Dm. Mason et Rr. Kerswell, Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown, J FLUID MEC, 396, 1999, pp. 73-108
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.