filament is clearly observed. However, after the some frames, for example, 3 $\sim$ 7 frames,
because the original edge of filament dilute or dissipates or disappear, the uncertainty in measurement may not
be constant. In this case, measurement is stopped at that frame.

\subsection{Linear regression with the first order polynomial}

\subsubsection{\textit{curve fit function}}

The heaviside step function is defined as the integral of the Dirac delta function as follow:

\begin{equation}
 \label{e:eq611}
H(t) = \int^{t}_{-\infty} \delta(\xi) d \xi  \ .
\end{equation}

The ramp function is the antiderivative of the Heaviside step function:

\begin{equation}
 \label{e:eq611a}
R(t) = \int^{t}_{-\infty} H(\xi) d \xi = t H(t)  \ .
\end{equation}

In a discrete form, it is defined as an alternative form for our linear regression model as follow:

\begin{equation}
 \label{e:eq611b}
R(t) = \left \{
         \begin{array}{ll}
           y_{m} = b, \ \  t \leq t_{om} \\
           y_{m} = b + v_{m} ( t - t_{om} ),  \ \ t > t_{om} \ \  ,
         \end{array}
         \right.
\end{equation}

where $y_{m}$, $b$, $v_{m}$, and $t_{om}$ denote the measured position of the filament as projected
onto the $y$ axis in image, the measured position of jet surface before the filament are developed,
the apparent velocity of the filament along the $y$ axis, and measured the earliest time that a filament
can be seen via shadow photography respectively.

\subsubsection{\textit{parameter estimation using multiple position of filaments}}

Shot 11019 is chosen for illustration.
Using Eqn.~(\ref{e:eq611b}) for linear regression model with measured data points $y_{m}$ and $t$, minimizing
$R^{2}$ yields $b$, $v_{m}$, and $t_{om}$. Figure~\ref{fig:velocity_fitexample} (a) shows the illustration of multiple
data points where the intercept of $x$ axis and slope estimate the onset time of filament via shadow photography
and apparent velocity projected on $y$ axis in image, which are $t_{om}$ = 43.6 $\pm$ 4.5 $\mu$s and $v_{m}$ = 55.5 $\pm$ 0.8 m/s respectively.
The reduced $R^{2}$ value and adjusted $\bar{R}^{2}$ values are 1.749 and 0.998 respectively.
Based on Eqn.~(\ref{e:eq611b}), the fit function to data points is as follow:

\begin{equation}
 \label{e:eq611c}
y_{m} = c_{1} (t - b_{1}) + a_{1} \ ,
\end{equation}

where $t$ and $y_{m}$ denote the measured time and position of filament respectively and
the units for $a_{1}$, $b_{1}$, and $c_{1}$ are pixels, $\mu$s, and pixel/$\mu$s respectively.
The parameterized values of coefficients and error values to fit function are given in
Table~\ref{tab:fitdata_ch6}.

In case of larger velocity of filaments, maximally measurable data points are limited to
$\sim$ 2 $\sim$ 3 points due to the limited field of view in optical diagnostic image. Figure~\ref{fig:velocity_fitexample} (b) shows the illustration for the case of 3 data points.
The onset time from regression model yields underestimated value such as negative
time delay because the data points are equal or smaller than the number of parameters in fit function.
Thus, assumption is that the real onset time for such a large velocity should be between typical onset
time 50~$\mu$s and 0~$\mu$s, which corresponds to the onset time of 25 $\pm$ 25~$\mu$s.
Therefore, the slope of fit curve is determined by fixing the assumed onset time accordingly, which yields
the filament velocity of 148 $\pm$ 24.5~m/s. The error is determined directly by dividing approximated
filament velocity of the cases of $t_{om}$ = 0 and $t_{om}$ = 50~$\mu$s by 2.
The shot 10008 is chosen for the case of illustration of parameter estimation of 3 data points.
The fit function to data points for the case of negative onset time (black solid line)
in Fig.~\ref{fig:velocity_fitexample} (b) is as follow:

\begin{equation}
 \label{e:eq611d}
y_{m} = c_{1} t + a_{1} \ .
\end{equation}

And Eqn.~(\ref{e:eq611c}) is employed for the case of having fixed $b_{1}$ = 0~$\mu$s and
$b_{1}$ = 50~$\mu$s onset time (blue and red solid line) in in Fig.~\ref{fig:velocity_fitexample} (b).
As one expects, this approach for a special case yields large uncertainty.

\subsubsection{\textit{filament velocity distribution on jet surface}}

Figure~\ref{fig:photo_dropvel_point} (b) shows the velocity distribution of filaments
over the jet surface shown in Fig.~\ref{fig:velocity_photo} according to the location of filaments
from Fig.~\ref{fig:photo_dropvel_point} (a), whose filaments are used for Fig.~\ref{fig:relativevelocity_filament_14pt} and Fig.~\ref{fig:velocity_filaments_surface}.
Figure~\ref{fig:relativevelocity_filament_14pt} (a) and (b) show calculated estimation and measured estimation
of filament velocity as a function of observed onset time of filaments respectively. $v_{o}$=60~$\mu$s and
$t_{o}$=40~$\mu$s for upwards filaments are used with Eqn.~(\ref{e:eq65}) for calculated estimation.
$t_{o}$=70~$\mu$s for downwards filaments are used.
As the estimated apparent velocity of filaments projected on $y$ axis in image increases, the estimated onset
time of filaments decreases. This shows the evidence of the geometric effects of viewing filaments.
Assuming the filaments are generated perpendicular to the jet surface, as the filaments leaves farther
from the jet surface, it takes more time to make an initial observation in images. Thus,it is possible to consider
the low velocity of filaments with large onset time leaves from more close to the center of jet normal to the
side view shown in images. Note that the velocity of each filament is approximated with uncertainty by doing
linear regression using the fit function in order to give one representative velocity according to each filament.
Low velocity of filaments close to 0 showed larger error of approximation of onset time due to the uncertainty of
the very small observed traveling distance of filaments.

According to the notation in Fig.~\ref{fig:photo_dropvel_point} (a),
Fig.~\ref{fig:velocity_filaments_surface} (a) shows the velocity of filaments on the upward free surface of jet
as a function of time and Fig.~\ref{fig:velocity_filaments_surface} (b) shows the velocity of filaments on
the downward free surface of jet as a function of time. Note that the instantaneous velocity as defined
in Eqn.~(\ref{e:eq622}) is used for velocity measurement in Fig.~\ref{fig:velocity_filaments_surface}.
The onset time of filament increases as the peak velocity of filament decreases, which
indicates the possible evidence of the geometric effect of viewing filaments.

\section{Filament Velocity on Mercury Jet Surface}

\subsection{Magnetic dissipation of energy}

As a conducting liquid moves through a static magnetic field, electric currents are
generated. This, in turn, leads to ohmic heating such as Joule dissipation. As the thermal energy of the fluid rises, there is a corresponding decrease in its kinetic energy, and thus the fluid decelerates.
This results in a suppression of the motion of filament.
According to Davidson's approximation \cite{Davidson}, the Eqn.~(\ref{e:eq314z}) shows the energy decay
with respect to time depending on the magnetic damping time constant, where $\tau = \rho / \sigma B^{2}$.
The implication is that the filaments decelerates on a time scale of $\tau$.
Figure~\ref{fig:energy_decay} (a) shows the decay of the normalized energy of flow in magnetic fields
with respect to time due to the magnetic damping. Figure~\ref{fig:energy_decay} (b) shows the integral
calculation of Fig.~\ref{fig:energy_decay} (a) with respect to time.

\subsection{Time response of filaments in a magnetic field}

Since the camera is triggered before beam arrives at the Hg jet and the laser pulse width is
150~ns, the first collected image shows the condition of Hg jet before beam interacts. Thus, the velocity
of filament can always be judged as 0~m/s in the following Fig.~\ref{fig:filament_instvel}.
Since the joule damping dissipates the energy with an exponential factor,
the energy dissipation arises rapidly in the beginning depending on the magnetic field term $B^{2}$.
Thus, higher magnetic field will have higher damping effect so that it takes more rising time to peak velocity.
The magnitude of steady peak velocity is reduced by increased applied magnetic field strength,
which is possible indication of the magnetic damping role induced by the joule damping dissipation.

Figure~\ref{fig:filament_instvel} shows the time response of instantaneous filament velocity as
a function of magnetic field with 14~GeV, 20~Tp beam and 24~GeV, 10~Tp beam respectively.
The expression for the calculation of instantaneous velocity assuming $\Delta t_{n}$ is small enough is

\begin{equation}
 \label{e:eq622}
v_{n} = \frac{y_{m} (t_{n}) - y_{m} (t_{n-1}) }{\Delta t_{n}} \ .
\end{equation}

\subsection{Proton beam induced filament velocity in a magnetic field}

\subsubsection{\textit{filament velocity with 14~GeV beam in a magnetic field}}

Figure~\ref{fig:velocity_14GeV} (a) shows the filament velocity as a function of 14~GeV beam intensity
and magnetic field corresponding to the observed onset time of filaments shown in Fig.~\ref{fig:velocity_14GeV} (b). Note that the data points without having onset time data is reported
by crude measurements of 2 positions of filament from 500~$\mu$s frame rate shot images since the
estimation of onset time by fitting is inadequate. The filament velocity increases with the beam intensity.
However, the magnetic field suppresses the filament velocity. At low intensity of proton beam,
the charged beam may be fluctuating depending on the initial conditions at experiment.
Thus, the observed onset time of filaments is large at low intensity of beam and it decreases as the
intensity of proton beam increases, see Fig.~\ref{fig:velocity_14GeV} (b).
Therefore, there are scattering distributions of filament velocity at lower intensity of beam over the
resulting data points. The slope of the data points in velocity at higher magnetic fields decreases
comparing with that associated with lower magnetic field.
All velocities are less than 50~m/s regardless of magnetic fields.
The filament velocity for the case of 14~GeV beam with 30~Tp in 10~T is $\sim$ 30~m/s.

\subsubsection{\textit{filament velocity with 24~GeV beam in a magnetic field}}

Figure~\ref{fig:velocity_24GeV} (a) shows the filament velocity as a function of 24~GeV beam intensity
and magnetic field corresponding to the observed onset time of filaments shown in
Fig.~\ref{fig:velocity_24GeV} (b).
Again, at low intensity of proton beam, the charged beam may be fluctuating depending on
the initial conditions at experiment. Thus, the observed onset time of filaments is large at low intensity of beam and it decreases as the intensity of proton beam increases, see Fig.~\ref{fig:velocity_24GeV} (b).
The filament velocity increases with the beam intensity.
The slope of the increase in velocity is $\sim$ 4 $\times$ larger that that for the 14~GeV beam case,
where the ratio of peak energy deposition between 14~GeV and 24~GeV beam energy is $\sim $ 2.3
based on the calculation given in Fig.~\ref{fig:energydep_nproton} (a), which implies the relationship
of peak energy deposition to maximum filament velocity.
However, the magnetic field suppresses the filament velocity. At relatively low intensity of beam
as in the 14~GeV beam case, the charged beam is unstably fluctuating depending on the event conditions
at experiment. Thus, the observed onset time of filaments is large at low intensity of beam and
it decreases as the intensity of proton beam increases, see Fig.~\ref{fig:velocity_24GeV} (b).
All velocities are less than 180~m/s regardless of magnetic fields, and the filament velocity for
the case of 24~GeV beam with 30~Tp in 15~T is $\sim$ 60~m/s.

\subsubsection{\textit{measurement of filament velocity in pump-probe condition as a check}}

Figure~\ref{fig:velocity_pumpprobe} shows the measured filament velocity of multiple events with pump-probe
conditions as a check. The conditions of each group in pump-probe events are given in
Table~\ref{tab:measure_pumpprobe}. There are 2 groups at 14~GeV and each group has different number
of bunches and time delay between pump and probe. Figure~\ref{fig:velocity_pumpprobe} (a) shows the histogram of filament velocity and Fig.~\ref{fig:velocity_pumpprobe} (b) shows statistics summary such as average,
minimum, maximum, and median value. In group 2, qualitatively meaningful distribution of measurements are shown,
which is 10.2 $\pm$ 3.6~m/s. The pump condition is meaningful due to the delay of beam delay, though
there is no significant difference in sub-category of group 2.
However, This check shows low velocity comparing with the results shown in Fig.~\ref{fig:velocity_14GeV} (a).
One thing to evaluate is that there is another error that should be considered in filament velocity
analysis, so called uncertainty distribution of filament velocity under repetition with same condition of
experiment. This is judged by $\sim$ 40~$\%$ of the measured velocity, which is integrated in
the following key result shown in Fig.~\ref{fig:velocity_peakenergy}.

\section{Filament Velocity on Jet Surface by Energy Deposition}

The energy deposition according to the interacting number of protons at both 14~GeV and
24~GeV beam energy is calculated by Fig.~\ref{fig:energydep_nproton}. Thus, Fig.~\ref{fig:velocity_14GeV} and Fig.~\ref{fig:velocity_24GeV} could be combined as a function
of energy deposition, which shows the combined result of experiment in maximum filament velocity together.
Figure~\ref{fig:velocity_peakenergy} shows the filament velocity
in magnetic fields as a function of peak energy deposition and fit is according to Eqn.~(\ref{e:eq623}).
Figure~\ref{fig:velocity_totalenergy} shows the filament velocity
in magnetic fields as a function of total energy deposition and fit is according to Eqn.~(\ref{e:eq623}).
The same threshold values of peak energy and total energy deposition with those in
Fig.~\ref{fig:disrupt_totalenergy} and Figure~\ref{fig:disrupt_peakenergy} in various magnetic fields
are used in order to connect mutual interplay between results.
In other words, the threshold peak energy deposition for filament velocity uses the same value with that
for disruption length in order to keep consistency between the onset of disruption and filament.
$\chi^{2}$ values between Fig.~\ref{fig:velocity_peakenergy} and Fig.~\ref{fig:velocity_totalenergy} are not
significantly different, although Fig.~\ref{fig:disrupt_totalenergy} has lower $\chi^{2}$ value possibly due
to effects of the forcefully adopted threshold values from Fig.~\ref{fig:disrupt_totalenergy}
and Fig.~\ref{fig:disrupt_peakenergy}.

As discussed, the filament velocity on jet surface is dominated by the distribution of
energy deposition interacting with proton beam. Hence,the peak energy deposition plays a role in
determining the maximum filament velocity in viewpoint that the velocity distribution on jet surface
can be normalized using the peak energy deposition. The employed global fit with multi-variables for
filament velocity using the measured filament velocity is:

\begin{equation}
 \label{e:eq623}
v_{m} = a_{1} (E_{p} - (b_{1} + b_{2} B^{b_{3}}))^{c_{1} + c_{2} B + c_{3} B^{2}} \ ,
\end{equation}

where $E_{p}$ and $B$ are energy deposition and magnetic field respectively. The parameterized
values of coefficients and errors of the fit functions are provided in Table~\ref{tab:fitdata_ch6}.
Note the error of each measured maximum filament is adjusted by $\sim$ 40~$\%$ of the measured velocity
in order to expect somewhat improved fit result with reduced $\chi^{2}$, as discussed previously in
multiple events analysis with pump-probe condition.
According to Fig.~\ref{fig:velocity_peakenergy}, the threshold of filament velocity increases in 1.2
power of magnetic field, and it is  $\sim$ 10~J/g
of peak energy energy deposition with no magnetic field. The filament velocity increases
in linear ($\sim$ 1.24) power of peak energy deposition with no magnetic field, but it is reduced in
$\sim  \ 1.24 - 0.015 B$ power of peak energy deposition with magnetic field.

For muon collider in the future, higher beam intensity equivalent with 80~Tp in 20~T of 24~GeV proton
beam energy is required. The peak energy deposition at 80~Tp with 24~GeV beam is $\sim$ 255~J.
The total energy deposition for the case of 80~Tp with 24~GeV beam is $\sim$ 20.7~kJ.
The maximum filament velocity at 255~J of peak energy in 20~T is expected to be $\sim$ 119~m/s.
The maximum filament velocity at 20.7~kJ of total energy in 20~T is expected to be $\sim$ 129~m/s.

\section{Comparison of Filament Velocity with Numerical Calculation As a Check}

This chapter provides explicitly an comparison of filament velocity measurement based on the results of
numerical calculation as a check purpose.
Considering that the major and minor axes of the elliptic jet shape are 0.8~cm and 0.3~cm respectively,
numerical calculation of filament velocity is performed with 14~GeV beam with 10~Tp in 0~T using Frontier
MHD code based on front tracking method.
The calculated energy deposition into the Hg jet from MARS code is used for calculation of filament
velocity, and the velocity of 20 filaments in different positions are evaluated \cite{Bo20091}.
It is reported that the filament velocity along minor axis is much larger than that along the major axis
and the filaments grow in the normal direction to the surface \cite{Bo20092}.

Figure~\ref{fig:filament_velsim_check} (a) shows the initial pressure contour in Hg jet from
energy deposition caused by proton beam interaction, where the peak pressure reached at $\sim$ 14000
bar \cite{Bo20092}. 3 positions for filament velocity measurement in simulation are picked, which is
schematically depicted in Fig.~\ref{fig:filament_velsim_check} (b) \cite{Bo20091}.
The filament velocity resulted in depending on the distance from position of peak energy deposition.
Thus, different results of filament velocity are calculated in simulation considering the location of
picked filaments, which is shown in Fig.~\ref{fig:filament_velsim_check} (c) \cite{Bo20091}.
The filament velocity $v_{o}$ for position 1,2 and 3 at $\sim$ 80 $\mu$s are estimated to be 18, 33,
and 28~m/s respectively from Fig.~\ref{fig:filament_velsim_check} (c).
Approximating that the angles $\phi$ for point 2 and point 3 are 10 $^{\circ}$ and 20 $^{\circ}$ followed by the
notation given in Fig.~\ref{fig:schematic_drop}, the apparent filament velocity along the $y$ axis and
the earliest visible time delay on images via shadow photography are estimated using Eqn.~(\ref{e:eq63})
and Eqn.~(\ref{e:eq64}) respectively, which is shown in Fig.~\ref{fig:filament_velsim_check} (d) with
measured maximum filament velocity and correspondingly estimated time delay from measurement.
Note that 60 $^{\circ}$ and $v_{o}$=28~m/s from calculated result are used for the case of velocity at
$\sim$ 800 $\mu$s with experimental measurement, and 40 $\%$ of the measured filament velocity is used
for giving the velocity measurement error for consideration of uncertainty from multiple measurement
under same condition.
Since the apparent filament velocity varies depending on the estimated angle $\phi$, appropriate angle is
chosen based on the estimated value of the earliest visible time delay of filament.
Simulation with no magnetic field estimates somewhat larger velocity than experimental measurement under 5~T of
magnetic field due to magnetic field and estimated angle $\phi$.

\setlength{\baselineskip}{12pt}

\newpage
\clearpage

\begin{landscape}
\begin{table}[!hbt]
\centering
\tablecaption{\label{tab:fitdata_ch6} Parameterized coefficients, its error, and statistics summary of fit function in figures.}

\begin{supertabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\hline
 $\textbf{Figure}$ & $\textbf{1}$ & $\textbf{2}$ & $\textbf{3}$ & $\textbf{4}$ & $\textbf{5}$ & $\textbf{6}$ & $\textbf{7}$ & $\textbf{8}$ & $\textbf{9}$  & $\textbf{10}$ \\
\hline
~\ref{fig:velocity_fitexample}(a)  & 128 & 0.93517 & 43.57 & 4.44411 & - & - & - & - & -0.26374 & 0.00392 \\
~\ref{fig:velocity_fitexample}(b)(black)  & 112.1 & - & - & - & - & - & - & - & -0.52 & - \\
~\ref{fig:velocity_fitexample}(b)(blue)  & 122 &  0 & 0 & 0 & - & - & - & - & -0.5865 & 0.01587 \\
~\ref{fig:velocity_fitexample}(b)(red)  & 122 & 0 & 50 & 0 & - & - & - & - & -0.81911 & 0.10777 \\
~\ref{fig:velocity_peakenergy}  & 0.76998 & 0.65104 & 9.93998 & 0 & 0.98744
 & 0 & 1.21081 & 0 & 1.23776 & 0.398 \\
~\ref{fig:velocity_totalenergy}  & 0.02454 & 0.0425 & 338.243 &  0 & 115.38 & 0 & 0.82899 & 0 & 1.00378 & 0.29245 \\
%  &  &  &  &  &  &  &  &  &  & \\
\hline
\hline
 $\textbf{Figure}$ & $\textbf{11}$ & $\textbf{12}$ & $\textbf{13}$ & $\textbf{14}$ & $\textbf{15}$ & $\textbf{16}$ & $\textbf{17}$ & $\textbf{18}$ & $\textbf{19}$ &  \\
\hline
~\ref{fig:velocity_fitexample}(a)  & - & - & - & - & 15 & 12 & 1.74908 & 0.99773
  & 0.0505 & \\
~\ref{fig:velocity_fitexample}(b)(black)  & - & - & - & - & 2 & 0 & 0 & 0 & 0 & \\
~\ref{fig:velocity_fitexample}(b)(blue)  & - & - & - & - & 3 & 2 & 12.31396 & 0.99622 & 0 & \\
~\ref{fig:velocity_fitexample}(b)(red)  & - & - & - & - & 3 & 2 & 281.74259 & 0.91351 & 0 & \\
~\ref{fig:velocity_peakenergy}  & -0.01468 & 0.01776 & 0 & 0 & 25 & 22 & 2.19995 & 0.08635 & 0.0009 & \\
~\ref{fig:velocity_totalenergy}  & -0.00671 & 0.0079 & 0 & 0 & 25 & 22 & 1.85595 & 0.22959 & 0.0082 & \\
%  &  &  &  &  &  &  &  &  &  & \\
\hline
\hline
\end{supertabular} \\

\ \\
%\normalsize
\begin{flushleft}

\shortstack[l]{\vrule width 8.5in height 0.4pt\\
        \ \ $\textbf{1}$ : a1 value, $\textbf{2}$ : a1 standard deviation,   \\
       \ \\
        \ \ $\textbf{3}$ : b1 value, $\textbf{4}$ : b1 standard deviation, $\textbf{5}$ : b2 value, $\textbf{6}$ : b2 standard deviation, \\
       \ \\
        \ \ $\textbf{7}$ : b3 value, $\textbf{8}$ : b3 standard deviation , $\textbf{9}$ : c1 value, $\textbf{10}$ : c1 standard deviation,
         \ \\
         \ \ $\textbf{11}$ : c2 value, $\textbf{12}$ : c2 standard deviation, $\textbf{13}$ : c3 value, $\textbf{14}$ : c3 standard deviation, \\
         \ \\
        \ \ $\textbf{15}$ : Number of points, $\textbf{16}$ : Degrees of freedom, $\textbf{17}$ : Reduced $\chi^{2}$, $\textbf{18}$ : Adjusted $\textbf{R}^{2}$, $\textbf{19}$ : $\chi^{2}$ probability.  }
       \ \\
\end{flushleft}

\end{table}
\end{landscape}

\newpage
\clearpage

\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=7in]{schematic_drop}
\caption{Schematic geometry of viewing filament on the Hg jet \cite{McDonald20082}. } \label{fig:schematic_drop}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{tof_beam2}
\caption{Time delay estimation of devices for triggered image calibration. a.) Measurement of characteristic response of 25 W laser used for high speed camera at Viewport 2 \cite{Tsang}. b.) Time structures between light source enabling and proton beam arrival.} \label{fig:tof_beam}
\end{figure}
\end{landscape}
\clearpage


\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{photo_splash_velocity4}
\caption{Photographs of filament evolution on the Hg jet surface as a function of time at 25~$\mu$s frame rate. The beam is 10~Tp with 24~GeV. The magnetic field is 10~T. The red circle on the 4$^{th}$ image of the top row points the filament that is used for velocity measurement in Fig.~\ref{fig:filament_instvel} (b).} \label{fig:velocity_photo}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{velocity_fitexample}
\caption{Illustration of bilinear fit for parameters estimation. a.) Multiple data points. b.) 3 data points.} \label{fig:velocity_fitexample}
\end{figure}
\end{landscape}
\clearpage


\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{photo_dropvel_point3}
\caption{Location of 18 points of filament on the Hg jet surface for velocity measurement and its velocity. The shot condition is same with Fig.~\ref{fig:velocity_photo}. The numbers above red circles point the filament that is used for velocity estimation in Fig.~\ref{fig:relativevelocity_filament_14pt} and Fig.~\ref{fig:velocity_filaments_surface}. a.) Location of filaments for measurement. b.) Filament velocity at each location. } \label{fig:photo_dropvel_point}
\end{figure}
\end{landscape}
\clearpage


\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{relativevelocity_filament_14pt5}
\caption{Estimation of velocity and onset time of filaments shown in Fig.~\ref{fig:velocity_photo}.
The beam is 10~Tp with 24~GeV and magnetic field is 10~T. a.) Calculated estimation of filament velocity
as a function of onset time of filaments. $v_{o}$=60~$\mu$s and $t_{o}$=40~$\mu$s for upwards filaments are used with Eqn.~(\ref{e:eq65}). $t_{o}$=70~$\mu$s for downwards filaments are used. b.) Measured estimation of filament velocity and onset time of filaments.} \label{fig:relativevelocity_filament_14pt}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{velocity_filaments_surface}
\caption{Time response of instantaneous filament velocity on jet surface for various filaments shown in Fig.~\ref{fig:photo_dropvel_point}. The beam is 10~Tp with 24~GeV. The magnetic field is 10~T. a.) Upper surface. b.) Lower surface.} \label{fig:velocity_filaments_surface}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{filament_instvel2}
\caption{Time response of instantaneous filament velocity as a function of magnetic field. Equation~(\ref{e:eq622}) is used for measuring instantaneous filament velocity. The half of elapsed time between each frame is used to indicate the time at each filament velocity. a.) 14~GeV, 20~Tp beam. b.) 24~GeV, 10~Tp beam.} \label{fig:filament_instvel}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{splash_velocity_14GeV5}
\caption{Maximum observed filament velocity as a function of 14~GeV beam intensity in magnetic fields. a.) Maximum observed filament velocity. b.) Onset time of that filament. } \label{fig:velocity_14GeV}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{splash_velocity_24GeV5}
\caption{Maximum observed filament velocity as a function of 24~GeV beam intensity in magnetic fields. a.) Maximum observed filament velocity. b.) Onset time of that filament. } \label{fig:velocity_24GeV}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{velocity_pumpprobe}
\caption{Filament velocity measurement in same conditions. Pump-probe conditions with harmonic 8 and 16 bunches are used. The conditions of each group in pump-probe events are given in Table~\ref{tab:measure_pumpprobe}. a.) Histogram of maximum observed filament velocity in each group. b.) Maximum observed filament velocity of each group. } \label{fig:velocity_pumpprobe}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=7in]{velocity_peakenergy2}
\caption{Maximum observed filament velocity as a function of peak energy deposition in various magnetic fields
and fit is according to Eqn.~(\ref{e:eq623}). } \label{fig:velocity_peakenergy}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=7in]{velocity_totalenergy}
\caption{Maximum observed filament velocity as a function of total energy deposition in various magnetic fields
and fit is according to Eqn.~(\ref{e:eq623}). } \label{fig:velocity_totalenergy}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=6.5in]{filament_velsim_check}
\caption{Comparison of calculated filament velocity with measurement. 10~Tp with 14~GeV in 0~T is considered in simulation, but 5~T in measurement. a.) Initial pressure contour in Hg jet by energy deposition \cite{Bo20092}. b.) Schematic location of picked filament for velocity calculation \cite{Bo20091}. c.) Calculated filament velocity on the position 1, 2, and 3 \cite{Bo20091}. d.) Apparent filament velocity in simulation and measurement. The apparent filament velocity and the earliest visible time delay in simulation are estimated using Eqn.~(\ref{e:eq63}) and Eqn.~(\ref{e:eq64})} \label{fig:filament_velsim_check}
\end{figure}
\end{landscape}
\clearpage

