Three-dimensionality of the triadic resonance instability of a plane inertial wave

TL;DR

This paper combines theoretical, numerical, and experimental approaches to demonstrate that the triadic resonance instability of a plane inertial wave in a rotating fluid is inherently three-dimensional, with secondary waves propagating out of the primary wave's vertical plane.

Contribution

It provides the first experimental evidence of the three-dimensional nature of TRI in inertial waves, supported by theoretical and numerical analysis of wavevector orientations.

Findings

Maximum growth rate occurs for secondary waves not in the same vertical plane as the primary.

Inviscid analysis shows the horizontal propagation angle of secondary waves varies from 60° to 90°.

Experimental results confirm the three-dimensional TRI and match theoretical predictions.

Abstract

We analyze theoretically and experimentally the triadic resonance instability (TRI) of a plane inertial wave in a rotating fluid. Building on the classical triadic interaction equations between helical modes, we show by numerical integration that the maximum growth rate of the TRI is found for secondary waves that do not propagate in the same vertical plane as the primary wave (the rotation axis is parallel to the vertical). In the inviscid limit, we prove this result analytically, in which case the change in the horizontal propagation direction induced by the TRI evolves from $60^\circ$ to $90^\circ$ depending on the frequency of the primary wave. Thanks to a wave generator with a large spatial extension in the horizontal direction of invariance of the forced wave, we are able to report experimental evidence that the TRI of a plane inertial wave is three-dimensional. The wavevectors of the secondary waves produced by the TRI are shown to match the theoretical predictions based on the maximum growth rate criterion. These results reveal that the triadic resonant interactions between inertial waves are very efficient at redistributing energy in the horizontal plane, normal to the rotation axis.