Emergence of Kinetic Behavior in Streaming Ultracold Neutral Plasmas
P. McQuillen
J. Castro
S. Bradshaw
and T.C. Killian
P. McQuillen
T.C. Killian
We create streaming ultracold neutral plasmas by tailoring the photoionizing laser beam that creates the plasma. By varying the electron temperature, we control the relative velocity of the streaming populations, and, in conjunction with variation of the plasma density, this controls the ion collisionality of the colliding streams. Laser-induced fluorescence is used to map the spatially resolved density and velocity distribution function for the ions. We identify the lack of local thermal equilibrium and distinct populations of interpenetrating, counter-streaming ions as signatures of kinetic behavior. Experimental data is compared with results from a one-dimensional, two-fluid numerical simulation.
Draft20pngStreamingPlasmas.tex
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\begin{document}

\title{Emergence of Kinetic Behavior in Streaming Ultracold Neutral Plasmas}% Force line breaks with \\

\author{P. McQuillen, J. Castro, S. Bradshaw, and T.C. Killian}
 %\email{Jose.Castro@rice.edu}
%\email{killian@rice.edu}
 %\email{Jose.Castro@rice.edu}
%\author{P. McQuillen, T.C. Killian}
\affiliation{%
Rice University, Department of Physics and Astronomy and Rice Quantum Institute, Houston, Texas 77005.}%
\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
We create streaming ultracold neutral plasmas by tailoring the photoionizing laser beam that creates the plasma.   By varying the electron temperature, we control the relative velocity of the streaming populations, and, in conjunction with variation of the plasma density, this controls the ion collisionality of the colliding streams. Laser-induced fluorescence is used to map the spatially resolved density and velocity distribution function for the ions.
 We identify the lack of local thermal equilibrium and  distinct populations of interpenetrating, counter-streaming ions as
  signatures of  kinetic behavior. Experimental data is compared with results from a one-dimensional, two-fluid numerical simulation.
\end{abstract}

%\pacs{Valid PACS appear here}


\maketitle

\section{Introduction}

The behavior and theoretical description of a plasma depends critically on the ratio of the mean free path for constituent particles  to the length scale of plasma inhomogeneities ($\left| n/\nabla n \right|$ for plasma density $n$, or $\left| T_{i,e}/\nabla T_{i,e} \right|$ for plasma ion or electron temperature $T_{i,e}$). For collisional systems in which  the mean free path is very small, particles remain close to local thermal equilibrium, and transport and collective motions can be described with a hydrodynamic treatment.
Because of the increasing mean free path with
particle kinetic energy, the situation can already change for mean free path  on
the order of or longer than 1\% of the length scale of plasma gradients \cite{gki80}. This gives rise to kinetic effects such as streaming plasmas and non-local transport, which are intensely studied because of their importance in many plasma environments, such as  high-density laser-produced plasmas \cite{pdd71,koo71,dms71,mob72,cgk73,gki80,lmp85,awb86,bcd88,ukm95,brt95,snb00,ggk04,bbr06,nfs06,rbb08,zrb09,mta13} and astrophysical plasmas like the solar wind \cite{pdf13}, solar atmosphere \cite{bra13,wbc08}, solar flares \cite{kde87}, and supernovae \cite{cga04}.

Here we show the emergence of kinetic behavior for ions in an ultracold neutral plasma \cite{kkb99,kpp07} by creating colliding plasma streams with controllable relative velocity and density.
It is the first observation in ultracold neutral plasmas of ion transport \cite{kkb00,lgs07,cmk10,kmo12,mcs13} that can not be explained with a hydrodynamic treatment.
%for increasing ratio of mean free path to the density-gradient lengthscale.
%Using laser-induced fluorescence, we measure the spatially resolved density and velocity distribution function for the ions \cite{cgk08}.
The emergence of kinetic effects is heralded by the absence of local thermal equilibrium for ions and interpenetrating streaming plasma populations.  This compliments recent work on acoustic-wave phenomena in ultracold neutral plasmas in the purely hydrodynamic regime \cite{cmk10,kmo12,mcs13}. Results are compared with a two-fluid hydrodynamic code \cite{bma03,bkl11}.



%A kinetic description of a plasma is much more challenging than hydrodynamic because it requires tracking of the full velocity and density distributions of constituents.

%Streaming ion beams are often created in double plasma devices \cite{gdo75,nni77,nbi04,sss05} through the application of external potentials.
%This technique is not effective for ultracold neutral plasmas due to their small size and transient nature. Ultracold neutral plasma  dynamics is dominated by electron pressure and ion inertial effects, similar to many laser-produced plasmas \cite{pdd71,mob72,bcd88,rbb08,koo71,dms71, cgk73}, but the initial density distribution  and electron temperature  are much more easily controlled in the ultracold regime than at high energies, providing significant advantages and opportunities for experiments.

%For example, there is a large body of experimental \cite{mld94,ggk04,ylw10} and theoretical \cite{gki80,lmp85,awb86,ukm95,brt95,bbr06,zrb09} work on nonlocal transport in plasmas produced from intense-laser-matter interactions.  Heating is often nonuniform, and understanding thermal transport is necessary for interpreting experimental results. In inertial confinement fusion experiments, it is studied intensely because of its impact on the design of ignition targets \cite{ggk04}. Nonlocal transport can also be important in astrophysical plasmas like the solar wind \cite{pdf13} and solar atmosphere \cite{bra13}.

 %that takes moments of the Boltzmann equation \cite{bal88}, with a collision term appropriate for Coulomb collisions \cite{csr50,sha53}. In highly collisional plasmas, with small density and temperature perturbations, classical expressions for the transport rates can be derived \cite{bra65}, for example in the Spitzer-H\"{a}rm theory for thermal transport in the presence of a temperature gradient \cite{sha53}. This is a local theory in which fluxes depend on gradients at the same location in space and time.





%, which are orders of magnitude colder than any other neutral plasma \cite{kpp07}.




%There are many possible instabilities in streaming plasmas, depending upon whether electrons, ions or both are streaming, the relative drift velocities and densities of the steams, and temperatures of the ions and electrons, which can differ in one stream compared to the other. When ions are streaming and the relative drift velocity, $v$, between two streams exceeds the ion-acoustic-wave phase velocity but is small compared to the thermal electron velocities, the dominant instability is the growth of ion sound waves \cite{mel86,gar93}.
%%, known as the ion-ion instability \cite{nbi04} or ion-ion acoustic instability \cite{gom87,gve90}. %The mechanism is inverse Landau damping and is similar to the %bump-in-tail instability for Langmuir waves occurring in the %presence of streaming electron beams. This describes the situation %in ultracold neutral plasmas.
%
%Early investigation of the instability criteria  was performed using a kinetic description in \cite{str64,fwo66}. Subsequent studies have analyzed the fluid-like versus kinetic-like properties in different regimes \cite{nbi04,gom87}, or presented systematic overviews of instability regions and mode properties depending upon  plasma parameters \cite{fwo66,gdo75,gom87,gve90,nbi04}.
%
% When viewed in the frame moving with one stream, the  streaming plasmas configuration  can be described as an ion beam-plasma system, in this case the density of streaming ions equals the density of background ions.
%At large beam velocity $v$ compared to the ion acoustic wave phase velocity $c_i$, there are three branches of the low-frequency electrostatic mode \cite{gdo75}- the standard ion acoustic wave in the stationary component, and fast and slow ion modes, which correspond to ion acoustic waves of the beam component moving with or against the beam.  (Electrons are typically assumed to neutralize both beam and background ions and have a uniform temperature, which is appropriate for our situation of cold ions and slow beams. Whether there are electron currents or not can affect the result.)
%%At large beam velocity compared to the ion acoustic wave phase velocity ($C_i$) but small compared to electron thermal speeds, all three modes are stable with normal Landau damping. As
%When the beam velocity approximately satisfies $c_i\leq v \leq 2c_i $, the  ion acoustic wave in the stationary component and the slow mode merge, and the resulting mode becomes unstable due to the interaction of the moving and stationary ions \cite{gdo75,nni77,gom87,gve90,gar93}, which is often termed the ion/ion acoustic instability \cite{gar93}.
%At moderately higher velocities, $v > 3 c_i $, the slow mode is unstable due to interaction between electrons and moving ions \cite{nni77,gom87,gve90,gar93}, which is often termed the electron/ion acoustic instability \cite{gar93}.
%At beam velocities below $c_i$, only a single mode exists, which is stable. It connects to the fast mode but becomes the ion acoustic wave as beam velocity goes to zero. The details of this spectrum depend upon many parameters, \cite{fwo66,gom87,nbi04}, and it has been studied experimentally in double \cite{gdo75} and triple plasma devices \cite{nni77}.
%
%
%%oblique gain
%
%
%
%%space plasmas



%There are many questions relating to the instability as observed in streaming ultracold plasmas. What determines the growth rate and wavelength of the mode that grows, and what are the dominant determinants of the growth rate? What limits the mode amplitude? Is observed ion heating due to the instability or is their turbulence in the streaming system? Is it an electron/ion  or ion/ion acoustic instability? This last question is related to whether a fluid description is sufficient  or kinetic effects are essential for describing the instability since the ion/ion instability can be described with a fluid treatment but the electron/ion instability can only be described in a kinetic picture \cite{nni77,mel86,gar93}. It seems possible for us to cross between these two regimes by changing the electron temperature. (Higher electron temperature and higher beam velocity favors the electron/ion instability \cite{nni77,gom87,gve90,gar93}.) If it is an ion/ion instability, it may be possible excite modes at oblique angles to the gap normal by increasing the relative velocity of the streams \cite{gar93,gve90} before the electron/ion instability takes over.
%Investigation of the oscillation frequency of the excited mode is also interesting. No oscillations were observed in our initial study, but ref.\ \cite{gom87} found that the unstable mode satisfies $\omega/k\simeq c_i\left(\frac{n_c-n_b}{n_e}\right)$, which is close to zero for our symmetric streams.
%%As the relative velocity increases above twice $c_i$, the mode with $\mathbf{k}\mathbf{\times} \mathbf{v_0}$ becomes stable, but modes propagating at an angle  with $\mathbf{k}\mathbf\cdot \mathbf{v_0}\simeq \omega $ experience growth  \cite{gar93,gve90}.







 %Another interesting question that is best addressed with the comparison of experiments and detailed modes is whether strong coupling modifies the properties of streaming instabilities.
% To our knowledge this question has not been examined. %Dusty plasmas
%Streaming ions or electrons in complex dusty plasmas have been shown to excite ion and dust acoustic waves in complex dusty plasmas \cite{sma02}.
%The presence of dust particles has been show to damp the growth of ion-ion instabilities in a double plasma device \cite{sss05}.
%But the only work on streaming dust particles we know of is a theoretical kinetic treatment \cite{psc04}, which would not address possible  affects of strong coupling.



%We are missing the one or two paragraph description of the physics of streaming plasmas and a more detailed overview of other work. (Have streaming plasmas been studied in dusty plasmas? I don't think so...if there are strong coupling affects seen in laser-produced plasmas, that would be important.)...look at the IAW paper. This is important because PRL is supposed to be for a general audience.

%Recent advances in UNPs have allowed for the ability to sculpt the plasma density upon creation. This technique was first implemented to excite ion acoustic waves \cite{cmk10}. Here we implement this new technique to create two plasma hemispheres, separated by a small gap, that stream through each other during plasma expansion to study collective modes and instabilities in UNPs. We use laser-induced fluorescence \cite{cgk08} to study the evolution of the ion density distribution, and the spatially-resolved mean velocity and local temperature.

\begin{figure*}[ht]
%\includegraphics[width=8in]{Fig1.eps}
\includegraphics[width=7in]{Fig1.pdf}
\caption{
 Evolution of the ion density distributions.
 The time since ionization is indicated. Initial peak densities are $6\times 10^{14}$\,m$^{-3}$.
  (a-c)  False color plots of the density distribution in the crossover between hydrodynamic and kinetic regimes ($T_e(0)=25$\,K).
  (d) 1D slices through the density profile for the same data as (a-c), normalized and offset for clarity.  Note the emergence of the central feature as the gap splits. The hydrodynamic simulation overestimates the prominence of the central density enhancement and the density depletions propagating away from the gap region.
  (e) Same as (d)  but for $T_e=105$\,K  showing more kinetic behavior. Note the absence of a significant central feature at the confluence of the ions streaming into the gap and the poor agreement between the data and hydrodynamic simulation.
  %(h) Expanded view of the density profile at 0.3\,$\mu$s to display to dimensions of the gap and the analyzed regions.
  Distance is normalized to plasma size $\sigma_0=1.5\,$mm.
  }
\label{fig:ImagesandProfiles}
\end{figure*}

\section{Experimental setup}
\label{sec:methods}

Ultracold neutral plasmas are created through photoionization of strontium atoms from a magneto optical trap (MOT) as described in \cite{scg04}.
%30-60\% of the atoms are ionized to produce an UNP with adjustable
The initial electron temperature  is tunable ($T_e(0)\approx1-1000$\,K), and is determined by the excess energy of the ionizing photons above threshold. Ion temperatures are set to approximately 1\,K by disorder-induced heating \cite{mur01,csl04}, resulting in strongly-coupled ions in the liquid-like regime \cite{kpp07,ich04}.  The plasma density distribution is a spherical Gaussian, $n(r)=n_0\mathrm{exp}(-r^2/2\sigma_0^2)$, with  $n_0\sim$\,$6\times10^{14}$\,m$^{-3}$ and $\sigma_0\sim$\,1.5\,mm.

A gap in the center of the plasma is created by placing an opaque wire in the path of the ionizing beam, which results in
%The wire is $500\,\mu$m thick, 20\,cm away from the atoms.
 two plasma hemispheres that stream into each other during plasma expansion.
 Streaming plasmas are precursors to complex phenomena such as instabilities, shock waves, and solitons \cite{mel86,cap94,bcd88,rbb08}, and they have been studied in a wide variety of environments
such as the solar wind \cite{gom87},
supernovae \cite{cga04}, %[ref],
 double plasma devices \cite{gdo75,nni77,nbi04,sss05},
and laser-produced plasmas \cite{pdd71,mob72,bcd88,rbb08,koo71,dms71, cgk73}.
A partially transmitting strip, instead of an opaque wire, was recently used to create a smaller amplitude density depletion in the plasma center \cite{mcs13}. This led to hydrodynamic phenomena such as gap splitting and propagation of localized density depletions at the ion acoustic wave speed. A similar technique was  used to create periodic density modulations and excite ion acoustic waves \cite{cmk10}.



As a diagnostic, a tuneable, narrowband  laser induces fluorescence from the primary Sr$^+$ transition, $^{2}$S$_{1/2}-^{1}$P$_{1/2}$ at $\lambda=$\,422\,nm \cite{lgs07}. This excitation beam is masked by a 1\,mm slit to illuminate only a central sheet of the plasma \cite{cgk08}. Fluorescence emitted close to perpendicular to the plane of the ionizing and 422\,nm beams is imaged onto an intensified CCD camera with a pixel size of $52\,\mu$m.
The resulting image, $F(x,y,\nu)$, is frequency dependent due to the natural linewidth and Doppler-broadening of the transition \cite{cgk08}.
Images are combined to obtain a map of
the density distribution of the ions in the illuminated plane, $n_i(x,y,z\approx 0)$
or the spatially resolved,  distribution function for  velocity perpendicular to the gap \cite{cgk08,cmk10}.

We also model the plasma evolution with a one-dimensional two-fluid simulation \cite{bma03,bkl11} in which we treat bulk flows (transport, compression, and rarefaction), energy exchange between particle species by collisions and the work done by  small-scale electric fields. We found that thermal conduction and viscous interactions made negligible contributions to the plasma energy balance in the temperature regimes explored here. The initial density distribution is matched to the profile observed experimentally within 100\,ns of plasma creation. Ion temperature is set equal to the value  after disorder-induced heating \cite{csl04} measured with LIF spectroscopy \cite{cgk08}. A one-dimensional simulation will not correctly reproduce the adiabatic cooling of electrons and ions caused by the three dimensional expansion of the ultracold plasma \cite{lgs07}. But the deviation is small and does not affect the phenomena discussed here.
%in the So in the simulation, we set the electron temperature using the known functional form for a self-similar expansion \cite{lgs07}, which is determined by the initial electron temperature and size of the plasma. (Details are given in the supplemental material \cite{SuppMaterial})  The ion hole propagation is insensitive to the ion temperature for our conditions of $T_i\ll T_e$.


The critical parameter that demarcates  hydrodynamic (collisional) and kinetic (collisionless) regimes is the ratio between the mean free paths of the particles and a characteristic scale-length of the plasma (the Knudsen number). When the mean free path is small compared with this scale length then the plasma is collisional and the particle velocity distribution functions are near Maxwellian. When the mean-free-path exceeds the characteristic scale-length then kinetic phenomena can become important.



 Mean free paths and collision timescales for ion-ion and electron-electron collisions respectively for conditions relevant to strontium ultracold plasmas  are defined in terms of the generalized collision frequency for a test particle ($\alpha$) streaming with speed $v_{\alpha}$ through a background of field particles ($\beta$)  with and  mass $m_{\alpha,\beta}$:
\begin{equation}
\nu^{\alpha,\beta} = \left(1 + m_{\alpha} / m_{\beta} \right) \psi\left( x^{\alpha,\beta} \right) \nu_0^{\alpha,\beta},
\label{eqnSB10}
\end{equation}
\noindent where
\begin{equation}
\psi(x) = \frac{2}{\sqrt{\pi}} \int_0^x t^{1/2} \exp(-t) dt,
\label{eqnSB11}
\end{equation}
\begin{equation}
x^{\alpha,\beta} = \frac{m_{\beta} v_{\alpha}^2}{2 k_B T_{\beta}},
\label{eqnSB12}
\end{equation}
\noindent and
\begin{equation}
\nu_0^{\alpha,\beta} = \frac{ e_{\alpha}^2 e_{\beta}^2 \lambda_{\alpha,\beta} n_{\beta}}{4\pi \epsilon_0^2 m_{\alpha}^2 v_{\alpha}^3}.
\label{eqnSB13}
\end{equation}
$\lambda_{\alpha,\beta}$ is the Coulomb logarithm for the interaction.
In the absence of a gap, at the density of the unperturbed plasma,  the ion-ion mean free path for ions with four times the thermal speed for $T_i=1$\,K  is 40\,$\mu$m for screening by  100\,K electrons. This is small compared to the plasma size $\sigma_0$, and we expect predominantly hydrodynamic behavior for the ion motion such as overall plasma expansion. Electron mean free paths are much longer, which might become important in higher temperature plasmas where the streaming population could influence the energy balance and, consequently, the dynamics of the plasma via high order moments of the distribution function, such as the heat flux. However, for the parameter space and phenomena explored here, we are not sensitive to these effects.

%For ion-ion collisions of four times the thermal velocity ($v_{th}$) for a typical ion temperature of $T_i=1$~K, the mean-free-path is 73\,$\mu$m, much less than the length scale of the gap in our experiments ($l=200 \mu m$). At $4v_{th}$ when $T_i=2$~K the mean-free-path is 300\,$\mu$m, on the order of the length scale of the gap. So kinetic effects due to the ions should be small in the parameter space of interest to us, which is supported by our observations. For collisions involving electrons, however, the mean-free-path approaches the gap dimension for one $v_{th}$ at $T_e=25$~K and so electrons should behave kinetically, but these effects appear to be unimportant at the electron energies encountered here. They might become important in higher temperature plasmas where the streaming population could influence the energy balance and, consequently, the dynamics of the plasma via high order moments of the distribution function, such as the heat flux. However, within the parameter space explored here the behavior of the plasma is dominated by the ion dynamics, which are firmly within the collisional, hydrodynamic regime.






\section{Results and Discussion\label{sec:RnD}}

A qualitative understanding of the plasma dynamics when a gap is formed can be gained from looking at the evolution of the ion density distributions. Figure \ref{fig:ImagesandProfiles} shows data for conditions that display phenomena characteristic of the crossover regime from hydrodynamic to kinetic ($T_e(0)=25$\,K) and the fully kinetic regime ($T_e(0)=105$\,K). The initial density distributions are similar for both cases.
Figures \ref{fig:ImagesandProfiles}(a)-(c) shows false color images of the two-dimensional central density slice $n_i(x,y,z\approx 0)$, and
Figs.\  \ref{fig:ImagesandProfiles}(d) and (e) show 1D central density traces at various times for initial electron temperature (d) $T_e(0)=25$\,K and (e) $T_e(0)=105$\,K.
%Figure \ref{fig:ImagesandProfiles}(a)-(c) shows false color images of the two-dimensional central density slice $n_i(x,y,z\approx 0)$ at various times after plasma creation, and
%Fig.\  \ref{fig:ImagesandProfiles}(f) shows 1D central density traces at various times for initial electron temperature $T_e(0)=25$\,K.
%Figure \ref{fig:ImagesandProfiles}(d) shows a 2D image for $T_e(0)=105$\,K, and Fig.\  \ref{fig:ImagesandProfiles}(g) shows the 1D central density traces.
(Due to optical access constraints, the plasma is slightly larger than the region illuminated by the fluorescence beam, creating the apparent oblong shape in the 2D images.)

Immediately after plasma formation, a gap is evident (Figs.\ \ref{fig:ImagesandProfiles}(a), (d), and (e)) with
%a width of $\sim$\,940\,$\mu$m and
a length scale for the sides of the gap of about $l \equiv\left| n/\nabla n \right|\sim$\, 400\,$\mu$m, which is determined by diffraction of the ionizing beam and is much less than $\sigma_0$.
 %(Figs. \ref{fig:ImagesandProfiles}b and \ref{fig:ImagesandProfiles}e)
The subsequent movement of ions into the gap %, and the behavior where these streams collide reflects the collisionality in the gap.
 can be understood hydrodynamically using the coupled ion-electron momentum balance equations \cite{kkb00}. In the presence of any density gradient, ions are accelerated ($a_0$)  by the electron pressure ($P_e$) according to $Mna_0\approx-\nabla P_e\approx-\nabla\left(nk_BT_e(0)\right)$. $T_e(0)$ is expected to be close to uniform \cite{rha03}, and a characteristic ion velocity after acceleration is $v_f\approx\sqrt{k_BT_e(0)/M}$. This suggests a timescale for motion of
\begin{equation}
\tau\approx \frac{v_f}{a_0}\approx -\frac{n}{\nabla n}\sqrt{\frac{M}{k_BT_e(0)}}.
\end{equation}
At the sides of the gap, this timescale is fast and can be identified with the timescale for ion streams to form and interact in the center of the gap, $\tau_{fast}\approx\sqrt{Ml^2/[k_BT_e(0)]}$. Here,   $\tau_{fast}\sim$\,8\,$\mu$s for $T_e(0)$=25\,K and $\tau_{fast}\sim$\,4$\mu$s for $T_e(0)$=105\,K, which approximately matches the timescales seen in the experiment.
Note that this argument also explains the overall expansion of the plasma, for which the length scale is that of the spherical gaussian, $-n/\nabla n\approx\sigma_0$. This gives the characteristic expansion time of a self-similar expansion, $\tau_{exp}\approx\sqrt{M\sigma_0^2/[k_BT_e(0)]}$ \cite{kpp07}
($\tau_{exp}= 15$\,$\mu$s and 31\,$\mu$s for $T_e(0)$=25\,K and $T_e(0)$=105\,K respectively).

\subsection{Signatures of the Kinetic Regime in the Density Evolution \label{sec:DensityEvolution}}

Whether the ion streams behave hydrodynamically or kinetically when they collide depends upon the inter-stream mean free path at the confluence. The relevant density-gradient length scale for this interaction  can be approximated as something between the  length scale of the side of the gap ($l$) and the gap width. For short mean free path, the streams cannot penetrate, and a density enhancement emerges at the confluence. The gap splits and two density depletions  propagate away from plasma center as seen in  \cite{mcs13}. There is indication of this behavior for $T_e(0)=25$\,K (Figs.\ \ref{fig:ImagesandProfiles}(b-c)). However, the central feature never rises to the height that the plasma would have had in the absence of an initial gap, and the amplitude of the features moving away from the center are much smaller than the initial gap (Fig.\ \ref{fig:ImagesandProfiles}(d)). These two caveats indicate that the colliding streams in the $T_e(0)=25$\,K  plasma are not fully in the hydrodynamic regime and kinetic effects are important. Experiments in \cite{mcs13} accessed and characterized the purely hydrodynamic regime for $T_e(0)=25$\,K by starting with a very small density depletion and approximately five times greater density in the gap. In this case, the central
feature rose to full height, and the amplitude of the traveling density depletions showed no significant attenuation.

To estimate the mean free path $\ell_{mfp}$ for an ion in one stream moving through the other, we use the density in the center at time $\sim \tau_{fast}/2\sim$\,4\,$\mu$s, which is about $10^{14}$\,m$^{-3}$, and take the velocity as $v_f\approx\,50$\,m/s. Equation \ref{eqnSB13} yields $\ell_{mfp}=600\,\mu$m.





To highlight the difference between the density evolution in the experiments and what one would predict for hydrodynamic behavior, we show in Fig.\ \ref{fig:ImagesandProfiles}(d) the results of a 1-D hydrodynamic simulation of the experimental data. For the hydrodynamic plasmas studied in \cite{mcs13}, the  simulation accurately captured all the features seen in the experiment. Here, there is obvious discrepancy between experiment and the hydrodynamic simulation in the gap region. The simulation overestimates the prominence of the central feature and traveling depletions.


The mean free path is further increased by increasing the electron temperature and relative velocity of the colliding streams, as shown in Fig.\ \ref{fig:ImagesandProfiles}(e) for $T_e(0)$=105\,K. Making the same approximations as before yields $\ell_{mfp}=7$\,mm.
In this case, the hydrodynamic phenomena related to the gap (central
density enhancement and traveling density depletions) are barely visible, indicating the dominance of kinetic effects.
There is also very large discrepancy between experiment and the hydrodynamic simulation in the gap region.


%Kinetic effects become important when the mean free path approaches the relevant spatial dimension of the system, which in this space is set by the gap.
%The data presented here is in a crossover regime in which hydrodynamics is starting to break down and kinetic effects are evident.
%As discussed below, we clearly see streaming of ions, which can only be explained in kinetic terms. However, the feature emerging at the confluence shows that collisions are still playing a role. For higher electron temperatures and lower plasma density in the gap, no feature appears, and we can study plasmas fully in the kinetic regime.


\subsection{Measuring the Velocity Distribution Function \label{sec:VelDistribution}}

To probe the effect of the gap on the plasma dynamics more quantitatively, and to support our statements regarding the emergence of kinetic behavior, we analyze the fluorescence excitation spectrum.
We obtain the spectrum of a region of the plasma, $ S(\nu)_{region}$, by summing the fluorescence from that region,
\begin{equation}
\label{eq:Fluorspec}
 \sum_{x,y\in region} F(x,y,\nu)\propto S(\nu)_{region}.
\end{equation}
We will first discuss the spectrum of the entire illuminated plasma sheet, which is obtained by expanding the region to encompass the entire field of view. Then  we will discuss the spectra of resolved regions of the plasma near the gap.
%What about plasma out of beam?


\begin{figure}[ht]
\begin{center}
%\includegraphics[width=3.21in]{CompareWholeCloudSpecSubset2TePlusScaled.eps}
%\includegraphics[width=3.21in]{Fig2.eps}
\includegraphics[width=3.4in]{Fig2.pdf}
\caption{Spectra of entire plasma sheet. (a) Effect of the gap and changing $T_e(0)$. Frequency has been converted to velocity ($\mathrm{v}$) through the Doppler shift $\mathrm{v}=\nu/\lambda$, and time since ionization is indicated.   Note the emergence of bimodal structure, corresponding to streaming plasmas. %Initially, the spectrum of the plasma is unaffected by the presence of the %gap. However at intermediate times a wider, bimodal shape appears, %corresponding to streaming ions and a fast timescale for the conversion of %electron thermal energy to directed ion velocity. At later times, the effect %of the gap is no longer significant. For plasmas without a gap, the %increasing Doppler-width results from the overall plasma expansion, and this %effect also contributes when a gap is present.
(b) Spectra of plasmas with gaps  with scaled velocity and  $t/\tau_{fast}\sim1.3$. The universal behavior highlights the emergence of the new timescale, $\tau_{fast}$.
(c) Same as (b) but for $t/\tau_{exp}\sim 0.7$. At later times, structure has vanished, and universal behavior indicates the scaling of the transfer of electron energy into plasma expansion velocity.}
\label{fig:WholeCloudSpec}
\end{center}
\end{figure}

The spectral shape is dominated by Doppler shifts and broadenings resulting from ion velocities, and for simplicity we can neglect the  small broadening of the spectra due to the natural linewidth of the transition. This allows us to interpret the spectrum
 as the distribution of ion velocities perpendicular to the gap by plotting the spectra versus velocity, where velocity is calculated as $\rm{v}=\delta \nu \lambda$. $\delta \nu$ is the detuning of the fluorescence excitation laser from resonance and $\lambda$ is the wavelength. (The FWHM,  natural linewidth of the transition $\gamma/2\pi=20$\,MHz leads to a width
 in velocity of $\delta \mathrm{v}=\gamma\lambda/2\pi=8.4$\,m/s.)



\subsection{Velocity Distribution of the Entire Plasma \label{sec:DensityEvolution}}


Using this approximation, Fig.\ \ref{fig:WholeCloudSpec}a shows the velocity distribution for the entire plasma sheet, which reflects directed ion motion (plasma expansion or streaming) and thermal ion motion.  At early times, the distribution is unaffected by the existence of the gap, and the width reflects the ion temperature after disorder-induced heating (DIH) \cite{mur01,csl04,cdd05}.
%Could cut next sentence but leave for now.
DIH occurs as excess potential energy in the initially random ion spatial distribution is converted to ion kinetic energy. The timescale for DIH is the inverse of the ion plasma oscillation frequency, $1/\omega_{pi} =\sqrt{M\epsilon_0/ne^2}\sim\,1\,\mu$s, where $M$ is the ion mass. This is much faster than the hydrodynamic timescale on which the plasma reacts to density inhomogeneities.
After DIH, ion velocities increase as electron thermal energy is converted to directed ion motion through the hydrodynamic process discussed above.  The  distribution for a plasma without a gap broadens as the ions accelerate radially, but it retains a generally featureless Gaussian shape characteristic of a self-similar expansion \cite{lgs07}.

The spectrum of the perturbed plasma broadens more quickly and develops a bimodal shape. We will show below that bimodality reflects ions from the two sides of the gap streaming into the gap and through each other. This motion is still driven by the conversion of electron thermal energy into ion expansion energy, but it occurs on the timescale $\tau_{fast}$. Figure \ref{fig:WholeCloudSpec}b shows spectra for the perturbed plasma at the same scaled time, $t/\tau_{fast}\sim1.3$ and with velocity scaled by $v_f$. The universality of the scaled data emphasizes how the gradient in the electron thermal pressure drives the dynamics.
Note that at later times %$t /\tau_{exp}\sim 0.67$
(Fig. \ref{fig:WholeCloudSpec}a top panel) the distributions for the perturbed and unperturbed plasmas display the same scaled width with no sign of bimodality.
%Scaled spectra at  (Fig. \ref{fig:WholeCloudSpec}c) also display a universality at later times.
The structure in the global velocity distribution has vanished even though some structure in the density distribution still persists (Fig. \ref{fig:ImagesandProfiles}).

\subsection{Spatially Resolved Velocity Distributions and Streaming Plasmas \label{sec:DensityEvolution}}

To probe the local dynamics of the plasma, we analyze spectra and extract the 1D velocity distributions of smaller spatial regions  in the vicinity of the gap, which measure local ion velocities and temperatures \cite{gls07,lgs07,cgk08} (Figs. \ref{fig:RegionSpectra25} and   \ref{fig:RegionSpectra105} for initial electron temperatures 25\,K and 105\,K respectively).
Due to the symmetry of our plasma with respect to the gap, corresponding regional spectra in the two hemispheres mirror each other. We only show the spectra of the central region and the regions to the right of the gap. Spectra are obtained from Eq. \ref{eq:Fluorspec}, where $region$ refers to the area covered by regions 0 to 3 in Fig. \ref{fig:Regions}.  The dimensions of these regions are 190\,$\mu$m perpendicular to the gap and 760\,$\mu$m parallel to it.
 The shift of a distribution represents the mean velocity of an ion population, and the width represents the rms velocity spread \cite{cgk08}. We typically quote the width in terms of an effective local temperature, but the width is also affected by spread in average ion velocity across the region and lack of thermal equilibrium.




\begin{figure}[ht]
\begin{center}
%\includegraphics[width=3.2in]{InnerSpectrasize0.0625sigma25KSubset2.eps}
\includegraphics[width=3.5in]{Fig3.pdf}
\caption{Spatially-resolved regional spectra for  plasmas with $T_e(0)=25$\,K. Time since ionization is indicated on the right. The solid lines are data with and without the gap while short dashed lines are fits to perturbed velocity distributions. Shifts represent ion mean velocities and widths represent ion temperature. For plasmas without the gap, positive velocities reflect expansion \cite{cgk08}. For plasmas with the gap, multiple populations appear moving left and right.  Time since ionization is indicated on the right.  The long dashed lines show results of the hydrodynamic simulation.
}
\label{fig:RegionSpectra25}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
%\includegraphics[width=3.2in]{InnerSpectrasize0.0625sigma105KSubset2.eps}
\includegraphics[width=3.5in]{Fig4.pdf}
\caption{Same as Fig.\ \ref{fig:RegionSpectra25}, but for $T_e(0)=105$\,K. Compared to Fig. \ref{fig:RegionSpectra25}, in the perturbed plasma, average velocities are much higher and populations moving to the right are more prominent.
}
\label{fig:RegionSpectra105}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.2in]{Fig5.pdf}
\caption{Regions used in analysis of local ion velocities and temperatures.}
\label{fig:Regions}
\end{center}
\end{figure}


Without the gap, regional spectra show expansion velocities that increase with time and distance from center  as expected \cite{kpp07}.
 For regions far from the gap and especially at early times, the distribution shape, shift, and width are essentially unchanged by introduction of a gap.
 But there are dramatic differences between spectra of perturbed and unperturbed plasmas for inner regions.

 Signal at positive velocities represent ions moving away from plasma center, whereas, negative velocities represent ions moving towards it. Positive and negative velocity groups in region 0  indicate ions moving through the center of the gap. It is important to note that ions move through the regions, so spectra for a given region at different times may reflect different ions. For instance, neglecting collisions and given the dimension of the regions, particles in a particular region with mean velocity of 60\,m/s, will be in a contiguous region in approximately $\sim3\,\mu$s.

There are many common features in $T_e(0)=25$\,K and $T_e(0)=105$\,K data that illustrate the general  behavior of streaming ultracold neutral plasmas.
At early times ($0-3\,\mu$s), the density gradient in regions 1 and 2 produces an acceleration of ions  towards the gap. This is followed by the appearance of two beams of high velocity ions crossing each other in region 0.
The stream velocities closely match the characteristic velocity  introduced earlier ($v_f=\sqrt{k_B T_e(0)/M}=100$\,m/s and 50\,m/s for 105\,K and 25\,K respectively). The bulk velocity of each population greatly exceeds the ion thermal speed and is comparable to the  phase velocity of ion acoustic waves.
At later times ($7-10\,\mu$s) these populations have crossed region 0, and ions  that originated on the other side of the gap now appear in regions 1-3 as ions moving away from the gap. These interpenetrating streams and local velocity distributions that cannot be defined by a single Maxwellian with a well-defined temperature are indicators that the plasma is in the kinetic regime. Data from hydrodynamic plasmas studied in \cite{mcs13} showed no deviation from local Maxwellians.

In Figs.\ \ref{fig:RegionSpectra25} and  Figs. \ref{fig:RegionSpectra105}, it is interesting to note that ion populations remain cold %and strongly coupled
until they have passed through the gap, after which their velocity spread greatly increases (Fig. \ref{fig:VelAndTemperture}).
Ions remaining in the gap or moving in at later times (10\,$\mu$s) eventually merge into one population of hot ions.
 %($T_i=2$\,K is $\sim\Gamma_i=1$)


Important differences between the data sets reflect the variation in electron thermal pressure. For example, moving populations appear sooner for $T_e(0)=105$\,K ($0.5\,\mu$s) than $T_e(0)=25$\,K, and velocities are higher for higher $T_e(0)$ (Fig. \ref{fig:VelAndTemperture}). The ion temperature in region zero at later times is also hotter for $T_e(0)=105$\,K than $T_e(0)=25$\,K.


%For $T_e(0)=105$\,K, (Fig. \ref{fig:RegionSpectra105}) at early times ($t=0.5\,\mu$s) a population of ions near the edge of the gap (region 3) is almost identical to the unperturbed plasma. Regions 1 and 2 start to show populations of ions with negative velocity (moving into the center), while region 0 shows a small unmoving population.  Initially, no populations appear with positive velocities (moving to the right) because the density gradient does not produce acceleration to the right. However, at $3\,\mu$s, most of the regions show two populations of ions moving in opposite directions, and at $7\,\mu$s regions 2 and 3 show the appearance of large positive velocity ions.  This indicates particles that have traveled from the left hemisphere through the center and across several regions. At $10\,\mu$s, two distinct populations of ions are still clear in regions 2 and 3, but in region 0 and 1, and at 15\,$\mu$s in region 2, we are no longer capable of distinguishing two populations. If we fit these distributions as one population we obtain hot ions with $T_i\approx8\pm1$\,K.

%We have similar behaviour for $T_e(0)=25$\,K, (Fig. \ref{fig:RegionSpectra25}) with some clear differences. At $0.53\,\mu$s we have an initially unmoving population of ions in all regions. At 3\,$\mu$s,  we see regional spectra that are very similar to those at 0.53\,$\mu$s and $T_e(0)=105$\,K (Fig.  \ref{fig:RegionSpectra105}), showing that negative velocity starts to develop more slowly for lower $T_e(0)$. Regions 1 and 2 show distinct populations with negative and  $\sim$\,0 m/s mean velocity, and almost no right going populations; while region 0, shows three groups of particles with a central group at 0 m/s mean velocity.  At $7\,\mu$s, regions 0 to 2 show signs of streaming plasmas, however not as clearly as Fig.  \ref{fig:RegionSpectra105}. At $10\,\mu$s in region 0 and at 15\,$\mu$s in regions 1 and 2, we are no longer capable of distinguishing 2 populations, resulting in fitted ion temperatures of  $T_i\approx5\pm1$\,K.

There is also significant difference in the penetration depth of ions from one side into the plasma on the other side of the gap. We estimate the penetration length in the plasma from the farthest excursion of right-moving populations. For $T_e(0)=105$\,K, right-moving ions clearly appear in region 3 (10\,$\mu$s), while for $T_e(0)=25$\,K right-moving ions appear in region 1, and perhaps region 2, and the population is less distinct than for $T_e(0)=105$\,K. $T_e(0)=48$\,K (not shown) displays intermediate behavior.
%The farthest right moving population for $T_e(0)=25$\,K  is much smaller population than the equivalent for  $T_e(0)=105$\,K
No clear right-moving population can be found at times later than $\sim$10\,$\mu$s for any $T_e(0)$ (data not shown). One expects a greater penetration length for higher $T_e(0)$ because the Coulomb collision cross section decreases with  collision velocity, $\sigma \propto 1/v^4$. From these measurements, it should also be possible to estimate the stopping power of electrons and strongly coupled ions in an ultracold plasma for a penetrating ion stream \cite{pme91,bps05,tbd08,ggm08}.
%Bigger cross sections (calculate Rutherford cross sections) are expected for slower moving particles (colder electrons). A larger mean free path is expected for faster particles (calculate Spitzer times). It is expected that higher $T_e(0)$ results in longer penetration length as this length is related to the stopping power \cite{pme91,bps05,tbd08,ggm08}. (strong coupling ...stopping power..interesting)

%Can we relate this to the mean free path for an ion with velocity equal to the stream velocity?

Hydrodynamic simulations are included in Figs.\ \ref{fig:RegionSpectra25} and  Figs. \ref{fig:RegionSpectra105} to emphasize the breakdown of the hydrodynamic approximation. But it is also interesting to note  the nature of the discrepancy between the data and the simulation. In the central region, when two symmetric, relatively cold, streaming plasma populations are present, the hydrodynamic code predicts a single cold population with zero average velocity, which is what one would expect given the constraint in the simulation of a local thermal equilibrium for the  ions. However, in regions away from the center,  the distributions from the hydrodynamic simulation are close the the distribution for the population of ions that have not crossed the gap, but the contributions from ions that have streamed across the gap are absent in the hydrodynamic simulation. This reflects the inability of the hydrodynamic code to account for non-local transport. The hydrodynamic code also tends to underestimate the ion temperature at later times. 

%For comparison, we show the region-resolved velocity distributions for a plasma with a small initial density perturbation showing purely hydrodynamic behavior in Fig.\ \ref{fig:RegionSpectra25Hydro}.


\begin{figure}[ht]
\begin{center}
%\includegraphics[width=3.1in]{IonVelocityTemperaturePlot6Panels2TeLimited2.eps}
%\includegraphics[width=3.1in]{IonVelocityTemperaturePlot6Panels2TeLimited3.eps}
\includegraphics[width=3.5in]{Fig6.pdf}
\caption{ Ion velocities and temperatures for the fast moving streams in Figs. \ref{fig:RegionSpectra105} and \ref{fig:RegionSpectra25}. (top) Velocities of the fast moving streams. Note that streams moving towards the gap (velocity $<0$) appear first in region 2. Streams moving away from the gap velocity $>0$) appear first in region 0. (middle) Temperature of left-moving streams moving towards the gap. (bottom)  Temperature of right-moving streams that have crossed the gap and are moving away from it. Note the different temperature scales. }
\label{fig:VelAndTemperture}
\end{center}
\end{figure}

\subsection{Trends in the Ion Temperature and Bulk Velocity} \label{sec:TempandVelocity}

To accurately extract ion temperatures and velocities we fit regional spectra (Figs. \ref{fig:RegionSpectra105} and \ref{fig:RegionSpectra25}) to  one or the sum of two Voigt profiles.
The Voigt profile is the convolution of a Lorentzian with homogeneous width  $\gamma$ and shifted Gaussian for Doppler effects \cite{kpp07}. In this way we take spectral broadening due to the natural linewidth of the transition into account for quantitative analysis. As mentioned above, the temperature is extracted from the width of the Gaussian. Temperature measurements are sensitive to many systematic effects,
%  The shift of a Gaussian represents the mean velocity of the ion population and the width represents the rms velocity spread \cite{cgk08}, which we typically interpret as reflecting the local temperature.  The  temperature measurement is affected by spread in average ion velocity across the region and lack of thermal equilibrium, so its accuracy is limited.
but it still an informative parameter.
Data that fit well to two profiles give clear indication of different populations of streaming ions with different mean velocities and temperatures. In these cases, the dotted lines show the individual Voigt profiles of each population, and the sum of both these curves is in good agreement with the data.

Ion velocities and temperatures for the  streams are shown in Fig. \ref{fig:VelAndTemperture} whenever they are resolvable.
%Figure \ref{fig:VelAndTemperture} shows the ion velocities and temperatures for the fast moving streams obtained from the fitted spectra in Figs. \ref{fig:RegionSpectra105} and \ref{fig:RegionSpectra25}.
For higher $T_e(0)$ we obtain faster streaming velocities, which is consistent with the separation of the bimodal distribution in Fig. \ref{fig:WholeCloudSpec}a.  In regions 1 and 2, populations that are moving into the gap (negative velocities and middle panel) appear to be relatively cold ($T_i\lesssim 2$\,K), while populations that have traveled across the gap  and moved an appreciable distance through a counter-streaming plasma (positive velocities and bottom panel), have higher temperatures ($T_i\gtrsim 20$\,K). In the center, region 0, particles moving to the left and right show very similar ion temperatures.

For all regions, velocities decrease with time. Decreasing plasma gradient and decreasing $T_e$ \cite{lgs07} with time imply that ions moving through the gap at later times have experienced less acceleration during their trajectories than at early times. But velocities also decrease for an individual population as it moves through the regions.  This may reflect that after ions move past the gap, the local electric field opposes the average motion for that population.  Damping or collisions may also play a role, which is supported by the considerably higher temperatures measured for populations that have crossed the gap.

When only one distinct population of ions is visible in the velocity distribution function, such as in region 1 at early times ($<3\,\mu$s), the hydrodynamic simulation does a reasonable job of predicting the average ion velocity (\ref{fig:VelAndTemperture}(top)). When two streaming populations are present, however, such as in region 1 at later times or in region 0, the velocity distribution is far outside the regime that can be treated by a hydrodynamic treatment. The simulation tends to track close to the average velocity, which is always zero in region 0. Similarly, in regions farther from the gap (region 2), the temperature in the simulation agrees reasonably well with the temperature of the population moving towards the gap (Fig.\ \ref{fig:VelAndTemperture}(middle)), indicating these ions are relatively unaffected by the small population of high velocity ions that have crossed the gap and are streaming through the region. This highlights the phenomena that characterize departure from the hydrodynamic regime.






%(I don't know yet where to put this: In region 0 the is a small 0 m/s population from the start, could be a residual from not clean (sharp) density gap. This is tied to: The apparent offset in the fit for $T_e(0)=105$\,K in region 0 at 3\,$\mu$s represents a small population of hot ions with 0 mean velocity. )

%\begin{figure}[ht]
%\begin{center}
%%\includegraphics[width=3.1in]{IonVelocityTemperaturePlot6Panels2TeLimited2.eps}
%%\includegraphics[width=3.1in]{IonVelocityTemperaturePlot6Panels2TeLimited3.eps}
%\includegraphics[width=3.1in]{Fig7.pdf}
%\caption{ Ion velocities and temperatures for plasmas with small initial density perturbations and hydrodynamic behavior. Note the absence of significant heating and indications of streaming plasmas and the good agreement between data and hydrodynamic simulation and the (From \cite{mcs13}.) }
%\label{fig:VelAndTemperture}
%\end{center}
%\end{figure}

\section{Conclusion \label{sec:Conclusion}}
In this work we have used the ability to create streaming populations of particles in ultracold neutral plasmas to demonstrate the crossover from hydrodynamic to kinetic behavior and predominantly kinetic behavior. This work is complimentary to a recent paper in which we studied plasmas with a similar initial geometry in the hydrodynamic regime \cite{mcs13}. The ratio of the ion mean free path to the length scale of plasma inhomogeneities   as ions move through the gap and the populations collide is the crucial parameter for determining the regime of behavior, and it can be tuned by varying the ion density in the gap and the electron temperature, which changes the stream velocity. The hydrodynamic regime is characterized by a density enhancement at the confluence of the plasma streams, gap splitting, and localized density depletions that propagate away from the initial gap. The kinetic regime lacks these features and displays velocity distributions that strongly deviate from local thermal equilibrium and show distinct populations of interpenetrating, counter-streaming plasmas.


There are many future directions in which this work can go. This study has focused on a qualitative description of plasma behavior in the various regimes, but it would be informative to  quantitatively map the boundary between different regimes in the parameter space of stream velocity and plasma density in the gap. One could use, for example, the appearance of streaming populations or visibility of a localized density enhancement in the confluence region as an indicator of the plasma regime.

A full kinetic description will allow a much more detailed and quantitative discussion of the observed effects. Alternatively, data such as this can provide a stringent test of kinetic codes. Kinetic codes, however, are much more computationally intensive.
Rather than move to a full kinetic description, a common strategy is to generalize transport coefficients in order to extend the usefulness of hydrodynamic treatments.  This is often done to treat
nonlocal transport in plasmas with low collisionality, in which particle and energy fluxes at a given location are affected by conditions in distant regions of the plasma, which represents an important and complex problem.
%For example, there is a large body of experimental \cite{mld94,ggk04,ylw10} and theoretical \cite{gki80,lmp85,awb86,ukm95,brt95,bbr06,zrb09}
Often, \textit{ad hoc} schemes are used in which the flux  is simply limited to a fraction  of its free-streaming value \cite{mld94,ggk04,ylw10,gki80,awb86,lar93}. Data such as presented here can provide a measurement of fluxes and thermodynamic gradients with high temporal and spatial resolution. This could provide a  test of flux-limiting treatments or inform improved schemes to extend hydrodynamic models to address nonlocal effects.

%Nonlocal transport of mass and energy is clearly significant in the data presented........
%
%This leads to the concept of nonlocal transport in plasmas with low collisionality, in which fluxes at a given location are affected by conditions in distant regions of the plasma, and it represents an important and complex problem. For example, there is a large body of experimental \cite{mld94,ggk04,ylw10} and theoretical \cite{gki80,lmp85,awb86,ukm95,brt95,bbr06,zrb09} work on nonlocal transport in plasmas produced from intense-laser-matter interactions.  Heating is often nonuniform, and understanding thermal transport is necessary for interpreting experimental results. In inertial confinement fusion experiments, it is studied intensely because of its impact on the design of ignition targets \cite{ggk04}. Nonlocal transport can also be important in astrophysical plasmas like the solar wind \cite{pdf13} and solar atmosphere \cite{bra13}.

%A full kinetic description of the plasma would in principle describe the breakdown of classical transport theory, but such an approach is very intensive computationally, and a common strategy is to generalize the transport coefficients in order to extend the usefulness of hydrodynamic treatments.  This is particularly done to treat
%nonlocal transport in plasmas with low collisionality, in which particle and energy fluxes at a given location are affected by conditions in distant regions of the plasma, and it represents an important and complex problem.
%%For example, there is a large body of experimental \cite{mld94,ggk04,ylw10} and theoretical \cite{gki80,lmp85,awb86,ukm95,brt95,bbr06,zrb09}
%Often, \textit{ad hoc} schemes are used in which the heat flux  is simply limited to a fraction  of its free-streaming value \cite{mld94,ggk04,ylw10,gki80,awb86,lar93}. Data such as presented here can provide a measurement of fluxes and thermodynamic gradients with high temporal and spatial resolution. This could provide a check of or test of flux-limiting treatments or inform improved schemes to extend hydrodynamic models to address nonlocal effects.


We have mentioned that it should be possible to estimate the stopping power of ions by studying the penetration depth of the ion streams. Away from the initial gap, the streams move into regions of cold ions that are strongly coupled, which gives greater interest to such a measurement \cite{pme91,bps05,tbd08,ggm08}.
With sharper density features in the initial plasma, it may also be possible to excite and study shock waves \cite{hua02,rha03}.
%Shock waves occur when particle flows exceed the speed of sound in a media \cite{mho80}, and they are typically associated with the formation of a shock front across which thermodynamics quantities such as the pressure, density, and temperature change very rapidly \cite{hua02}. Shocks  can be generated by a steep pressure gradient that accelerates part of the plasma, and they are very important in the study of astrophysical and interstellar objects \cite{mho80} and plasmas produced from intense laser interactions with clusters \cite{kds03,pmf07} and solid targets \cite{mor05physplasmas,mor05pre,gmo06}.


This work was supported by the Department of Energy and National Science Foundation (PHY-0714603). SJB acknowledges funding support from NASA.


%\bibliography{bibliography,library}
\begin{thebibliography}{55}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
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\expandafter\ifx\csname bibfnamefont\endcsname\relax
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