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\def\PRL {{\it Phys. Rev. Lett.}}
\def\PP  {{\it Phys. Plasmas}}
\def\PS {{\it Phys. Scr.}}
\def\NPG {{\it Nonlinear Processes in Geophys.}}
\def\erf {\mathop{\rm erf}\nolimits}

\definecolor{light-blue}{rgb}{0.75,0.9,1}
\pagecolor{light-blue}

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\begin{document}

\begin{center}
{\Huge\bf Isolated Electrostatic Structures Observed in the Auroral Zone and
Magnetosheath by the Cluster Spacecraft}
\vskip 0.1in
{\LARGE\bf J. S. Pickett, S. W. Kahler, and L.-J. Chen }
\vskip 0.05in
{\LARGE\it Department of Physics and Astronomy, University of Iowa, Iowa City,
IA 52242, USA}
\end{center}

\begin{multicols}{2}
\small

\noindent
We report on recent measurements of isolated electrostatic structures observed 
in the auroral zone and magnetosheath by the Wideband (WBD) plasma wave receiver 
located on each of the four Cluster spacecraft.  These structures are bipolar 
(electron holes) and tripolar (weak double layers) in form with time durations 
of tens of microseconds to tens of milliseconds and peak-to-peak amplitudes up 
to ~80 mV/m.  We show examples of the two types of structures and discuss the 
reasons for the large differences in their characteristics between the two 
regions.   We also present the details of a cross correlation study, the purpose 
of which is to identify the same structures on more than one spacecraft in order 
to determine whether they propagate and evolve as is suggested by simulations.  
We briefly comment on the global significance of these structures within the two 
different regions. 

\section*{\color{blue} Cross-Spacecraft Correlations}
\noindent Plotted below are data from the Cluser WBD experiments
showing the electric field amplitudes (peak to peak)
vs. time.  The angle of the antenna with respect to the direction
of the measured magnetic field for the chosen events is shown in
the middle of the panels.  The first plot shows a fairly large
amplitude bipolar structure that may have propagated from Cluster 3
to Cluster 2, in doing so having lost some of its amplitude.  The
separation of the spacecraft and the velocity of the structure
based on this separation are shown at the top of the plot.  The
second plot has the same format, but here we are correlating
the tripolar structures at fairly low amplitudes.  The derived
velocities for both structures are consistent with electron beam
velocities for this region.the magnetic field strength at the position where the electrostatic
solitary waves were observed. The observations indicate:
\centerline{\includegraphics[width=0.59\columnwidth]{correlation.eps}}
\centerline{\includegraphics[width=0.59\columnwidth]{correlation_1.eps}}

\section*{\color{blue} 3D azimuthally symmetric solutions}
With magnetic field-aligned electrons (zero cyclotron radius),
localized potentials such as Gaussian potentials given by
$\Phi(r,z)=\Psi \exp{(-z^2/2\delta_z^2)}\times
\exp{(-r^2/2\delta_r^2)}$
can be self-consistently supported by the plasma provided that
the peak potential amplitude $\Psi$ and the widths $\delta_z$ and $\delta_r$
satisfy the inequality relation [4,7]
\begin{eqnarray*}
\delta_z & \geq & \sqrt{{2(4\ln 2 -1)\over{{\sqrt{\pi}e^\Psi[1-\erf(\sqrt{\Psi})]/\sqrt{\Psi}}-{4/\delta_r^2}}}} \;.
\end{eqnarray*}
\vskip 0.05in
\begin{minipage}{0.35\columnwidth}

\end{minipage}
\begin{minipage}{0.60\columnwidth}
\noindent
The inequality width-amplitude relation (plotted on the left) allows
\begin{itemize}
\item 
to distinguish BGK electron holes with other fluid solitons whose
width-amplitude relation is of 1-1 mapping based on macroscopic variables.
\item a high probability of excitation as
for a fixed amplitude there is a wide range of permitted widths.
\end{itemize}
\end{minipage}
\noindent
(o): the allowed parameter range\\
(x): the forbidden

\section*{\color{blue} No localized solutions in 3D when B=0}
\noindent
The self-sustaining of electron holes relies on the decrease of the passing
electron density [5]. When there is no magnetic field to confine electron
motion, the passing electron density as a function of potential amplitudes
has qualitatively different behavior for different dimensionality.\\
\vskip 0.04in
\begin{minipage}{0.35\columnwidth}
\centerline{\includegraphics[width=\columnwidth]{bipolar_amp.eps}}
\end{minipage}
\noindent
\begin{minipage}{0.6\columnwidth}
\centerline{\includegraphics[width=\columnwidth]{tripolar_amp.eps}}

\end{minipage}
\begin{itemize}
\item solitary waves are present in regions of weak magnetic field,
and therefore neglecting
finite cyclotron radius effects is not justified in these regions.
\item
the largest amplitudes measured in a particular region
are higher for regions with higher background magnetic field strength.
\end{itemize}
\vskip 0.05in
\begin{minipage}{0.35\columnwidth}
\centerline{\includegraphics[width=\columnwidth]{bipolar_duration.eps}}
\end{minipage}
\noindent
\begin{minipage}{0.6\columnwidth}
\centerline{\includegraphics[width=\columnwidth]{tripolar_duration.eps}}

\end{minipage}

\vskip 0.2in
\noindent{\bf The reason in 3D $n_p$ increases with the potential amplitude:}
\vskip 0.05in
\begin{minipage}{0.65\columnwidth}
Plotted on the right: the charge density distribution for a localized
potential hump. Electrons incident from outside
the potential will not only be accelerated but their orbits
also bend toward the positive core
and result in a density concentration of passing
electrons near the core.
\end{minipage}
\begin{minipage}{0.3\columnwidth}
\end{minipage}

\vskip 0.08in
\noindent
Because of the density concentration of passing electrons and the repelling
of ions from the positive core, the plasma cannot self-consistently
support localized potential humps [4].
\section*{\color{blue} Conditions for coherence in 3D with finite B}
To take into account effects of finite cyclotron radius,
consider motion of an electron inside the solitary potential:\\
\begin{minipage}{0.9\columnwidth}
\end{minipage}
\vskip 0.1in
\noindent
  The above figures illustrate that the electron motion is influenced
  by the uniform magnetic field and a 3D electric field {\bf E}(x,y,z).
  To know if finite cyclotron radius would result in decoherence of
  electron holes, we need to know if the distance of the instantaneous guiding
  center to the symmetry axis (z-axis) would increase. For this purpose,
  the electron trajectory is projected onto the 2D plane (shown on the right)
  that is perpendicular to the magnetic field, and {\bf E}(x,y,z) becomes
  {\bf E}(x,y,t) where the time dependence comes from the z-dependence.
  The equations of motion for the coordinates of the instantaneous guiding
  center can be written down and solved analytically
  to the lowest order dependence on the cyclotron radius.
  It turns out that the instantaneous guiding
  center would spiral around the zeroth order guiding center trajectory
  (straight line along {\bf B}) provided that the frequency of electron
  bounce ($\omega_b$) inside the solitary potential $\Phi$ is much greater
  than the cyclotron frequency ($\omega_c$) and the inhomogeneity length
  ($r_U$) of {\bf E}(x,y,t) is much greater than
  the cyclotron radius ($r_c$) [7].

\vskip 0.01in
\noindent\begin{minipage}{0.5\columnwidth}
\end{minipage}
\noindent\begin{minipage}{0.45\columnwidth}
$\Psi$: peak potential amplitude\\
$\delta_z$: parallel size\\
$\delta_r$: perpendicular size\\
$B$: magnetic field strength\\
$e,m$: charge and mass of electrons
\end{minipage}

\vskip 0.01in
Significance:
\begin{itemize}
\item lifetime of electron holes depends on the above ratios. The
smaller the ratios, the longer lifetime. Within the lifetime,
the inequality width-amplitude relation that we derived
for 3D electron holes in strongly magnetized plasmas still applies,
and hence a high probability of electron hole excitation in driven
or turbulent plasmas.
\item the conditions provide a ground to
understand the observational trend shown in the very first figure.
In the auroral ionosphere where solitary waves are observed,
the above conditions are satisfied with $\omega_b/\omega_c \sim 0.1$ and
$r_c/r_U \sim 0.01$ (using the parameters reported in [8]).
When we move to a weaker magnetic
field region (as is the case for all observations presented in the first
figure), either the potential amplitude has to decrease or the size
has to increase in order to keep the same conditions satisfied,
which results in smaller electric
field amplitudes.
\end{itemize}
\section*{\color{blue} Conclusions}
\begin{itemize}
\item The strength of the magnetic field and solitary wave parameters have to
satisfy the above two conditions for electron holes to remain coherent.
Under the conditions, there is a high probability of electron hole
excitation. The presence of a large number of small scale electron holes
can alter the bulk properties of the plasma
such as the dc resistivity and temperature.
\item
The conditions are consistent with the observational trend that in weaker
magnetic field, the electric field amplitudes of solitary waves tend to
be smaller.
\item
Our results substantiate
the relevance of electron holes in weakly magnetized collisionless
plasmas such as astrophysical plasmas and the plasma involved in
collisionless magnetic reconnection processes.
\end{itemize}

{\scriptsize
\begin{itemize}
\item[{[1]}] J. P. Lynov, et al., \PS, 20, 328, 1979.
\item[{[2]}] J. S. Pickett, et al., \NPG, 10, 3, 2003.
\item[{[3]}] J. F. Drake, et al., {\it Science}, 299, 873, 2003.
\item[{[4]}] Li-Jen Chen, PhD thesis, University of Washington, 2002.
\item[{[5]}] Li-Jen Chen and G. K. Parks, \NPG, 9, 111, 2002.
\item[{[6]}] R. L. Morse and C. W. Nielson, \PRL, 23, 1087, 1969.
\item[{[7]}] Li-Jen Chen, David J. Thouless and Jian-Ming Tang, submitted to
\PRL, 2003.
\item[{[8]}] R. E. Ergun, et al., \PRL, 81, 826, 1998.
\end{itemize}
}

\noindent
{\bf Acknowledgment:}
We thank the PI of FGM on Cluster, Andre Balogh, for providing the magnetic
field data.
This work was supported under DOE Cooperative
Agreement No. DE-FC02-01ER54651, and NASA Grant No. NAG5-9974.

\noindent
{\small Contact email: li-jen-chen@uiowa.edu}
\end{multicols}
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