text:
\section{Introduction}\label{sec:introduction}

Bose-Einstein condensates (BECs) in the dilute gas limit and in the presence of many kinds of
external potentials are modeled very well by the nonlinear Schr\"odinger
equation (NLSE)~\cite{dalfovo1999,ketterle1999}.  BECs with potential barriers have many practical
applications, including atom lasers~\cite{anderson1998,hagley1,bloch2} and high-resolution holography~\cite{zobay1999,helmerson1}.  Two recent experiments at Washington State~\cite{engelsP2007} and Rice~\cite{dries2010} have considered the dynamics of a finite barrier dragged over an effectively 1D BEC, giving one the opportunity to investigate the Landau criterion~\cite{dalfovo1999} under the constraint of a 1D system.  Instead of the quantized vortices common to superfluids in two and higher dimensions, dark solitons appear in 1D.  As the stationary NLSE with a delta function potential can be solved exactly~\cite{hakim1,pavloff2002,carr2005a,witthaut2005}, it is convenient for theory to model localized potential barriers with delta functions. For instance, multiple delta functions have been used to investigate intriguing problems such as nonlinear band structure~\cite{carr2005d} and Anderson localization~\cite{paulT2007}.  However, as we show in this article, when the delta function is replaced with a potential of finite width new features arise, including a denumerably infinite series of bifurcations in the transmission resonances, and one or more dark solitons localized on the barrier.

Since the stationary NLSE can be solved exactly for any piece-wise constant potential, we take the potential of finite width to be rectangular in form~\cite{footnote1}.  In single-particle quantum mechanics, stationary solutions of the Schr\"odinger equation for a rectangular potential barrier or well is a classic problem taught in undergraduate quantum mechanics courses.  The nonlinearity inherent in the mean-field description of the BEC as captured by the NLSE substantially changes this well-known problem, as we will show.  The physical context of our problem is the steady-state transmission behavior of an atom laser incident on a barrier.  We assume that the BEC is
confined in the transverse directions by a harmonic oscillator trap, and that its behavior is quasi-one-dimensional~\cite{olshanii1998,petrov2000b,carr2000e,dunjko2001};
that is, the longitudinal direction of the BEC is much larger than the
transverse directions and the healing length, and the chemical potential is much larger than
the transverse excitation energy.  Further, we assume that the width of the
potential barrier is much less than the longitudinal dimension of the
BEC, so that the system is effectively longitudinally infinite and far-field effects may be neglected.  This physical situation was experimentally produced in the Washington State and Rice experiments~\cite{engelsP2007,dries2010}.

Past studies of the 1D stationary NLSE with various boundary conditions and potentials are very extensive, including both repulsive and attractive interactions; see~\cite{kevrekidisPG2008} and references therein.  In this article we focus on the more common case of repulsive interactions, although our techniques work for general interactions in the NLSE.  Our own work has often treated cases described by piecewise constant potentials, starting with a uniform potential with periodic or box boundary conditions~\cite{carr2000e,carr2000a,carr2000b}.  This work was later extended to a discontinuous potential step and to a delta-function potential~\cite{carr2005a}.  Other studies in this direction included the Kronig-Penney potential~\cite{carr2005d} and the bichromatic lattice~\cite{carr2005e}.

The potential barrier or well has been treated previously in restricted circumstances in three previous studies.  First, Carr, Mahmud, and Reinhardt~\cite{carr2001f} found particular classes of bound states in the potential well.  Such bound states are localized or partially localized, and it was found that experimental parameters could be tuned to achieve different regimes of tunneling.
Second, Rapedius, Witthaut and Korsch~\cite{rapedius2006} took the region outside the well to be strictly linear.  They found a bistable behavior in the transmitted flux for unbound states.  The critical nonlinearity at which bound states appear was discovered.  Third, Ishkhanyan and Krainov~\cite{ishkhanyan2009} considered the limit of very small nonlinearity.  They used perturbation theory to expand the NLSE, and a
multi-scale method to find the solutions.  The reflection
coefficient $R$ could then be determined by the usual methods of linear
quantum mechanics, decomposing the wave functions in each region into
left- and right-traveling waves via the superposition principle.  The latter two studies had in common that they retained the concept of the superposition principle outside the well.  We discard this concept in favor of full nonlinearity, solving the complete nonlinear problem without approximations of any kind.  Additional related work on scattering of a bright soliton on a potential barrier or well has treated a variety of applied and foundational issues in quantum mechanics, from bound state spectroscopy and resonant trapping~\cite{ernstT2010} to macroscopic superpositions~\cite{weissC2009,streltsov2009} (Schr\"odinger cat states).

In all studies of the stationary 1D NLSE solitons play a key role.  In the Washington State experiment~\cite{engelsP2007}, the barrier
was produced by an elliptical laser beam, then dragged through the
BEC.  For intermediate drag speeds a train of dark solitons was
observed in the presence of the barrier.  Such solitons have been
produced by other experimental methods, for example, by merging two
coherent BECs~\cite{denschlag2000}.  An experiment by Weller \textit{et al}. at Heidelberg studied
the dynamical behavior and interactions of such solitons
\cite{weller2008}.  Solitons are localized, persistent, robust
nonlinear structures which appear often in BEC experiments
\cite{burger1999, carr2002b}.  A key feature of solitons in quasi-1D in the absence of an external potential is that they interact
elastically and do not
dissipate~\cite{reinhardt1997, stellmerS2008}.  Such behavior was observed in the Heidelberg experiment,
and was found to agree with numerical simulations of NLSE solution
dynamics~\cite{weller2008}.

Our article is outlined as follows.  In Sec.~\ref{sec:equation} we present the fundamental equations and a general solution method.  We emphasize that our work is completely analytical and symbolic up to one numerical integration used to characterize transmission; the NLSE solution itself is analytical and is determined rigorously~\cite{RRMthesis}.  The appendices contains some key brief proofs to support this rigor, as well as numerical studies to support our analysis and a brief overview of Jacobi elliptic functional relationships used in our analysis.  In Sec.~\ref{sec:soliton} we apply our solution method to scattering on the rectangular barrier and obtain characteristic solutions, highlighting cases for which solitons are localized on the barrier.  In Sec.~\ref{sec:transmission} we study transmission of the BEC for both wide and narrow barriers.  In Sec.~\ref{sec:resonances} we focus in on the transmission resonances, where the barrier is invisible, and discover a series of bifurcations in such resonances. Finally, in Sec.~\ref{sec:conclusions} we conclude.

\section{Equation and Methods}\label{sec:equation}

\subsection{The 1D Nonlinear Schr\"odinger Equation}

The 1D NLSE is
\begin{multline}
  \label{eq:unitsNLS}
  \left[-\frac{\hbar^2}{2M}\frac{\partial^2}{\partial\tilde{x}^2} + \tilde{g}\vert\tilde{\Psi}(\tilde{x},\tilde{t})\vert^2 +
    \tilde{V}(\tilde{x})\right]\tilde{\Psi}(\tilde{x},\tilde{t})\\ =
  i\hbar\frac{\partial}{\partial\tilde{t}}\tilde{\Psi}(\tilde{x},\tilde{t}),
\end{multline}
where $\tilde{g}=2 a_s \hbar \omega $ is the interaction strength or nonlinearity renormalized to 1D~\cite{carr2000e}, $a_s$ is the $s$-wave scattering length for binary contact interactions, $\omega$ is the oscillation frequency of the transverse trap as described in Sec.~\ref{sec:introduction}, $M$ is the mass of the atoms or molecules that are Bose-condensed, and
$\tilde{V}(\tilde{x})$ is an external potential.  We take the Landau interpretation of the wavefunction or order parameter: $\tilde{\rho}\equiv |\tilde{\Psi}|^2$ is the local BEC number density and $\tilde{v}=(\hbar/m)\partial_{\tilde{x}} \mathrm{Arg}(\tilde{\Psi})$ is its local velocity.  Here and throughout this
paper, a tilde denotes a dimensional quantity.

Non-dimensionalizing~\eqref{eq:unitsNLS} by scaling everything to
harmonic oscillator units
leads to the dimensionless or scaled NLSE,
\begin{multline}
  \label{eq:NLStime}
  \left[-\frac{1}{2}\frac{\partial^2}{\partial x^2} +
    g\vert\Psi(x,t)\vert^2 + V(x)\right]
  \Psi(x,t) \\=
  i\frac{\partial}{\partial t}\Psi(x,t),
\end{multline}
where
\begin{eqnarray}
  \label{eq:xnd}
  x &=& \frac{\tilde{x}}{\ell},\\
  \label{eq:tnd}
  t &=& \omega\tilde{t},\\
  \label{eq:gnd}
  g &=& \frac{\tilde{g}}{\hbar\omega\ell},\\
  \label{eq:Vnd}
  V &=& \frac{\tilde{V}}{\hbar\omega},\\
  \label{eq:psind}
  \Psi &=& \tilde{\Psi}\ell^{1/2},
\end{eqnarray}
and $\ell = \sqrt{\hbar/(m\omega)}$ is the harmonic oscillator length.

Equation~\eqref{eq:NLS} can
be solved~\cite{carr2000a} for stationary states of the form
\begin{equation}
  \label{eq:genansatz}
  \Psi(x,t)=\sqrt{\rho(x)}\,e^{i[\phi(x)-\mu t]},
\end{equation}
where $\mu$ is the eigenvalue of the stationary 1D NLSE.  Then the 1D NLSE becomes
\begin{equation}
  \label{eq:NLS}
  \left[-\frac{1}{2}\frac{\partial^2}{\partial x^2} +
    g\vert\Psi(x,t)\vert^2 + V(x)\right]
  \Psi(x,t) =
  \mu \Psi(x,t).
\end{equation}
We work with the non-dimensionalized stationary 1D NLSE, Eq.~\eqref{eq:NLS}, throughout the rest of our article.

\subsection{Exact Solution for a Constant Potential}
\label{ssec:exact}

When the potential $V(x)=V_0$ is a constant,
one can prove that the dimensionless density $\rho\equiv |\Psi|^2$
and the phase $\phi(x)$ have the form
\begin{eqnarray}
  \label{eq:gendens}
  \rho(x) &=& A\text{sn}^2\,(bx+\delta_0|m)+B,\\
  \label{eq:genphase}
  \frac{\partial\phi}{\partial x} &=& \frac{\alpha}{\rho},
\end{eqnarray}
where $A$ is the density scaling, $b$ is the translational scaling,
$m$ is the elliptic parameter, $\delta_0$ and $B$ are offsets, and $\alpha$ is an integration constant.  The function $\text{sn}$ is one of twelve Jacobi elliptic
functions~\cite{abramowitz1964}, which can be interpreted geometrically as the
elliptical analog of the circular trigonometric functions.  In this interpretation, the square root of the elliptic parameter $m$
represents the eccentricity of an ellipse.  In the limit that $m\to0$,
the Jacobi functions become the circular functions; in the limit
$m\to1$, the Jacobi functions become the hyperbolic functions.

Relationships between parameters may be obtained by substituting the solutions~\eqref{eq:gendens} and
\eqref{eq:genphase} into Eq.~\eqref{eq:NLS} and equating coefficients of linearly
independent powers of Jacobi elliptic functions.  The relevant proofs
of linear independence are given in Appendix~\ref{app:proofs}.  We
obtain the relationships
\begin{eqnarray}
  m&=&\frac{A}{b^2}g,  \label{eq:meq}\\
  \mu &=& \frac{1}{2}[b^2+(A+3B)g]+V_0,  \label{eq:mueq}\\
  \alpha^2 &=& B(A+B)(b^2+Bg).
  \label{eq:alpha2eq}
\end{eqnarray}
Since $\alpha$ appears only in its square, and $\rho(x)$ does not
depend on $\alpha$, all solutions with $\alpha \neq 0$ are doubly
degenerate.  That is, $\pm\alpha$ result in solutions with the same
density $\rho(x)$ and eigenvalue $\mu$, but phases of opposite sign.

Mathematical and physical considerations lead us to conclude that not
all possible parameter values are relevant to this problem
\cite{RRMthesis}.  The relevant parameter space is summarized in Table
~\ref{tab:reparams}.

 \begin{table}[H]
   \centering
   \caption{Mathematical conditions on both NLSE parameters and solution parameters.}
    \begin{tabular}{l@{\hspace{0.7cm}}l}
      Parameter & Condition\\\hline
      $A$ & Real\\
      $B$ & Real\\
      $b$ & Either $\text{Re}(b) = 0$ or $\text{Im}(b) = 0$\\
      $\delta_0$ & No constraints\\
      $m$ & Real\\
      $g$ & Real\\
      $\mu$ & Real\\
      $\alpha$ & Real
    \end{tabular}
   \label{tab:reparams}
 \end{table}

\subsection{Calculation of Transmission}
\label{sec:Tcalc}

We consider a potential barrier of form
\begin{equation}
  \label{eq:potlbarr}
  V(x) =
  \begin{cases}
    0, \quad &x < x_1,\\
    V_0, \quad &x_1 < x < x_2,\\
    0, \quad &x_2 < x.
  \end{cases}
\end{equation}
We will take $V_0$ to be positive definite.  The barrier can have any width and height, with the boundaries $x_1$
and $x_2$ located
at any positions along the $x$-axis.  We define regions I and III to
be the left and right sides of the barrier, respectively, and region
II as the region over the barrier.  We apply our fully general solution from Sec.~\ref{ssec:exact} to
a numerical study of the transmission of the BEC across the barrier.
We note that the solution and our code can be generalized to arbitrary
piecewise-constant potentials with a finite number of jump
discontinuities.

Using the solution to the constant potential one can treat any piecewise constant potential by the use of appropriate boundary conditions, similar to the well-known method for finding stationary solutions of the linear Schr\"odinger equation with a rectangular well/barrier.  Our explicit boundary conditions, expressed in terms of density and phase, are
\begin{eqnarray}
\rho(x_i^-)&=&\rho(x_i^+),\\
\left.\frac{d\rho(x)}{dx}\right|_{x_i^-}&=&\left.\frac{d\rho(x)}{dx}\right|_{x_i^+},\\
\phi(x_i^-)&=&\phi(x_i^+)+2\pi n,\,n\in\mathbb{Z},\\
\alpha^-&=&\alpha^+\\
\mu^-&=&\mu^+
\label{eq:boundary}
\end{eqnarray}
where $x_i$ is the location of the $i$th boundary and $\pm$ indicates the right/left side of the boundary~\cite{footnote2}.

The effective potential experienced by the BEC in the region over the
barrier is
\begin{equation}
  \label{eq:muVeff}
  V_{\rm eff} \equiv V_0 + g\rho(x).
\end{equation}
In the regime $\mu > \text{max}(V_{\rm eff})$, transmission over the
barrier is classically allowed.  For $\mu < \text{max}(V_{\rm eff})$, transmission is classically forbidden.  In both regimes, our scattering displays quantum or wave-like effects, modified by the nonlinearity.  The effective potential is why the linear and nonlinear problems are so different, in addition to the lack of a superposition principle.

We briefly remind the reader of the linear case.  For $g=0$ only in Eq.~\eqref{eq:NLS}, the wave function in each region can be split into a sum of two terms
representing left- and right-traveling waves, using the principle of superposition.  We can take the left-hand side as the incident
one.  On this side, the solution contains both right- and
left-traveling waves, i.e., incident and reflected waves, with only right-traveling waves on
the transmission side of the barrier.  In this case, we may define the
transmission coefficient as
\begin{equation}
  \label{eq:LinearTransmission}
  T =
  \frac{\langle\vert \Psi_{\text{trans}} \vert^2\rangle}{\langle\vert
    \Psi_{\text{inc}} \vert^2\rangle},
\end{equation}
where $\Psi_{\text{trans}}$ is the transmitted wave function and
$\Psi_{\text{inc}}$ is the incident wave function.  The angle brackets,
$\langle\hspace{1\jot}\cdot\hspace{1\jot}\rangle$, denote an average
value over one period of the function.  The definition given by
Eq.~\eqref{eq:LinearTransmission} is standard in linear quantum
mechanics.  In this interpretation, $T\leq1$ over the entire domain of
the system, and $T$ represents the probability that a given particle
will be transmitted across the barrier.

However, in the nonlinear case superposition does not apply.  We
cannot define separate left- and right-traveling waves in this case, and
thus the transmission coefficient is defined simply as
\begin{equation}
  \label{eq:NonlinearTransmission}
  T=\frac{\langle\vert\Psi_{III}\vert^2\rangle}{\langle\vert\Psi_{I}\vert^2\rangle} =\frac{\langle \rho_{III}\rangle}{\langle \rho_I\rangle},
\end{equation}
where $\Psi_{I}$ and $\Psi_{III}$ are the total wave functions in regions I
and III, with $\rho_{I}$ and $\rho_{III}$ the corresponding number densities.  In Eq.~\eqref{eq:LinearTransmission} this would be equivalent to replacing the denominator with the sum of incident and reflected amplitudes and then squaring to get the total probability density to the left hand side of the barrier.
In the nonlinear case, since neither region I nor region III can be said to be the incident side, we could just as easily have replaced the definition in Eq.~\eqref{eq:NonlinearTransmission} with its inverse.

Thus, in the nonlinear case, we find that $T$ may exceed unity, according to Eq.~\eqref{eq:NonlinearTransmission}.  Physically,
output cannot exceed input, and we conclude that the ``transmission''
coefficient as we define it contains information for atom lasers incident on either
side of the barrier.  Since we cannot choose only right- or
left-traveling waves in the solution, due to the nonlinearity, we
cannot define $T$ to restrict incidence to only one side of the
barrier, and must consider both circumstances in the same solution set.

To calculate the average over the densities, we note that the period of $\text{sn}^2\,(bx+\delta_0|m)$ is $2K(m)/b$, where $K(m)$ is the
complete elliptic integral of the first kind~\cite{abramowitz1964}.  Thus we
obtain the average density by
\begin{equation}
  \label{eq:AvgDensIntegral}
  \langle \rho \rangle = \frac{b}{2K(m)} \int \!dx\,\rho(x),
\end{equation}
where the integral is taken over one period of $\rho(x)$.
Using the properties of Jacobi functions and elliptic integrals, one
can show that if $0\leq m \leq1$,
\begin{equation}
  \label{eq:AvgDensExplicit}
  \langle\rho\rangle = B + A\left[\frac{1}{m} - \frac{E(m)}{mK(m)}\right],
\end{equation}
where $E(m)$ is the complete elliptic integral of the second kind.  If $m \notin [0,1]$, we must first apply a transformation to write the
density in terms of Jacobi elliptic functions which depend on a
parameter $m' \in [0,1]$, then use the methods above to simplify the
integral.  The relevant transformations are given in Appendix~\ref{app:JEfns}.

\subsection{Numerical Methods}

We use Mathematica to compute the transmission coefficient,
Eq.~\eqref{eq:NonlinearTransmission}.  We require an internal precision
of 100 digits; our main reason for using Mathematica is its feature of arbitrary internal precision, which is absolutely necessary for the highly singular problem of nonlinear scattering.  Besides internal precision, we use a uniform numerical tolerance of $10^{-5}$ for parameters.  If a quantity
is smaller than this value, it is taken to be zero; for example, we take $g < 10^{-5}$ to be zero.  We consider only repulsive interactions, i.e., $g \geq 0$, in our
numerical analysis.  In addition, we restrict analysis to positive
barriers $V_0 > 0$, for consistency with the idea of scattering by an
atom laser.  However, as mentioned previously, our mathematics are completely general, and our
solution may be applied to attractive interactions and potential wells.

To find a scattering solution, we take $A$, $B$, and
$\delta_0$ on the left side of the barrier, denoted by a subscript
$I$.  These parameters can be calculated uniquely from the average number density, momentum, and energy for a physical atom laser, as described in~\cite{carr2005a}.  We consider the physical quantities $g$
and $\mu$, as well as the barrier parameters, to be input parameters for an experiment, and take them as known.  Other
parameters on the left side of the barrier may be obtained from
Eqs.~\eqref{eq:meq}--\eqref{eq:alpha2eq}.  We then use the physical boundary conditions~\eqref{eq:boundary} to
solve for parameters in regions II and III in terms of the known input
parameters from region I.

Our code is completely symbolic, with the exception of one numerical
integration used in computing the transmission coefficient.  In order
to keep the code symbolic wherever possible, we used Mathematica's
pure function routines.  These are functions whose arguments are
defined in terms of their position rather than being given a specific
variable name.  This construction avoids the possibility of errors
arising from multiply-defined variables, while still allowing us to
maintain a consistent convention for functional dependencies.

The analytical solution process using boundary conditions yields
twelve solutions for ($A$, $B$) in regions II and III.  Six of these
are extraneous solutions, obtained as a result of squaring both sides
of an equation.  Since they are not true solutions, they do not
satisfy the boundary conditions, and this is used as a filter in the
code to discard these extraneous solutions.  The remaining six solutions all correspond to the same functional form
of the density, and we need only consider one.  In our construction of
Eq.~\eqref{eq:gendens} as the polar form of a complex function, we may
take $\rho(x)$ to be real without loss of generality.  Solutions
with $\text{Im}(\rho)\neq0$ are mathematically invalid, and these
are encountered in the code as a result of one or more assumptions
breaking down.  Therefore, we take only real solutions for ($A$, $B$).

To obtain the average densities in
Eq.~\eqref{eq:NonlinearTransmission}, we numerically integrate the
densities using Simpson's rule~\cite{press1993}.  The
transmission coefficient is computed as in
\eqref{eq:NonlinearTransmission}, and is treated as a resonance if
it is within an interval of $\pm 10^{-4}$ around unity.

In the linear limit $g \to 0$, exact analytical expressions may be
obtained for the boundary conditions.  We use these exact expressions,
rather than Mathematica's Solve routine, in the limit of small
$g$.  By doing this, we avoid problems with infinities due to internal
Mathematica processes.

We also made use of several other methods to ensure the correctness of
the computations performed by Mathematica.
In our experience, Mathematica does not always handle $\sqrt{-1}$ correctly.
Therefore, we kept $i$ as a symbol, using replacement tables to handle
powers of $i$, and avoiding Mathematica's internal
complex-number routines wherever possible.  In addition, we included
several identities for Jacobi elliptic functions via replacement
tables, as Mathematica's default processes do not make use of
these identities for simplification.  Finally, rather than relying on
Mathematica to handle the special cases of complex argument and
$m>1$ internally, we used known transformations to write equivalent
expressions in terms of Jacobi functions with real argument and
parameter smaller than unity.  These expressions were used in the code
for computations.

Convergence was verified using several methods.  These are detailed in
Appendix~\ref{app:convergence}.

\section{Soliton Localization}\label{sec:soliton}

 \begin{figure}[t]
   \begin{center}
   \subfloat[][]{\label{fig:densScatt}\includegraphics[width=0.4\textwidth]{nonlinearScattering_fig1a.eps}}\\
   \subfloat[][]{\label{fig:phaseScatt}\includegraphics[width=0.4\textwidth]{nonlinearScattering_fig1b.eps}}
   \end{center}
\caption{(Color online) Density (upper panel) and phase (lower panel) of a solution to the NLSE in which an incident near-sinusoidal wave produces a single dark soliton localized over the barrier.  The three colors in the phase curve denote the three scattering regions.  The red dashed curve shows the potential barrier for reference.}
   \label{fig:densphaseScatt}
 \end{figure}

\begin{figure}[t]
   \centering
   \subfloat[][]{\label{fig:solitontraindens}\includegraphics[width=0.4\textwidth]{nonlinearScattering_fig2a.eps}}\\
   \subfloat[][]{\label{fig:solitontrainphase}\includegraphics[width=0.4\textwidth]{nonlinearScattering_fig2b.eps}}
   \caption{(Color online) Density (upper panel) and phase (lower panel), similar to Fig.~\ref{fig:densphaseScatt}, but with a soliton train localized on the barrier instead of a single soliton.  The input parameters are identical to those of Fig.~\ref{fig:densphaseScatt}; only the barrier width has been changed, from 20 to 90.  We observe a train of 9 very deep solitons localized on the barrier.}
   \label{fig:solitontraindensphase}
 \end{figure}

A large variety of solution types appear after following the methods of Sec.~\ref{sec:equation}.  We present here some particularly interesting cases which are very far from the kinds of solutions found for the linear Schr\"odinger equation.  We recall that, in the linear Schr\"odinger equation, the wavelength of the plane wave components across regions I, II, and III is the same; it is only the spatial offset in the arguments of the exponentials and the amplitudes that can be different.  In the nonlinear case the wavelength can be quite different in all three regions.  We present two such cases, in which a phonon-like standing wave is transformed into a strongly localized soliton or soliton train over the barrier, and then returns to a phonon-like profile.  We recall that phonons are a limit of Bogoliubov quasi-particles for BECs, and appear as a small modulation on top of a large offset, qualitatively speaking, as ripples on a pond.  Since we work with the fully nonlinear problem our phonons are not constrained to be small oscillations, but continue to appear as ripples of amplitude $A$ and translational offset $\delta_0$ on top of an offset of magnitude $B$.

Figure~\ref{fig:densphaseScatt} shows the density and phase for a localized soliton.  A well-localized solution can be
seen above the potential barrier.  The barrier, indicated with a red dashed box in the figure, has height $V_0=1$ and width
20.  The nonlinearity is $g=2.02$ and the chemical potential $\mu=2.404$.  We choose $A_I=1$, $B_I=1$, and $\delta_{0I}=0$.  In regions I and III, outside the barrier, the elliptic parameter $m\ll 1$. In region II, over the barrier, $m$ approaches unity and the peaks of the
$\text{sn}^2$ function broaden significantly.  The local minimum of the density over
the barrier is a dark soliton, called ``grey'' because it does not have a node.  Such a soliton is always moving.  Examining the phase profile, we observe two areas in region II: away from the dark soliton is a background superflow with a shallower slope, while over the soliton there is a characteristic phase jump displaying a sharp increase in slope.  In this solution the superflow flows to the right with velocity $v=\partial_x \phi(x)$, while the dark soliton moves to the left with an equal and opposite velocity; the two velocities cancel each other, leading to a stationary state.

The phase has an overall linear envelope on either side of the
barrier, with clearly visible oscillations on this background.
The background slopes differ slightly between regions I and III, and the slope over
the barrier in region II is smaller by a factor of 3.  This shows that
the velocity profile of the BEC need not be the same on either side of
the barrier.  Although we fixed parameters on the left-hand side in order to find this solution, as described in Sec.~\ref{sec:equation}, from the physical perspective of an atom laser the BEC is incident on the right-hand side.  Thus, as happens approximately half the time in our solution method, we have in fact fixed the output rather than the input parameters.  The overall transmission right to left is about 50\%, and $T>1$ according to our definition from Eq.~\eqref{eq:NonlinearTransmission}.

With a longer barrier of width $x_2-x_1=90$ and the same nonlinearity $g=2.02$, chemical potential $\mu=2.404$, and input parameters $A_I=1$, $B_I=1$, and $\delta_{0I}=0$ fixed on the left hand side, we obtain a soliton train localized on the barrier, as shown in Fig.~\ref{fig:solitontraindensphase}.  In Fig.~\ref{fig:solitontraindensphase} we again display the potential with a dashed red line for reference.  The location of the dark solitons can be observed in both the density and the phase, similar to Fig.~\ref{fig:densphaseScatt}.  Multiple solitons have been observed in a variety of BEC experiments~\cite{strecker2002,engelsP2007,weller2008}.  However, such solitons have so far not been phase-locked into a train in BECs.  Our barrier method constitutes a totally new way of producing multiple solitons locked into a train.  The length of the barrier can be used to control the number of solitons in the train.  We note that dark soliton trains have been produced in nonlinear spin waves in thin magnetic films~\cite{kalinikos2000,wuMingzhong2004}, but not via our scattering technique; also spin waves are in fact a damped driven system which are best modeled by an open-system version of the NLSE quite different from Eq.~\eqref{eq:NLS}~\cite{carr2010g}.

\section{Atom Laser Transmission}\label{sec:transmission}

We turn now to a more systematic exploration of the solution space.  The nonlinear Schr\"odinger equation has a substantially larger parameter space to explore than the linear case.  We divide this exploration into wide and narrow barriers.  The barrier size $x_2-x_1$ can be compared to the BEC healing length, $\xi \equiv 1/\sqrt{8\pi\langle\rho\rangle a_s}$,
with $\langle\rho\rangle$ the average linear number density.  However, the healing length only provides a useful comparison for low-energy excitations, as it describes how a uniform ground-state BEC is perturbed by a localized potential, e.g. a hard wall.  This corresponds to well-separated dark solitons, as in region II in Figs.~\ref{fig:densphaseScatt} and~\ref{fig:solitontraindensphase}.  As we treat a wide variety of excitations, many of which take standing-wave form which is very far from the ground state, e.g. regions I and III in Figs.~\ref{fig:densphaseScatt} and~\ref{fig:solitontraindensphase}, the correct comparison to identify wide and narrow barriers is in fact the wavelength of the excitations,
\begin{equation}
\lambda\equiv \frac{2K(m)}{b} = \frac{2K\left(\frac{Ag}{2(\mu-V_0)-(A+3B)g}\right)}{\sqrt{2(\mu-V_0)-(A+3B)g}},
\label{eq:wavelength}
\end{equation}
as can be calculated from Eqs.~\eqref{eq:meq} and~\eqref{eq:gendens}.  Thus $x_2-x_1 \ll \lambda_j$ is the narrow barrier case and $x_2-x_1 \gg \lambda_j$ is the wide barrier case, where the subscript $j\in\{I,II,III\}$ refers to the region.  For sufficiently small $\lambda$ all barriers are effectively wide.  Holding other parameters fixed, including the nonlinearity, the wide barrier limit occurs for large chemical potential, since the wavelength is a decreasing function of $\mu$, as can be verified by consideration of Eq.~\eqref{eq:wavelength}.

We found that the barrier height $V_0$ is less important, a statement which we will support further in Sec.~\ref{sec:resonances}.  This dependence on width more than height is another sense in which the nonlinear case is very different from the linear case; in the latter it is only the effective area of the barrier, $2M\tilde{V}_0(\tilde{x}_2-\tilde{x}_1)^2/\hbar^2$, that is important.  The dependence on width arises mainly from the fact that the density in the nonlinear case can have different wavelengths in different regions for the same solution, in contrast to the linear case, as previously mentioned.

For simplicity we will keep the input parameters $A_I$, $B_I$, and $\delta_{0I}$ fixed to the same values as those used in Figs.~\ref{fig:densphaseScatt}-\ref{fig:solitontraindensphase} and Sec.~\ref{sec:soliton}; after a massive exploration of the parameter space, we found all results to be qualitatively similar to those described below.

\subsection{Wide Barrier Case}\label{sec:wideV}

 \begin{figure}[t]
   \centering
  \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig3.eps}
   \caption{(Color online) \emph{Wide barrier, small nonlinearity}.  Transmission for wide barrier with
     increasing nonlinearity from bottom to top, starting with the linear case, $g=0$, in steps of $\Delta g = 0.01$. Even a very small nonlinearity changes the problem because the left-right symmetry of the effective potential is broken.}
   \label{fig:wideBarrierSmallNonlinearity}
   \end{figure}

 \begin{figure}[t]
   \centering
   \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig4.eps}
   \caption{(Color online) \emph{Wide barrier, medium nonlinearity}.  Same as Fig.~\ref{fig:wideBarrierSmallNonlinearity}, but with steps of $\Delta g =0.1$, from $g=0.1$ to $g=1.0$  An extended region of near-perfect invisibility of the barrier in the $\mu\simeq 20$ to $\mu\simeq 30$ range translates to the right as the nonlinearity increases.}
   \label{fig:wideBarrierMediumNonlinearity}
   \end{figure}

We consider a barrier whose width is much larger than its height.  We take a barrier of width $x_2-x_1=20$ and height $V_0=1$ for the purposes of illustration.
In Figs.~\ref{fig:wideBarrierSmallNonlinearity}-\ref{fig:wideBarrierMediumNonlinearity} transmission plots are laid out on the same set of horizontal axes.  The dashed red lines denote
$T=1$, and solid curves are the transmission coefficient as defined in Eq.~\eqref{eq:NonlinearTransmission}.  The
nonlinearity $g$ increases in steps of 0.01 in Fig.~\ref{fig:wideBarrierMediumNonlinearity} as we move upward on the
plot, and in steps of 0.1 in \ref{fig:wideBarrierSmallNonlinearity}.  Each transmission plot is at a convenient vertical offset for
illustration, but the plots are not otherwise scaled or shifted.  The
nonlinearity is shown next to each curve for reference.  As discussed in Section~\ref{sec:Tcalc}, the transmission coefficient may be greater than 1.  This is
not a novel physical feature; it arises due to the redefinition of
transmission as in Eq.~\eqref{eq:NonlinearTransmission} and the
invalidity of superposition in this problem.

Comparing the linear, i.e $g=0$, transmission curve in Fig.
\ref{fig:wideBarrierSmallNonlinearity} with the other transmission plots in the same figure, we observe that there is a significant change in transmission behavior when we enter the
nonlinear regime, even for very small nonlinearity, $g\sim \mathcal{O}(10^{-2})$.  The most significant change occurs when $\mu$ is small, meaning that the
potential barrier has a greater overall effect on the condensate.  In
the linear case, $T$ oscillates about evenly on either side of unity for small $\mu$, and the amplitude
of oscillations is similar in either direction.  In the nonlinear
case, we see that while transmission can still be less than unity, it does
not drop as far below unity as in the linear case.  We understand this difference to be due to the effective potential: for $g=0$ the operator in the NLSE is not biased, whereas for non-zero $g$ it is.  Thus, by choosing parameters on the left hand side, we fix the effective potential and bias the system towards particular parameter sets.

As $g$ increases, new peaks appear in the small-$\mu$ regime, and the behavior of $T$ changes
significantly for smaller $\mu$.  This regime does not appear for $g \gtrsim 0.2$.  The reason that the transmission curves depend strongly on $g$ in the small-$\mu$ regime is that $g$ and $\mu$ are
not completely independent.  For a given chemical potential $\mu$,
when the nonlinearity $g$ becomes larger than a certain cut-off value, solutions are generated
that have $\rho(x) \in \mathbb{C}$.  This is invalid because, when we
solved the NLSE, we assumed $\rho(x) \in \mathbb{R}$.  We may take $\rho(x)
\in \mathbb{R}$ without loss of generality, since Eq.~\eqref{eq:genansatz}
is a general polar representation of a complex number.  For large $\mu$ the cut-off in $g$ is pushed to a region off the top of our plot; see figures in Sec.~\ref{sec:resonances} for more details.  The appearance of complex $\rho(x)$ means that one or more of
our assumptions are breaking down in this regime.  A more detailed
analysis of this situation is considered in~\cite{RRMthesis}.
One such break-down is due to the appearance of quasi-bound states with complex $\mu$~\cite{carr2004b,carr2005c,dekel2009}.

For larger values of $\mu$, the overall behavior of the transmission does
not change significantly between each plot.  However, we do see a
shift in the transmission curve.  For higher $\mu$, the transmission
plot retains its shape and shifts to the right as $g$ increases.  This
feature is especially apparent when a computer is used to animate the
transmission plots for increasing $g$; in Fig.~\ref{fig:wideBarrierMediumNonlinearity} we have endeavored to lay such an animation out on the page.  In
Fig.~\ref{fig:wideBarrierMediumNonlinearity}, the regime of the same curve-shape translating gradually to the right with increasing $g$ occurs when $\mu \gtrsim 6$.

Another feature of note is that with the definition
\eqref{eq:NonlinearTransmission} of transmission, the amplitude of oscillations in $T$ does not decrease monotonically as in the usual linear interpretation.  In all of the transmission plots of Fig.~\ref{fig:wideBarrierMediumNonlinearity}, we
see significant oscillations on either side of a transition region
between $\mu\simeq 20$ and $\mu \simeq 30$.  In this region, we have almost perfect
resonance, i.e., $T$ is very close to unity.  Further analysis of transmission resonances will be presented in Sec.~\ref{sec:resonances}.

\subsection{Narrow barrier}\label{sec:narrowV}

 \begin{figure}[!htb]
   \centering
   \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig5.eps}
   \caption{Transmission for narrow barrier with small nonlinearity.
     Note the difference in behavior between the
     linear and nonlinear regimes.}
   \label{fig:narrowBarrierSmallNonlinearity}
 \end{figure}

 \begin{figure}[!htb]
   \centering
\includegraphics[width=0.4\textwidth]{nonlinearScattering_fig6.eps}
   \caption{Transmission for narrow barrier with increasing
     nonlinearity from bottom to top, from $g=0.1$ to $g=2.0$ in steps of $\Delta g = 0.1$.}
   \label{fig:narrowBarrierMediumNonlinearity}
 \end{figure}

To explore the narrow-barrier regime, we choose $x_2-x_1=0.1$ and $V_0=10$, so that the barrier is a factor of ten narrower than it is high; note that the height is the same as the wide-barrier case explored in Sec.~\ref{sec:wideV}.  Over the full range of values of $\mu$ and $g$ that we consider, the narrow-barrier requirement, $\lambda \gg x_x-x_1$, is satisfied, as can be verified from Eq.~\eqref{eq:mueq}; in the range we consider, $\lambda \geq 0.35$.  We first focus on transmission plots for small nonlinearity, in Fig.~\ref{fig:narrowBarrierSmallNonlinearity}.  The layout of the figure is the same as in Sec.~\ref{sec:wideV}: five transmission
plots are shown on the same set of horizontal axes, with vertical
offsets in $T$ for illustration, the plots are not otherwise shifted
or scaled, and $g$ is increased from zero in steps of $\Delta g = 0.01$.
Comparing the linear $g=0$ transmission curve with those for $g>0$, we again see a significant drop in the amplitude of oscillations as we go from the linear to the nonlinear regime.  Again,
the nonlinear transmission curves tend to stay further above unity than below unity,
although the difference is less pronounced
than in the wide-barrier case.  As a function of $\mu$ the transmission curve is much smoother as compared to Fig.~\ref{fig:wideBarrierSmallNonlinearity}.  This is because there is always less than one wavelength fitting into the barrier, and thus a small change in $\mu$ cannot suddenly cause an integer number of wavelengths to match the barrier width.

In Fig.~\ref{fig:narrowBarrierMediumNonlinearity}, we turn to the regime of medium nonlinearity, increasing $g$ from 0.1 in steps of $\Delta g = 0.1$ up to $g=2.0$.  The amplitude of oscillations decreases smoothly for increasing $\mu$, independent of $g$.  There is no region of near-perfect resonance as in the wide-barrier case: $T$ oscillates quasi-periodically about $T=1$.  In fact, the entire transmission curve simply translates smoothly to the right as $g$ increases, with no real abrupt behavior even for the smallest values of $\mu$ allowed for each curve.  Thus we can surmise that the abrupt behavior in Sec.~\ref{sec:wideV} is due to the barrier width, and it is the width, not the height, that mainly controls the transmission curves.  In the following section we will adduce further evidence to support this point.

\section{Transmission Resonances and Bifurcations}\label{sec:resonances}

\begin{figure}[t]
  \centering
  \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig7.eps}
  \caption{Transmission resonances for narrow barrier.  All behavior is described by regularly spaced straight lines except for very small $\mu$, where some combinations of $g$ and $\mu$ do not produce stationary solutions.}
  \label{fig:res1}
\end{figure}

To better understand the results of Sec.~\ref{sec:transmission}, in particular the extended region of near-perfect invisibility of the barrier, we focus on the transmission resonances only, i.e., the points in the $g$--$\mu$ plane for which $T=1$ to within a tolerance of $10^{-5}$.

We first treat the simpler case of the narrow barrier.  The transmission resonances from Fig.~\ref{fig:narrowBarrierMediumNonlinearity} are shown in Fig.~\ref{fig:res1}.  This clearly shows that all transmission resonances translate smoothly to the right for increasing $g$, with a constant slope.  This figure allows us to predict the location of all transmission resonances except for small $\mu$, where constraints in the allowed simultaneous values of $g$ and $\mu$ cut off certain regions, as discussed in Sec.~\ref{sec:transmission}.  We also observe in Fig.~\ref{fig:res1} that the spacing between curves increases with increasing $\mu$. A functional fit to this spacing can allow us to predict transmission resonances in the entire $g$--$\mu$ plane.  Since the spacing is the same starting from $g=0$, it is straightforward to find such a function for any values of $A$, $B$, and $\delta_0$.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig8.eps}\\
  \caption{\label{fig:res2}Transmission resonances for wide barrier. Bifurcations occur in an extended region of near-perfect invisibility of the barrier.}
\end{figure}

With the narrow-barrier case as a reference, we move on to the more intricate wide-barrier case.  Figure~\ref{fig:res2} shows the transmission resonances from Fig.~\ref{fig:wideBarrierMediumNonlinearity}.  We observe that the resonances are sparse for low values of $\mu$, with much more sporadic and abrupt behavior than the narrow barrier case.  This is because an integer number of wavelengths can fit into the barrier width, an effect which is more pronounced for small $\mu$; for instance, for $g=0.02$ and $\mu=2$, $\lambda=2.27$, while for $\mu=40$, $\lambda=0.355$.  In the regions where lines of resonance appear, the spacing between these lines is smaller than in Fig.~\ref{fig:res1}.  The
spacing increases as $\mu$ increases, though not as fast as in the
narrow-barrier case.

The resonances are sparse for lower values of $\mu$ and more uniform for
higher values of $\mu$.  For mid-range $\mu$, we see very different
behavior.  There are three regions between $\mu=20$ and $\mu=26$ where
the resonances are extremely dense.  These correspond to the
region of near-constant resonance seen in the transmission plots.  This region shifts to the
right as $g$ increases, as observed in the transmission plots.  Three bifurcations are present in this region.  Bifurcations are typical of nonlinear systems; the particular example we show here for this parameter set is expected to be a generic feature for all parameter sets.  Further numerical exploration found these three bifurcations continued in an apparently infinite sequence sloping away upwards to the right.  We found that a linear function was not sufficient to fit such bifurcations, and they occurred near but not precisely at a value of $\lambda = 1/2$.  We did not find bifurcations at other rational values of $\lambda$, so this may be a coincidence; the barrier length in this parameter set is $x_2-x_1 = 10$.  We explored up to $\mu=70$ and $g=10$.  Although our computationally intensive study was only performed thoroughly for this particular choice of parameters, we did spot checks through many regions of parameter space and observed similar extended regions of near-perfect invisibility of the barrier.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig9.eps}
  \caption{Slopes of resonance transmission lines, as a function of density offset, coarse sampling.  The curves show $B=0.2$, 0.5, 1.0, 1.5, 2.0 from bottom to top; a vertical offset is given to each curve for visualization, since they in fact all lie on top of each other.  The points represent actual data, while the curves are a guide to the eye.}
  \label{fig:coarseslopeplot}
\end{figure}

We return to the narrow barrier case for further analysis.  We
consider the slope $s$ of these parallel lines of resonance as a function of other physical
parameters.
For the parameter regimes considered in this study, the slope of the
resonance lines does not depend on the value of the density offset,
$B$, as seen in Fig.~\ref{fig:coarseslopeplot}.  In this plot, the
slope is shown for several values of $B$.  Each curve has been
vertically shifted by a convenient offset for illustration; all curves in fact lie on top of each other.  We note that the elliptic parameter
$m$, which is strongly governed by the nonlinearity of the system, depends on both $A$ and $b$, but does not depend on $B$, as described in Sec.~\ref{ssec:exact}.  Similarly, we found that the slopes $s$ do not depend on the spatial translational offset $\delta_0$.  However, the slopes do depend strong on the amplitude $A$.  The slope decreases as the amplitude of the input density increases. We find an exponential least-squares fit as shown in Fig.~\ref{fig:slopeplot2}:
\begin{equation}
  \label{eq:fiteqn}
  s(A_I) = 0.167 + 0.485 e^{-0.549 A_I},
\end{equation}
where we fixed $B_I=1.0$.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.4\textwidth]{nonlinearScattering_fig10.eps}
  \caption{Slope of resonance curves as a function of input amplitude.  Although the transmission resonances curves do not depend on $B$ and $\delta_0$, they do depend strongly on $A$.  Points are numerical calculations, while the solid curve is a least squares fit.}
  \label{fig:slopeplot2}
\end{figure}

\section{Conclusions}\label{sec:conclusions}

We developed a general method for obtaining stationary states of the nonlinear Schr\"odinger equation for any piecewise constant potential.  We applied this method to nonlinear scattering on a rectangular potential barrier, focusing on cnoidal waves, or dark soliton trains.  This problem differs greatly from the textbook linear quantum mechanics problem of scattering on a rectangular barrier.  Among novel nonlinear features are the following.  First, the wavelength in the three regions (left of the barrier, on the barrier, and right of the barrier) need not be the same.  As a consequence, such a barrier can be used to create one or more sharply localized dark solitons from a broad phonon-like input. Second, it is mainly the barrier width, not its area, that controls the kind of stationary states observed. For wide barriers in which the barrier width is larger than the wavelength extended regions of near-perfect transmission occur; the barrier is invisible over a range of interaction strength $g$ and chemical potential $\mu$.  Third, in wide barriers an apparently infinite sequence of bifurcations appears in this invisibility region.  Despite the parameter space for solutions being large, we showed that it is just the amplitude that determines slopes of transmission resonance lines, greatly simplifying this complex problem.

To relate our predictions to alternate physical input of average density, momentum, and energy of an atom laser formed from a Bose-Einstein condensate, the analysis laid out in Ref.~\cite{carr2005a} may be used directly; thus we do not repeat it here.  Future possibilities for this work include applying our method to various piecewise constant potentials of interest in applications, and a more detailed mathematical study of the bifurcations we found in transmission resonances.

We acknowledge helpful discussions with Paul A. Martin and Matthew
Heller.  This work was supported by the National Science Foundation under Grant PHY-0547845 as part of the NSF CAREER program.  L.D.C. thanks the Institute for Advanced Study at Tsinghua University for hosting him during completion of this work.

redpajama_set_name: RedPajamaArXiv


text:
'use strict';

Object.defineProperty(exports, "__esModule", {
  value: true
});
exports.incrementGregorianDays = incrementGregorianDays;
exports.getUTCDateForTwinBirthdaysOnYear = getUTCDateForTwinBirthdaysOnYear;
exports.getUTCDateForSunsetOnDate = getUTCDateForSunsetOnDate;
exports.getUTCDateForNextNewMoonFromDate = getUTCDateForNextNewMoonFromDate;
exports.getUTCDateForNawRuzOnYear = getUTCDateForNawRuzOnYear;

var _blueyonder = require('./extern/blueyonder');

var _blueyonder2 = _interopRequireDefault(_blueyonder);

var _LocationMap = require('./LocationMap');

var _LocationMap2 = _interopRequireDefault(_LocationMap);

var _stellafane = require('./extern/stellafane');

var _stellafane2 = _interopRequireDefault(_stellafane);

function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; }

var MillisPerDay = 1000 * 60 * 60 * 24;

/**
 * Increment a gregorian date by the number of days specified.
 */
function incrementGregorianDays(gregorianDate, days) {
  return new Date(gregorianDate.getTime() + days * MillisPerDay);
}

/**
 * Finds the time (sunset in Tehran) that twin birthdays start. Twin birthdays
 * occur at Sunset in tehran 8 new moons after Naw Ruz.
 */
function getUTCDateForTwinBirthdaysOnYear(gregorianYear) {
  var gregorianNawRuz = getUTCDateForNawRuzOnYear(gregorianYear);

  var UTCNewMoons = getUTCDateForSunsetOnDate(gregorianNawRuz, _LocationMap2.default.Tehran);
  for (var i = 0; i < 8; ++i) {
    UTCNewMoons = getUTCDateForNextNewMoonFromDate(UTCNewMoons);
  }

  var UTCSunset = getUTCDateForSunsetOnDate(UTCNewMoons, _LocationMap2.default.Tehran);
  if (UTCNewMoons < UTCSunset) {
    UTCNewMoons = incrementGregorianDays(UTCNewMoons, 1);
  } else {
    UTCNewMoons = incrementGregorianDays(UTCNewMoons, 2);
  }

  UTCNewMoons = getUTCDateForSunsetOnDate(UTCNewMoons, _LocationMap2.default.Tehran);
  return new Date(Date.UTC(UTCNewMoons.getUTCFullYear(), UTCNewMoons.getUTCMonth(), UTCNewMoons.getUTCDate()));
}

function getUTCDateForSunsetOnDate(gregorianDate, place) {

  var sunsetTime = _blueyonder2.default.SunRiseSet(gregorianDate.getUTCFullYear(), gregorianDate.getUTCMonth() + 1, gregorianDate.getUTCDate(), place.latitude, place.longitude)[1];

  var sunsetHours = Math.floor(sunsetTime);
  var sunsetMinutes = Math.floor((sunsetTime - sunsetHours) * 60);
  var sunsetSeconds = Math.floor(((sunsetTime - sunsetHours) * 60 - sunsetMinutes) * 60);

  return new Date(Date.UTC(gregorianDate.getUTCFullYear(), gregorianDate.getUTCMonth(), gregorianDate.getUTCDate(), sunsetHours, sunsetMinutes, sunsetSeconds));
}

/**
 * Takes a date and returns the date of the next new moon. This function is
 * recursive.  A client should call this function with two identical parameters.
 *
 * The detailed contract is this: find the first new moon during or after the
 * lunar cycle containing 'date' but occurring after the time 'min'.
 */
function getUTCDateForNextNewMoonFromDate(UTCDate) {
  return recurse(UTCDate, UTCDate);
}

function recurse(UTCDate, minUTCDate) {
  var quarters = _blueyonder2.default.MoonQuarters(UTCDate.getUTCFullYear(), UTCDate.getUTCMonth() + 1, // Month 0-based to 1-based indexing
  UTCDate.getUTCDate(), 0);
  var newMoonDateComponents = _blueyonder2.default.jdtocd(quarters[0]);
  var UTCNewMoon = new Date(Date.UTC(newMoonDateComponents[0], // Year
  newMoonDateComponents[1] - 1, // Month: 1-based to 0-based indexing
  newMoonDateComponents[2], // Day
  newMoonDateComponents[4], // Hour
  newMoonDateComponents[5], // Minute
  newMoonDateComponents[6]));

  if (UTCNewMoon > minUTCDate) {
    return UTCNewMoon;
  }
  return recurse(incrementGregorianDays(minUTCDate, 30), minUTCDate);
}

/**
 * Gets the day of Naw Ruz in a given gregorian year.  It returns the day at
 * 00:00:00 UTC.
 */
function getUTCDateForNawRuzOnYear(gregorianYear) {

  // Step 0: Follow to the UHJ
  // In 2026, the equinox is less than a minute from sunset. The
  // algorithms from Meeus which we're using don't give accuraccy
  // better than one minute, and don't give us the right answer. The
  // UHJ has said that Naw Ruz this day is on the 21st.
  if (gregorianYear == 2026) {
    return new Date(Date.UTC(2026, 2, 21));
  }

  // Step 1: find UTC time of the equinox
  var UTCEquinox = _stellafane2.default.vernal_equinox(gregorianYear);

  // Step 2: Find Tehran's sunset on the day of the equinox
  var UTCSunset = getUTCDateForSunsetOnDate(UTCEquinox, _LocationMap2.default.Tehran);

  // Step 3: Find the final day.
  if (UTCEquinox < UTCSunset) {
    return new Date(Date.UTC(UTCEquinox.getUTCFullYear(), UTCEquinox.getUTCMonth(), UTCEquinox.getUTCDate()));
  }
  return new Date(Date.UTC(UTCEquinox.getUTCFullYear(), UTCEquinox.getUTCMonth(),
  // NOTE: Date class handles the case where the day spills to the
  // next month.
  UTCEquinox.getUTCDate() + 1));
}

redpajama_set_name: RedPajamaGithub


text:
Q: How to determine an injectable angular service at runtime? I apologize in advance for not creating a plunker for this code, there is a number of third party controls that are needed for it to run.
My question is about selecting service, dependency injection, at run time, based on some conditions.
lets discuss only one service for this example.
import { Injectable } from '@angular/core';
import { Http, Response} from '@angular/http';

@Injectable()
export class Service1 {
    constructor (
        private http: Http) {
    }
}

In my component if I have a fixed one, all is well.
import {Component, ViewChild, OnChanges, Input, ReflectiveInjector, SimpleChange} from '@angular/core';
import {Observable} from 'rxjs/Rx';
import { Service1 } from './my.service.1';
@Component({
    providers: [
        Service1
    ],
    moduleId: module.id,
    selector: 'my-component',
    templateUrl: './my-component.component.html'
})
export class MyComponent {
    constructor(private service : Service1
        ) {
}

in the above case all is well, no runtime errors.
Now if I change to the dynamic version, I get an error
EXCEPTION: Error in http://localhost:3000/app/app.component.html:5:8 caused by: No provider for Http! (Service1 -> Http)
the version of code causing the error is:
I added a member variable for MyComponent: private service: any = '';
and the constructor now looks like
constructor() {

    // this does not work with the http module
    let injector = ReflectiveInjector.resolveAndCreate([Service1]);
    this.service = injector.get(Service1);
}

anyone sees what is causing the above mentioned error?

A: don't forget to add Http to the array passed in   ReflectiveInjector.resolveAndCreate :
let injector = ReflectiveInjector.resolveAndCreate([Service1,Http]);

redpajama_set_name: RedPajamaStackExchange


text:
Europe and India: Testing New Delhi's New Diplomacy
By Kabir Taneja
http://thediplomat.com/2015/07/europe-and-india-testing-new-delhis-new-diplomacy/
An underperforming relationship could benefit from some Indian reforms.
At a recent business event in a major European city, a senior executive was heard to explain why his company did not do business in China. His main argument was economic: The Chinese market was crowded with competitors to midsize companies such as his. But the executive also said that he felt more comfortable doing business in democratic economies such as India, even if it takes more effort and time for projects to come to fruition.
While that view may be in the minority among corporate investors, it offers one more reason why India, the world's largest democracy and a market economy set to be the world's fastest growing by next year, should become a global economic hub without any further delay. In fact, the government of Prime Minister Narendra Modi is now moving to realize this ambition, and the world is showing renewed interest in the India story following the change in government last year. Yet the European Union remains an elusive and highly uncelebrated entity in Indian polity, business, and diplomacy.
For Modi, who came to power in a landslide election in 2014, improving India's economy and its perception among foreign investors has been a top priority. During his first year, he traveled the world, including stops in France and Germany. The latter, of course, is Europe's financial powerhouse, while the former has helped India with its military needs in past decades.
The Europe-India dynamic over the past few years has seen its shares of ups and downs. Recently, the negatives were highlighted when Brussels fumbled an opportunity to get Modi to make a pit stop there during his European trip. Beyond this, EU-India relations have often been sidetracked by other issues, such as the ghosts of the Bofors scandal from the 1980s emerging every now and then, or the very public case of the Italian marines deployed on anti-pirate duty aboard the oil tanker MT Enrica Lexie allegedly shooting dead two Indian fishermen. The legal fallout saw the European Union back Rome, and the case is now heading to international arbitration. Italy, of course, is a prominent EU member and the case has been a serious thorn in relations. Rome is adamant that the Marines be tried in an international court, while New Delhi maintains that the fishermen were shot in Indian waters and any sort of international arbitration would be seen as political interference in India's internal affairs. The Lexie case was made all the more fraught for India domestically because it happened under the UPA government, led by the Congress party whose chief, Sonia Gandhi, is Italian born.
These prickly issues withstanding, the European Union has tried to take some modest steps to bring India closer both economically and politically. The word "strategic" is overused in modern diplomacy, usually in an aspirational sense, but the EU listed India in 2004 as one of the ten countries it has chosen as its "strategic partners" in the international community. Javier Solana, a distinguished fellow at the Brookings Institution and the former Secretary General of NATO, has called this selection of India as a "reasonable choice" given that it is the world's most populous democracy.
Narratives such as this are not unusual in European socio-political circles, but they paper over some real weakness in the bilateral relationship. The fact that India's economic potential, skilled worker base, business opportunities, and willingness to achieve rapid economic development are still largely overlooked explains why EU government and institutional understanding of the country remains largely poor, with some exceptions perhaps in the U.K., France, and Germany. Even within countries such as France and Germany, insight seems patchy.
Still, trade between Europe and India is rising steadily, from 28.6 billion euro ($31.8 billion) in 2003 to 72.5 billion euro in 2014. EU investment stock in India was 34.7 billion euro as of 2013, showcasing the gradually forward moving trend in EU-India relations. Nonetheless, the frustrations on the EU's side remain evident: On the European Commission's website, immediately after noting the above numbers, India's economy and regulatory environment are still described as "comparatively restricted" on the back of its poor showing on World Bank's Ease of Doing Business Index 2014.
The fact that the new government in Delhi has launched high-profile campaigns such as "Make In India" to boost foreign direct investment (FDI) and expand the economy, without orchestrating crucial institutional reforms within its ministries and bureaucracy to lay the groundwork exposes it to criticisms of indulging in mere PR. Despite this, India has attracted FDI ($34 billion according to reports) sufficient to enter the top ten list, and this only highlights the immense possibilities structural reforms in areas such as taxation could have for the Indian economy, specifically for sectors such as manufacturing, the only avenue for creating millions of needed new job opportunities.
The mooted changes to Indian diplomacy and bureaucracy include the strategic placement of industry experts in identified countries according to their industrial expertise, moving the Directorate General of Foreign Trade from under the Ministry of Commerce and Industry to the Minister of External Affairs (MEA), opening up hiring for the MEA to external expertise for full diplomatic placements in missions around the world, among others. While these reforms would seem obvious to many, they are easier said than done, particularly departing from the traditional hiring procedures for the foreign ministry, an idea that has met with considerable bureaucratic resistance.
India-EU relations as they stand today offer an ideal situation for the two sides to try out some path-breaking diplomacy to forward the relationship. The situation is particularly ripe for India, as reforms and modern diplomacy could work well in partnership with the EU, which in return offers very low political risk and potentially big economic and technological dividends.
Kabir Taneja specializes in South Asian foreign affairs, energy security and defense. He is currently a Visiting Researcher with the Fridtjof Nansen Institute, Norway. These are his personal views.
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Rob Jones Is Lighting Up The Streets Through Two Games
In his entire football career, including pee-wee, high school, college and pros, Rob Jones had never had a five touchdown game. That was until this past Saturday night when he found the end zone five times in a monster performance during the Streets' 70-31 win over the Orlando Predators in the inaugural home opener at the Westchester County Center.
The Tampa, Florida native had seven catches for 130 yards and five touchdown receptions.
"It was good to see," said Jones. "It was a sign that we were clicking. Players are always trying to keep that kind of rhythm throughout the season."
Through two games, Jones is in some kind of rhythm with fifteen catches for 285 yards and six touchdowns as the expansion Streets have come out of the gates with a 2-0 record. A big reason why is the reunion between Jones and quarterback EJ Nemeth who has thrown eleven touchdown passes so far this season. The fact that they've enjoyed instant chemistry is not a surprise.
The duo played together with the York Capitals in 2015 and won an American Indoor Football championship. Now both players are teammates again after some time away from football but they have picked up where they left off.
"I have some experience with him and a little bit of chemistry so us coming back and working together and being that dynamic duo is a great thing," said Jones.
The Streets have put together an impressive receiving corp led by Jones, Devonte Jenkins and Edgar Allen Poe. When Streets Head Coach Rick Marsilio began to assemble this team, it was a no-brainer for him to reach out to Jones and Nemeth to try and rekindle the magic that they had a few years ago in York.
"We're really excited to have Rob Jones in New York," said Marsilio. "He played for me in the past and has a chemistry with EJ that has been very important for our early success. Both EJ and Rob bring veteran leadership to our young guys on offense."
The former University of Delaware star has proven to be a big acquisition for the Streets. Arena Football is back in New York after an eleven year absence and Jones has made an immediate impact while also becoming a fan favorite. He was able to enjoy some interaction with the crowd during the home opener thanks to the intimacy of the Westchester County Center.
The fans are right on top of the game and Jones loves it.
"I liked that," said Jones. "It was very different. I felt like the fans were very into the game. They have views like no other. They're right there next to the action. I was telling one of my teammates that was the first time I had those kind of interactions with fans being there close to the game. As soon as I scored a touchdown, people are grabbing my face so that was pretty cool."
In the first two games, the Streets outscored their opponents 122-72 and Jones has played a huge role in that success. He has been a threat with the short passing game as well as with Nemeth firing the deep ball down the field. One of Jones' touchdown passes this past week came on a shovel pass and he's also the Streets' kick returner.
For an expansion team to have an offensive weapon like this is unique.
"Rob is a special athlete and we are utilizing his talents," said Marsilio. "His fast success will open up opportunities for our younger receivers as we move deep into the season. Rob always becomes a fan favorite as shown this past week in our home opener. Rob has the ability to have a breakout season and we are only seeing about half of what he can add to our team."
The Streets are unbeaten through two games and enjoying a bye week before returning to the field next Saturday May 4th in Carolina. Jones an company may be 2-0, but it's clear that they have received Marsilio's message that this team has not even scratched the surface of what they can do.
"Like coach said, we're not even a good team yet," said Jones. "We aren't even clicking on all cylinders how we should be. The chemistry is there but we're all knocking off a little bit of rust."
An expansion team that is 2-0 with a high-octane offense but isn't even "a good team yet"?
That's bad news for the rest of the National Arena League because right now Rob Jones and the New York Streets are on fire. How do you get hotter than being on fire? The Streets are hard at work during the bye week figuring out just how to do that before returning to the practice field next week.
The Streets' next home game is on Saturday May 18th against the Massachusetts Pirates at 7pm. For season and group ticket information, please call 1-833-697-8733 and choose extension one. Individual tickets for Streets home games are available at www.ticketmaster.com.


We anticipate the state of the natural world to improve the quality of our own lives. The anticipation comes from expectations we make based on experiencing the natural world and it's various rhythms that occur over time. Those who invest the time to understand weather might anticipate a sunset if the joy of experiencing sunsets is something that moves them. Now that I've experienced the beauty of a blood-red harvest moon rising through the ocean, I've developed a more keen interest in the moon phase cycle come September and October each year. Those who get giddy over the annual flyover patterns of bird migration invest in considering a wide range of factors that affect the natural world. Some people coordinate their anticipation through a long list of facts and heuristics they maintain in their personal knowledge base. Others, like me, have found that systems simulations provide a more efficient mechanism for anticipating nature by more efficiently storing a model for rules generation than the long list method. In that case, even the list-items that do attach themselves seem less cumbersome to remember with a simulation at their base.
This chapter is the first of three chapters that provide representative case studies into the power of system simulations to provide understanding and insight. Simulations are often built for their power of prediction and decision-support. Let's focus on understanding and insight since those metrics offer us the opportunity to explore without having specific reasons ahead of time. We can build, run, and investigate simulations and see where our thought processes turn before they turn to known benefits. Only afterwards, once we've had our own independent thoughts and experiences, can we consider the experts' knowledgeable use of the simulation. At that point, I ask myself whether those considerations are natural ramifications of my own thoughts or if they are exceptions. I invest time into understanding the exceptions for they are most likely to seem foreign to me most often. Those that are natural tend to be remembered much easier than those that seem foreign. And, of course, I don't shortchange my right and opportunity to challenge the exceptions — even though those challenges are often unfounded by the time I understand better what the experts already understand. The process itself seems meaningful.
Let's start with the mother of natural systems simulations. Meteorologists have made great strides in predicting weather during my lifetime. As a result, meteorology is a great case study in the potential of simulation for understanding nature. Weather is complex (and fickle) friend of ours as we find conditions changing over time and place in ways that force us to anticipate future weather based on past experience. The collective knowledge of all meteorologists is a brain trust that allows us to inject weather consideration into our daily lives — which has also been great for starting up conversations with strangers we might not start up otherwise. Such conversations often feel more meaningful when I am in unfamiliar surroundings.
As we look at the dynamic range of natural conditions on the planet, we find that some areas are harder to predict weather for than others. There are natural convergence zones where weather in higher latitudes mixes with weather in lower latitudes and creates larger gradients of behavior over smaller distances. We have coastal zones where relevant weather data is harder to acquire as no one lives on the ocean consistently to report the data — and our ocean-based sensors confront rough conditions to keep reporting reliably. We have regions where cloudy conditions are much more prevalent — limiting our ability to gather highly relevant data by remote sensing satellites way up in orbit. And then, of course, we have more reliably uniform weather regions on the planet where weather forecasting is significantly simpler — although longer-term climate change moves those regions ever so slightly from year to year.
Weather is a significant driver (also called a forcing agent) in modeling other natural systems we encounter on our planet — and thus critical for many natural system simulations. As a result, we have put a lot of time and effort (and research money) into simulating weather systems in order to understand and better predict weather in the future. Meteorologists have become recognizable community figures as they have learned to communicate model outputs to others.
One interesting follow-on simulation we can use as a case study looks at the effect weather has had when investigating water movement on the planet: How does water move from atmosphere to land to water network to ocean to atmosphere again? The behavior of water as it passes by us in streams and rivers might matter most when flooding is involved. But, the behavior of water is a fascinating place to invest one's thought and insight-generation powers. We might take water for granted thanks to nature's competent stewardship on a day-to-day basis. Or, we might live in a community that can't afford to be as cavalier since water issues affect the community more noticeably more often. We can build compassion for those communities that have struggled with water issues by growing our understanding and insight of water through interaction with visual simulations.
If we use the often-used word watershed as the geographical area where water exists and yet moves downhill to the same eventual exit locale, we can then say that the hydrology modeling effort to simulate water movement is a key component of all watershed simulation — simulation that looks at any behavior of man-made or other process within the bounds of a geographical watershed.
My first exposure to natural systems simulation started with hydrology modeling as an impressive cadre of natural systems simulation experts introduced me to the hydrology component of watershed simulation. Although those experts had each come to prominence by focusing on a specific natural domain regardless of watershed geography, the university was asking them to coordinate their research on behalf of the greater local watershed known as the Puget Sound region, for a better integrated understanding of residents' actions on the health of systems that provide fresh water inputs to the sound — the body of water that dominates the local economy and recreation activities.
As it turns out, hydrology modeling is often one of the easier components to simulate in a watershed domain. In many situations on our planet, hydrology is straightforward and floods, drinking water supply, and minimal water levels to support life year-round can be predicted to a necessary precision quite consistently if the weather forecast is accurate — in other words, the hydrology is mainly a result of weather precipitation, temperature, wind, and evaporation (each of which is predicted in a weather forecast). It's harder to model the interaction of that water with plants and animals (including us humans). Sometimes the modeling effort is more difficult due to convergence zone weather issues and complex terrain.
The Puget Sound area is unique in that it sits between two mountain ranges that amass water content as snow pack for much of the year. The mountains are high in elevation and include dramatic slopes as the elevation lowers from mountaintop to sea level over a relatively short distance (way shorter than the Mississippi-Missouri waterway, for example). The temperature at the mountaintops can be quite different from the temperature at sea level and the geographical spread of the temperature gradient can vary significantly from day to day. Rain at sea level does not mean rain at the mountaintop and sun at sea level does not mean sun at the mountaintop. Wind effects add complexity. The diversity of behavior over the course of a year provides a complex environment for simulating water flow — and water flow in times of potential flood levels is very valuable to those living near river systems.
We take a look at the hydrology-modeling component of a watershed simulation in this chapter. We compare a simple model that is effective for a stable environment to a complex model that is effective for a dynamic environment. It's an ideal case study for learning how to focus on simulation development by differentiating inputs, processes, and outputs — since the community of hydrology experts has evolved their identification and conventional naming significantly over time. And, of course, each input, process, or output can help us practice our simulaton life skills as we have likely interacted with each input, process, and output often in our lives and can readily conjure up each in our imagination on demand.
HSPF simulates for extended periods of time the hydrologic, and associated water quality, processes on pervious and impervious land surfaces and in streams and well-mixed impoundments.
Rainfall is a primary input — something most of us would anticipate as being critical to hydrology simulation.
Elevation is a primary input as it defines the terrain over which rain will fall — most of us can easily envision how gravity pulls water downhill faster over a steep elevation change than a gradual one.
Soil moisture is often a primary input as it defines whether more water can be absorbed by soil within the watershed or whether the water will run downhill without the opportunity to be caught.
Even soil moisture can become irrelevant to the model should the ground be so impervious (for example when covered with bedrock at the surface) that soil moisture becomes immaterial. Most of us can imagine a watershed within a rocky mountain range where all the water falls on rock and moves downhill to a lake at the bottom of the elevation. Our simulation model in those situations is likely going to be as good as the weather forecast. If we can predict the rainfall (and the evaporation of water back to the atmosphere on sunnier days), and the terrain doesn't change by some unlikely geological event (a mass wasting event like a rock slide or interruption by volcanic activity), we don't need much software support to create a solid simulation of water behavior in that watershed.
The processes involved in that simplified environment include moving rainfall from the atmosphere to the land area that drains to the lake, moving water back to the atmosphere through evaporation, and moving water downhill from higher elevations according to the elevation profiles along the way. The output is a lake volume that consistently can be evaluated as a lake depth and a shoreline detail.
Vegetation is often a primary input when the watershed upon which rain falls is filled with plant life that uses the water for life process — most of us can envision plants taking up water through its roots to support life processes within the plant.
The vegetation adds significant complexity to our simulation. The vegetation is dependent on the soil for its ability to live in that location. The vegetation is dependent on access to sunlight that is varied based on elevation and the direction of slope. And, of course, the vegetation is dependent on the rainfall for it's ability to stay alive. There are interactions among vegetation and all other inputs as soon as we add vegetation as an input to the simulation. The complexity behavior between soil moisture and vegetation is partially driven by root depth and partially driven by the water requirements of the vegetation. Some plants take up a lot more water than others. Soil that is saturated with plant roots behaves very different from soil that is porous but deeper than any root depth.
Adding vegetation as an input also means we can benefit from simulating the use of water by the plants. New processes help us become more precise in our simulation: plant transpiration moves water through a plant and evaporates water from aerial parts — especially from leaves but also from stems and flowers; plant photosynthesis uses water to produce sugars from sunlight; plant respiration burns sugars for energy when sunlight is not providing input to photosynthesis. Thankfully, the plant transpiration is a dominant process compared to photosynthesis and transpiration and we can add plant transpiration to overall evaporation of water from the watershed surface to create a single simulated process called evapotranspiration.
These vegetation effects to our hydrology simulation can vary significantly by time of year if the watershed experiences dramatic seasons. Plants don't take up water in winter if they are a species that goes dormant, for example. Some plants lose their leaves in preparation for winter. The effects of fire can dramatically affect the vegetation inputs being simulated. It's powerful to dwell on the effects life has on the behavior of water. Plants are just the beginning as of course animals, and particularly us humans, use water in life processes as well. Animals are not as limited by watershed boundaries and can move water from one watershed to another in the course of daily living.
With the addition of vegetation, we are already at a level of complexity in which we can continue to have insights and develop deeper understanding through simulating hydrology. In many cases, the effects are subtle but we as humans have the opportunity to explore the subtle effects, as the watershed we experience is the sum total of all effects and nothing else. If we plant a tree, we change the hydrology in the watershed to some degree. If we plant 100,000 trees as has been happening in some Scottish watersheds, we can change the watershed significantly. If we change the soil within the watershed, we change the opportunity plants have to thrive. Or, on the flip side, we increase the chances they won't. Clear-cutting a forest has been a much more recent change to the hydrology of a watershed than planting the same number of trees. Simulation tools have taught many forest managers a reasonable approach to managing tree harvest and replanting.
Snowpack forms from layers of snow that accumulate in geographic regions and high altitudes where the climate includes cold weather for extended periods during the year.
The snowpack grows as a season gets colder and shrinks when a season gets warmer (0 degrees Celsius being a reasonable starting point for expecting growing versus shrinking). A snowpack can accumulate tremendous amounts of water and keep that water from heading downhill for significant periods of time. The snowpack usually happens most where the potential energy of water is at its highest (where the momentum for downhill movement is greatest). We can imagine a watershed without soil and without vegetation that would be affected greatly by snowpack. Antarctica and Greenland are two large geographical regions that have been in the news often for their dominant effects of snowpack within their watersheds. As we add soil and vegetation to the model, we see how their effects are dwarfed by snowpack that prohibits water from reaching their contribution to the simulation.
Wind becomes important as an input because snowpack interacts with wind to increase evaporation across the snowpack — the greater the wind, the increase in evaporation. Wind increases evaporation in places besides those covered in snowpack as well.
Air temperature becomes important as an input because snowpack melts faster for each additional degree in temperature as the air temperature increases well beyond the freezing temperature of water.
Shading becomes important as shade cools regions that can maintain snowpack more easily as a result. As snowpack changes the surface elevation away from the basic elevation input, shading changes as well.
Snow state becomes important as different snow types have different properties just as soil types have different properties.
We can also start to differentiate our previous inputs into more specific inputs. The soil input can be decomposed to include separate inputs as to the soil type, soil depth, and soil state — the latter as one that most readily changes throughout the simulation. The vegetation can include separate inputs as to vegetation type, root depth, and vegetation state — the latter as one that most readily changes throughout the simulation. Even the precipitation can be decomposed into amount and type (light, moderate, heavy, sleet, hail, snow).
The stream channel input is a network of all the channels within the watershed that carry surface water downhill — we know these channels as streams and rivers — organized into a network of segments mapped to location and connectivity.
The stream channel is an observed phenomenon and so needs to be authentic to map the outputs of the simulation to actual objects that can be evaluated. Of course, if all the other inputs were perfect and of necessary resolution, the stream channel would be a pure output of the simulation (the elevation model would suggest where those streams should be). The cost of maintaining such a high-level of resolution in the simulation is more often not justified since the probability of additional insight is not enough to justify it. And so, the simulation must map its view of the world to water in the channel — something it does quite well, thankfully, once the stream channel input is made available.
The left image represents a situation where elevation change is gradual and elevation is quite flat in most places. The right represents a situation where elevation change is rapid in places and elevation is quite sloped overall — blue is the highest elevation and red is the lowest. One insight to be gained through experience with hydrological simulations is that the complexity for simulating the hydrology in the left situation can be a lot simpler than the right situation for getting the same level of correlation with the physical world — the closeness of the simulated volume of water in each channel to the actual.
The left image shows the soil depth input within a watershed while the right image combines the stream channel network input as a blue presentation on top of the elevation input that is colored and shaded to suggest height in relief.
Each input to a hydrology simulation has its own natural temporal rhythm — the time period over which values tend to change. The elevation input is unique in that it is usually considered fixed throughout the simulation period — unless it is part of a scenario looking at elevation change for some simulated event that would change elevation. Soil depth, vegetation type, and soil type are examples of inputs that change slowly in a simulation. Soil moisture, precipitation, and wind are examples of inputs that change more rapidly — rapidly enough that the simulation keeps changing them continuously to move the simulation forward through time. When an input changes often through a shorter time rhythm of change, the input makes a prime candidate for animation to visually explore its input to the simulation. Here are a few examples.
Air temperature animations show the change of air temperature over time. Blue color below is cooler than the red temperature. The warming trend between the left and right provides detail when animated over short time increments. Animations that cover a longer time period show patterns of how the air mass above a watershed cools and warms near the surface. Our brain's ability to pick out patterns provides insight into the nature of the temperature change.
Precipitation animations show patterns of rainfall over the same period of time. Our brains can explore the relationship between precipitation and air temperature when both are provided over the same time period. In this case, heavy rainfall is falling on the western half of the watershed with a very small transition zone to the dry half of the watershed to the east.
We can explore one of the more intricate inputs, soil moisture, through visualizing soil moisture animation over time. Soil moisture levels change across the watershed as the simulation runs. The mid-blue color below represents soil that has been saturated by moisture. Other colors represent percents of saturation that are less than 100%. In this case saturation moves from west to east, which is not surprising given that precipitation is moving from west to east as well.
Snow pack animation provides insight into the effect of other inputs and processes on the amount of snow covering the terrain. Warm temperatures, rainfall, wind, and other inputs all affect the snow pack and an animation of snowpack simulates the rate of change over time. The dark blue color represents areas within the watershed with the most snow pack. Lighter colors represent values of less snow pack. White areas contain no snow pack. White areas increase as overall snow pack reduces over time.
Each input to the simulation is involved in at least once dynamic process that determines the flow of water. If the input were not involved in the computation of water movement, it wouldn't be directly relevant to the simulation. Some processes include many input types while others include just two or a few. The design of the simulation takes into account the relevance of each input to the authenticity of the result of computations. Some inputs may have an impact that is so small that the designers consider it immaterial for the typical range of system behavior seen in the physical world. Computation time reduces when inputs can be removed from being involved in a process. From that point of view, we can consider some of the processes often included in hydrology models and consider the impact of each input involved.
Gravity plays a dominant role in hydrology. Water moves downhill consistently in response to gravity as a force. Friction of water as it moves against and over surfaces slows the rate at which water moves. Different soils have different grain sizes. The smallest grains don't let water through once fully saturated. The largest grains let water through with least friction thanks to the open spaces available between grains. Other soil types vary from predominantly smaller grains to predominantly larger grains. Some are a complete hodge-podge of small and large grains all within a small volume. As a purely physical process, gravity moves water through its environment in predictable ways. Digital elevation inputs, soil inputs, vegetation inputs, and channel inputs all contribute to the computation of water movement due to the physical force of gravity. The simulation of the force of gravity plays a dominant part of predicting hydrology behavior in many environments. Other environments include relevant chemical and biological forces.
Evaporation plays a significant role in hydrology. Water returns to a vapor state from a liquid state over time in many hydrological environments. Without replenishing evaporated water through new precipitation or other phenomena that turn water vapor back into liquid, a watershed dries out over time through the evaporation process. The surface area of water available for evaporation is derived from the soil, vegetation, and channel inputs. The rate of evaporation is derived from temperature and wind inputs. A hydrology simulation computes the movement of water back into the air as a contribution to overall hydrology movement — a contribution that works against the force of gravity more than along with it.
Saturation plays a significant role in hydrology. When soil gets saturated with water, water can move over the soil instead of being drawn into it by gravity and other forces. When plants get saturated with enough water for their biological needs, they no longer need to take water out of the soil. The specifics of the soils and vegetation inputs to the hydrological model affect the simulation of water movement. Once a soil is saturated, soil type becomes less significant from a water run-off perspective. Water begins to run downhill much faster without the effects of soil and vegetation damping the movement.
Other processes vary widely depending on the characteristics of the watershed. Urban watersheds, or those that provide drinking water for urban watersheds, can be dramatically affected by human actions. Dams can change the movement of water downhill by creating reservoirs that catch the water without letting it progress. Pipes and culverts can redirect water. Pumps can drive water uphill against the natural pull of gravity. An urban hydrology simulation often benefits from the expertise of a civil engineer who studies the behavior of water in man-made environments. An urban behaviorist can help when simulating the draw human beings make of water reservoirs over time. A hydrology simulation can get significantly more complex when human behavior and man-made objects are active in the watershed. The outputs from one time step's worth of computation include parts of the input to the next time step's worth. As a result, the inputs and outputs available for investigating a hydrology simulation as it runs are tightly coupled and the artifacts that the simulation make available for insight and understanding are often similar enough to benefit from similar perception and cognition processes. A soil moisture animation artifact includes a view of soil moisture at each time step the simulation creates a discrete computation. The snow pack artifact includes a view of snow pack at each time step the simulation creates a discrete computation. The channel volume artifact includes a view of water volume in each channel for each time step in the simulation.
With enough artifacts providing changing views of key components to the hydrological simulation process, we have a wide variety of mental responses that can occur. Each mental process can be highly unique to each individual. As a result, a rich discussion can come from groups of people who gain access to the simulation with the goal of gaining insight and understanding.


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/ Modified sep 11, 2019 4:19 p.m.
Tucson Sector border apprehensions are up, national numbers down
Migrant apprehensions across much of Arizona are higher than they were a year ago, even while the national numbers have dropped considerably.
by Ariana Brocious TWEET SHARE
The U.S.-Mexican border fence near Nogales.
Customs and Border Protection announced Monday apprehensions have dropped by about 56% from the highest levels in May, while apprehensions in the Tucson Sector have risen.
"We attribute the majority of that [overall] drop to some of the efforts that are being made by the government of Mexico and also the Northern Triangle countries, who are starting to treat this now as a regional migration crisis," Border Patrol Tucson Sector Special Operations Supervisor Pete Bidegain said.
The Yuma, El Paso and Rio Grande Valley areas have seen drops in border crossings. But in the Tucson Sector, Bidegain said apprehensions in August were up about 25% over last year, due almost entirely to an increase in families and children who cross the border seeking asylum.
"Over the last few months in May, June, July, we've actually seen that traffic shift from the Ajo area near Quitobaquito Springs to east of Sasabe," he said.
Bidegain said while traditionally there's about a 10% drop in border crossings during the summer, in the last couple years monthly apprehensions have remained fairly consistent in his sector. Roughly a third of Tucson Sector apprehensions are family units or unaccompanied children who turn themselves in to agents, he said, but the majority are adult males actively evading arrest.
Hundreds of thousands more people have been apprehended at the southern border this year than in years past, largely because of the rise in families and unaccompanied children from Central America.
VIEW LARGER U.S. Government Accountability Office map showing Border Patrol sectors.
GAO/Flickr
MORE: Border, Immigration, News
Tucson Sector Border Patrol chief on trends along southern border US sending Mexican migrants 1,000 miles from border Trial over treatment of migrants in Border Patrol custody begins


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alberto albor
Burgos, 1977. A graduate in Fine Arts from the UPV, Alberto Albor has been a recipient of scholarships from, for example, Juan de Otaola, Bilbao Arte and the Provincial Council of Bizkaia. He has participated in several exhibitions, both group and individual. He is currently continuing to develop his pictorial work.
[Project]
"I present my paintings as windows through which to access an artificial reality which is strange, but no less plausible for that, and which serves as the base for a series of objects that make up a narrative, whose content appeals directly to the collective memory."
Open programme for the production of context in contemporary art.
Department of Culture and Language Policy of the Basque Government.
Receiving entity of the eremuak programme: Tabakalera.