



\section{Stochastic Spin-up by Internal Gravity Waves}
\label{spinup}

A stochastic influx of IGW into the core will also enforce a minimum expectation value of the core spin rate. Consider a non-spinning core being irradiated by IGW wave packets emitted from an overlying convective shell. The spherical symmetry of the background structure implies that the wave packets will contain randomly oriented AM vectors. Since we expect incoming waves to non-linearly dissipate before being reflected, the core will absorb their AM. At the end of the convective shell phase, the total core AM is the sum of the randomly oriented wave packet AM vectors emitted by the shell. Each wave packet will contain an AM vector of length
\begin{equation}
\label{eqn:AMvec}
J_w \sim \frac{2 \pi}{\bar{\omega}} \dot{J}  \sim \frac{2 \bar{m} \pi}{\omega_{\rm c}^2} \mathcal{M} L_{\rm c} \, ,
\end{equation}
which we have calculated using equations \ref{eqn:Ewaves} and \ref{eqn:Jwaves}, and assuming typical wave packets have angular frequency $\omega_{\rm c}$ and last for an eddy turnover time, $2 \pi/\omega_{\rm c}$. The expected magnitudes of the $x$, $y$, and $z$ components of the AM vector are $J_w/\sqrt{3}$. Since our goal is to find a minimum rotation rate, we use $\bar{m}=1$ to minimize the AM carried by each wave packet.
