  We study the stability of steady convection rolls in 2D Rayleigh--B\'enard
convection with free-slip boundaries and horizontal periodicity over twelve
orders of magnitude in the Prandtl number $(10^{-6} \leq Pr \leq 10^6)$ and
five orders of magnitude in the Rayleigh number $(8\pi^4 < Ra \leq 3 \times
10^7)$. The analysis is facilitated by partitioning our modal expansion into
so-called even and odd modes. With aspect ratio $\Gamma = 2$, we observe that
zonal modes (with horizontal wavenumber equal to zero) can emerge only once the
steady convection roll state consisting of even modes only becomes unstable to
odd perturbations. We determine the stability boundary in the $(Pr,Ra)$-plane
and observe remarkably intricate features corresponding to qualitative changes
in the solution, as well as three regions where the steady convection rolls
lose and subsequently regain stability as the Rayleigh number is increased. We
study the asymptotic limit $\Pr \to 0$ and find that the steady convection
rolls become unstable almost instantaneously, eventually leading to non-linear
relaxation osculations and bursts, which we can explain with a weakly
non-linear analysis. In the complementary large-$\Pr$ limit, we observe that
the stability boundary reaches an asymptotic value $Ra = 2.54 \times 10^7$ and
that the zonal modes at the instability switch off abruptly at a large, but
finite, Prandtl number.
