Shear induced breaking of large internal solitary waves

The stability properties of 24 experimentally generated internal
solitary waves (ISWs) of extremely large amplitude, all with minimum
Richardson number less than 1/4, are investigated. The study is
supplemented by fully nonlinear calculations in a three-layer fluid. The
waves move along a linearly stratified pycnocline (depth h₂) sandwiched between a thin upper layer (depth h₁) and a deep lower layer (depth h₃),
both homogeneous. In particular, the wave-induced velocity profile
through the pycnocline is measured by particle image velocimetry (PIV)
and obtained in computation. Breaking ISWs were found to have amplitudes
(a₁) in the range a₁>2.24 √h₁h₂(1+h₂/h₁), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 < h₂/h₁ < 1 and 4.14 < h₃/(h₁ + h₂) < 7.14 and stable waves for 0.36 < h₂/h₁ < 3.67 and 3.22 < h₃/(h₁ + h₂) < 7.25. Kelvin–Helmholtz-like billows were observed in the breaking cases. They had a length of 7.9h₂
and a propagation speed 0.09 times the wave speed. These measured
values compared well with predicted values from a stability analysis,
assuming steady shear flow with U(z) and ρ(z) taken at the wave maximum (U(z) horizontal velocity profile, ρ(z) density along the vertical z).
Only unstable modes in waves of sufficient strength have the chance to
grow sufficiently fast to develop breaking: the waves that broke had an
estimated growth (of unstable modes) more than 3.3–3.7 times than in the
strongest stable case. Evaluation of the minimum Richardson number (Ri_min, in the pycnocline), the horizontal length of a pocket of possible instability, with wave-induced Ri < 14, (L_x) and the wavelength (λ), showed that all measurements fall within the range Ri_min = −0.23L_x/λ + 0.298 ± 0.016 in the (L_x/λ, Ri_min)-plane. Breaking ISWs were found for L_x/λ > 0.86 and stable waves for L_x/λ < 0.86. The results show a sort of threshold-like behaviour in terms of L_x/λ. The results demonstrate that the breaking threshold of L_x/λ
= 0.86 was sharper than one based on a minimum Richardson number and
reveal that the Richardson number was found to become almost
antisymmetric across relatively thick pycnoclines, with the minimum
occurring towards the top part of the pycnocline