\subsection{Gravitational Radiation}


The development of strong nonaxiymmetry instability leads
to a pulse of graviational wave (GW) radiation. GW strains for 
the "$\times$" and "+" GW polarization modes are calculated 
using the quadrupole approximation in the Newtonian limit
{\it e.g.}, see Finn \& Evans (1990). The strain $h_{\mu\nu}$ is 
represented as a perturbation to flat space,
\begin{equation}
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},
\end{equation}
where $g_{\mu\nu}$ is the metric tensor, $\eta_{\mu\nu}$ is the Minkowski 
metric, and $h_{\mu\nu}$, the strain tensor, satisfies
$|h_{\mu\nu}|$ $\ll$ $|\eta_{\mu\nu}|$.
The strains are given by
\begin{equation}
h_{+} = \frac{G}{rc^4} 
\left(\frac{\partial^2{\rm I}_{xx}}{\partial t^2}{\rm cos}^2\theta
+\frac{\partial^2{\rm I}_{zz}}{\partial t^2}{\rm sin}^2\theta
-\frac{\partial^2{\rm I}_{xz}}{\partial t^2}{\rm sin}2\theta
-\frac{\partial^2{\rm I}_{yy}}{\partial t^2}
          \right) 
\end{equation}
and
\begin{equation}
h_{\times} = \frac{2G}{rc^4} \left(\frac{\partial^2{\rm I}_{xy}}
{\partial t^2}{\rm cos}\theta
-\frac{\partial^2{\rm I}_{yz}}{\partial t^2}{\rm sin}\theta\right),
\end{equation}
where ${\rm I}_{ij}$ is the quadrupole moment tensor, 
\begin{equation}
{\rm I}_{ij} = \int\;\rho(x_i^{\prime}x_j^{\prime}-\frac{1}{3}
\delta_{ij}r^{\prime 2})
{\rm d}^3x,
\end{equation}
and $r$ and $\theta$ are spherical coordinates. The angle
$\theta$ is measured from the rotation axis of the fizzler to the
direction of the observer, and $r$ is the distance to the fizzler. 
Second time derivatives are difficult to determine accurately in 
hydrodynamic simulations and so we follow New, Centrella, \& Tohline (2000)
and use the partially integrated forms of the above relations, 
\begin{equation}
 \frac {\partial^2{\rm I}_{ij}} {\partial t^2} =
 \int\rho[2v_{i}v_{j}-\frac{2}{3}v^kv_k\delta_{ij}-x_j\nabla_i\Phi_g-
 \frac{2}{3}x^k\nabla_k\Phi_g\delta_{ij}]{\rm d}^3x.
\end{equation}
Here $\delta_{ij}$ is the Kronecker $\delta$-function and the Einstein
convention is used for repeated summation indices.