Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

A direct numerical simulation is presented of an elliptical instability obs erved in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifest s itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest 'subharmonic' form of the instability where the waves are a complex conjugate pair and wh ich at weakly supercritical elliptical distortion should ultimately saturat e at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down t o small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either un stable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The fo rmer scenario confirms earlier work (Kerswell 1999) which highlights the ge neric instability of inertial waves even at very small amplitudes. The latt er represents a first numerical demonstration of two competing elliptical i nstabilities co-existing in a bounded system.