index; Toomre $Q$ parameter at 2.2$R_{d}$ .

-body Simulations {#sec:nbodysimulations}
-----------------

![image](fig2.pdf){width="\linewidth"}

![image](fig3.pdf){width="\linewidth"}

We use <span style="font-variant:small-caps;">galactics</span> to generate $N$-body realizations for the two models just described. These realizations have $2.5M$ disc particles, $250k$ bulge particles, and $5M$ halo particles. The masses of particles within each component are identical. The initial conditions were evolved for $\sim\!10\,{\rm Gyr}$ using <span style="font-variant:small-caps;">gadget-3</span> [@springel2005] with a softening length of $40\,{\rm pc}$ for all particle types. The maximum time step was set to $0.2\,{\rm Myr}$, which is $\sim$ 0.2% of the Galactic dynamical time at a radius of 20 kpc, where the warp occurs, and $\sim$ 1% of a Galactic dynamical time at 4 kpc. Total energy was conserved to within 0.06%. We start with “pristine" equilibrium initial conditions, which allow us to focus on key dynamical processes such as the secular evolution of bending waves. Our simulations ignore gas dynamics and do not capture the physics of galaxy formation.

Over time the discs in our simulations rotate and drift slightly about the global coordinate origin. To account for this when analyzing face-on maps of our discs we center the disc and rotate it into the $x$-$y$ plane by way of the following iterative 