Instability mechanisms of a two-dimensional progressive internal gravity wave

We present a detailed investigation of the parametric subharmonic resonance mechanism that leads a plane, monochromatic, small-amplitude internal gravity wave, also referred to as the primary wave, to instability. Resonant wave interaction theory is used to derive a simple kinematic model for the parametrically forced perturbation, and direct numerical simulations of the Boussinesq equations in a vertical plane permit the nonlinear simulation of the internal gravity wave field. The processes that eventually drive the wave field to breaking are also addressed. We show that parametric instability may be viewed as an optimized scenario for drawing energy from the primary wave, that is, from a periodic flow with both oscillating shear and density gradient. Optimal energy exchange maximizing perturbation growth is realized when the perturbation has a definite spatio-temporal structure: its energy is phase-locked with the vorticity of the primary wave. This organization allows the perturbation energy to alternate between kinetic form when locally the primary wave shear is negative, then maximizing kinetic energy extraction from the primary wave, and potential form when the primary wave shear is positive, then minimizing the reverse transfer to that wave. The perturbation potential energy increases through the primary wave density gradient whether the latter is positive, that is when the medium is of reduced static stability, or negative (increased static stability). When the primary wave amplitude is small, all energy transfer terms are predicted well by the kinematic model. One important result is that the rate of potential energy transfer from the primary wave to the perturbation is always larger than the rate of kinetic energy transfer, whatever the primary wave. As the perturbation amplifies, overturned isopycnals first appear in reduced static stability regions, implying that the total field should become unstable through a buoyancy induced (or Rayleigh-Taylor) instability. Hence, a two-dimensional model is no longer valid for studying the subsequent flow development.