Abstract
A numerical model based on the incompressible two-dimensional Navier–Stokes
equations in the Boussinesq approximation is used to study mode-2 internal
solitary
waves propagating on a pycnocline between two deep layers of different
densities.
Numerical experiments on the collapse of an initially mixed region reveal
a train
of solitary waves with the largest leading wave enclosing an intrusional
‘bulge’.
The waves gradually decay as they propagate along the horizontal direction,
with
a corresponding reduction in the size of the bulge. When the normalized
wave
amplitude, a, falls below the critical value
ac=1.18, the wave is no longer able to
transport mixed fluid as it propagates away from the mixed region, and
a
sharp-nosed intrusion is left behind. The wave structure is studied using
a Lagrangian
particle tracking scheme which shows that for small amplitudes the bulges
have a
well-defined elliptic shape. At larger amplitudes, the bulge entrains and
mixes fluid
from the outside while instabilities develop in the rear part of the bulge.
Results
are obtained for different wave amplitudes ranging from small-amplitude
‘regular’
waves with a=0.7 to highly nonlinear unstable waves with
a=3.8. The dependence
of the wave speed and wavelength on amplitude is measured and compared
with
available experimental data and theoretical predictions. Consistent with
experiments,
the wave speed increases almost linearly with amplitude at small values
of a. As
a becomes large, the wave speed increases with amplitude at a
smaller rate, which
gradually approaches the asymptotic limit for a two-fluid model. Results
show that
in the parameter range considered the wave amplitude decreases linearly
with time
at a rate inversely proportional to the Reynolds number. Numerical experiments
are
also conducted on the head-on collision of solitary waves. The simulations
indicate
that the waves experience a negative phase shift during the collision,
in accordance
with experimental observations. Computations are used to determine the
dependence
of the phase shift on the wave amplitude.