omega_2$ the stable one. Because $b$ depends on $\omega$ through Eq.  and the eigenmode structure $\phi(z)$ depends on $b$ through Eq. , the two solutions $\omega_j$ correspond to two different eigenmodes $\phi_j(z)$. We identify $b_j$ and $\phi_j$ as the $b$ and $\phi$ corresponding to $\omega_j$. The eigenmodes are then $$\label{solved phi}
\phi_j(k,z) =
\begin{cases}
\left(e^{2|k|}+b_j\right)e^{-|k|z} & z > 1 \\
e^{|k|z}+b_je^{-|k|z} & -1 < z < 1 \\
\left(1+b_je^{2|k|}\right)e^{|k|z} & z < -1,
\end{cases}$$ where $$\label{bj}
b_j = e^{2|k|}\frac{2|k|(\omega_j+k)-k}{k}$$ satisfies $b_1(k) = b_2(-k) = b_2^*(k)$ for $|k|<k_c$, and $b_j(k) = b_j(-k) = b_j^*(k)$ for $|k|>k_c$.

![Equilibrium (left column) compared with velocity profiles of the unstable $\phi_1$ (middle column) and the stable $\phi_2$ (right column) at wavenumbers $k = 0.4$ (top row) and $k=1$ (bottom row) plotted over one wavelength in $x$ and from $z=-2$ to $z=2$. Streamlines are plotted with color indicating flow speed. The first row is in the unstable range, where $\phi_1$ grows exponentially while $\phi_2$ decays exponentially. The second row is a marginally stable wavenumber, where both $\phi_1$ and $\phi_2$ oscillate without any growth \[see Fig. \[dispersion\_plot\]\].[]{data-label="eigenmode_plot"}](Fig2.eps){width="16cm"}

For $\omega^2<0$, the eigenmodes are nearly identical but satisfy $\phi_1(k,z) = \phi^*_2(k,z)$. Figure \[eigenmode\_plot\] shows the flows corresponding to these eigenmodes for four wavenumbers sampling the unstable and stable ranges. Previous work has shown that the physical mechanisms for instability of $\phi_1$ and stability of $\phi_2$ can be understood in terms of resonant vorticity waves in both the hydrodynamic[@Baines] and MHD[@Heifetz] systems.

In standard descriptions of turbulence and quasilinear transport calculations, it is common practice to neglect stable modes given their exponential decay from a small initial value. In this paper we account for the nonlinear drive of the stable mode by the unstable 