The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic
internal wavepackets are examined analytically and by numerical simulations. The
weakly nonlinear dispersion relation for horizontally periodic, vertically compact
internal waves is derived and the results are applied to assess the stability of weakly
nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of
constant phase make with the vertical, the wavepackets are predicted to be unstable
if [mid ]Θ[mid ] < Θc, where
Θc = cos−1 (2/3)1/2 ≃ 35.3°
is the angle corresponding to internal
waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of
finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets
with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with
[mid ]Θ[mid ] < Θc
increases initially, but then decreases as the wavepacket subdivides into a wave train,
following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable
in the sense that eventually it convectively overturns. Two new analytic conditions
for the stability of quasi-plane large-amplitude internal waves are proposed. These
are qualitatively and quantitatively different from the parametric instability of plane
periodic internal waves. The ‘breaking condition’ requires not only that the wave
is statically unstable but that the convective instability growth rate is greater than
the frequency of the waves. The critical amplitude for breaking to occur is found
to be ACV = cot Θ (1 + cos2 Θ)/2π,
where ACV is the ratio of the maximum vertical
displacement of the wave to its horizontal wavelength. A second instability condition
proposes that a statically stable wavepacket may evolve so that it becomes convectively
unstable due to resonant interactions between the waves and the wave-induced
mean flow. This hypothesis is based on the assumption that the resonant long
wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves
linearly in time, continues to amplify the waves in the fully nonlinear regime. Using
linear theory estimates, the critical amplitude for instability is
ASA = sin 2Θ/(8π2)1/2.
The results of numerical simulations of horizontally periodic, vertically compact
wavepackets show excellent agreement with this latter stability condition. However,
for wavepackets with horizontal extent comparable with the horizontal wavelength,
the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It
is proposed that these results may explain why internal waves generated by turbulence
in laboratory experiments are often observed to be excited within a narrow frequency
band corresponding to Θ less than approximately 45°.
Energy spectrum of the ensemble of weakly nonlinear gravity-capillary waves on a fluid surface
