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\title{IDBrillData}
\author{Carsten Gundlach, Gabrielle Allen}
\date{$Date$}
\maketitle

\abstract{This thorn creates time symmetric initial data for Brill
wave spacetimes.  It can create both axisymmetric data (in a 3D
cartesian grid), as well as data with an angular dependency.}

\section{Purpose}

The purpose of this thorn is to create initial data for a Brill wave
spacetime.  It does so by starting from a three--metric of the form
originally considered by Brill
\begin{equation}
ds^2 = \Psi^4 \left[ e^{2q} \left( d\rho^2 + dz^2 \right)
+ \rho^2 d\phi^2 \right] =\Psi^4 \hat{ds}^{2},
\label{eqn:brillmetric}
\end{equation}
where $q$ is a free function subject to certain regularity and
fall-off conditions, $\rho=\sqrt{x^2+y^2}$ and $\Psi$ is a conformal
factor to be solved for.

Substituting this metric into the Hamiltonian constraint gives an
elliptic equation for the conformal factor $\Psi$ which is then
numerically solved for a given function $q$:
\begin{equation}
\hat{\nabla} \Psi - \frac{\Psi}{8} \hat{R} = 0
\end{equation}
where the conformal Ricci scalar is found to be
\begin{eqnarray}
\hat{R} = -2 \left(e^{-2q} (\partial^2_z q + \partial^2_\rho q) + 
\frac{1}{\rho^2} (3 (\partial_\phi q)^2 + 2 \partial_\phi q)\right)
\end{eqnarray}
Assuming the initial data to be time symmetric means that the momentum
constraints are trivially satisfied.

The thorn considers several different forms of the function $q$
depending on certain parameters that will be described below.

Brill initial data is activated by choosing the {\tt CactusEinstein/ADMBase}
parameter {\tt initial\_data} to be {\tt brilldata}.

In the case of axisymmetry, the Hamiltonian constraint can be written
as an elliptic equation for $\Psi$ with just the flat space Laplacian,
\begin{equation}
\nabla_{flat} \Psi + \frac{\Psi}{4} (\partial_z^2 q + \partial_\rho^2 q) = 0
\end{equation}
If the initial data is chosen to be {\tt
ADMBase::initial\_data = "brilldata2D"} then this elliptic equation
is solved rather than the equation above.


\section{Parameters for the thorn}

The thorn is controlled by the following parameters:

\begin{itemize}

\item brill\_q (INT):  Form of the function $q$ [0,1,2] (default 2):

\begin{itemize}

\item brill\_q = 0:
\[
q = a \; \frac{\rho^{2+b}}{r^2} \left( \frac{z}{\sigma_z} \right)^2
e^{-(\rho - \rho_0^2)}
\]

\item  brill\_q = 1:
\[
q = a \left( \frac{\rho}{\sigma_\rho} \right)^b \frac{1}{1 + \left[
\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}}
\]

\item  brill\_q = 2:
\[
q = a \left( \frac{\rho}{\sigma_\rho} \right)^b e^{-\left[
\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}}
\]

\item If one specifies 3D data (see above), the function $q$ is multiplied
by an additional factor with an angular dependency:
\[
q \rightarrow q \left[ 1 + d \frac{\rho^m}{1 + e \rho^m}
\cos^2 \left( n \phi + \phi_0 \right) \right]
\]

\end{itemize}

\item brill\_a (REAL): Amplitude (default 0.0).

\item brill\_b (REAL):  $b$ in above expressions (default 2.0).

\item brill\_c (REAL):  $c$ in above expressions (default 2.0).

\item brill\_d (REAL):  $d$ in above expressions (default 0.0).

\item brill\_e (REAL):  $e$ in above expressions (default 1.0).

\item brill\_m (REAL):  $m$ in above expressions (default 2.0).

\item brill\_n (REAL):  $n$ in above expressions (default 2.0).

\item brill\_r0 (REAL):  $r_0$ in above expressions (default 0.0).
  
\item brill\_rho0 (REAL):  $\rho_0$ in above expressions
  (default 0.0).
  
\item brill\_phi0 (REAL):  $\phi_0$ in above expressions
  (default 0.0).
  
\item brill\_sr (REAL):  $\sigma_r$ in above expressions
  (default 1.0).
  
\item brill\_srho (REAL):  $\sigma_\rho$ in above
  expressions (default 1.0).
 
\end{itemize}

The elliptic solver is controlled by the additional parameters:

\begin{itemize}
  
\item {\tt solver} (KEYWORD): Elliptic solver used to solve the
  hamiltonian constraint [sor/petsc/bam] (default "sor").
  
\item {\tt thresh} (REAL): Threshold for elliptic solver (default
  0.00001).

\end{itemize}

\section{Notes}

Thorn {\tt IDBrillData} understands both the {\tt physical} and {\tt
static conformal} {\tt metric\_type}. In the case of a conformal
metric being chosen, the conformal factor is set to $\Psi$. Currently
the derivatives of the conformal factor are not calculated, so that
only {\tt staticconformal::conformal\_storage = "factor"} is
supported.

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\include{param}
\include{schedule}

\end{document}
