Short-wavelength instabilities in a finite-amplitude plane inertial wave

We perform a linear stability analysis of a finite-amplitude plane inertial wave (of frequency $\omega$ in the range $0\le \omega \le f$, where $f$ is the Coriolis frequency) by considering the inviscid evolution of three-dimensional (3-D), small-amplitude, short-wavelength perturbations. Characterizing the base flow plane inertial wave by its non-dimensional amplitude $A$ and the angle $\varPhi$ that its wavevector makes with the horizontal axis, the local stability equations are solved over the entire range of perturbation wavevector orientations. At sufficiently small $A$, 3-D parametric subharmonic instability (PSI) is the only instability mechanism, with the most unstable perturbation wavevector making an angle close to $60^{\circ }$ with the inertial wave plane. In addition, the most unstable perturbation is shear-aligned with the inertial wave in the inertial wave plane. Further, at large $\varPhi$, i.e. $\omega \approx f,$ there exists a wide range of perturbation wavevectors whose growth rate is comparable to the maximum growth rate. As $A$ is increased, theoretical PSI estimates become less relevant in describing the instability characteristics, and the dominant instability transitions to a two-dimensional (2-D) shear-aligned instability, which is shown to be driven by third-order resonance. The transition from 3-D PSI to a 2-D shear-aligned instability is shown to be reasonably captured by two different criteria, one based on the nonlinear time scale in the inertial wave and the other being a Rossby-number-based one.