Small-scale instabilities in a plane inertial wave

We analyze the linear stability of a finite-amplitude plane inertial wave by studying the inviscid evolution of three-dimensional, small-amplitude, and short-wavelength disturbances. The base flow plane inertial wave is described by its non-dimensional amplitude (Α) and the angle (Φ) that its wave vector makes with the horizontal axis. We solve the local stability equa-

tions for a wide range of perturbation wave vector orientations. When A is sufficiently small, the only instability mechanism observed is the three-dimensional parametric subharmonic instability (PSI), where the most unstable perturbation wave vector aligns at an angle close to 60^o with the inertial wave plane. Moreover, this most unstable perturbation is shear-aligned with the inertial wave within the inertial wave plane. Additionally, in the near-inertial regime, characterized by large Φ, there exists a broad range of perturbation wave vectors whose growth rates are comparable to the maximum growth rate. As we increase Α, the theoretical PSI estimates become less relevant in describing the instability characteristics. Instead, the dominant instability transitions to the two-dimensional shear-aligned instability, which is shown to be driven by third-order resonance. This transition from three-dimensional PSI to two-dimensional shear-aligned instability can be reasonably captured using a Rossby number-based criterion.