| Summary: | When two flows of different velocity and density meet, a shear layer with a density
gradient is formed. Under certain conditions this flow can be unstable. A statically
stable stratified shear flow in which the density interface is much thinner than the shear
layer thickness can be linearly unstable to two modes with equal growth rates and equal
and opposite phase speeds. The superposition of these two modes is called a Holmboe
instability. This instability is only possible when the flow is symmetric about the center of
the shear layer. We examine the effect of breaking this symmetry by allowing the center
of the density interface to be displaced with respect to the center of the shear layer. There
are three major components to this study: linear stability analysis, nonlinear numerical
simulations, and comparison with laboratory experiments.
Linear stability analysis is used to examine the effect of the density interface offset on
the overall stability of the flow. Both inviscid theory with piecewise linear background
velocity and density profiles, and viscous theory with smooth background profiles are
used. As in previous studies, it was found that the growth rate of one mode increases and
that of the other mode decreases as the density interface displacement is increased. The
precise behaviour depends on the relative thickness of the density interface with respect to
the shear layer thickness. For inviscid theory with piecewise linear background profiles,
we show that the initial perturbations must be two-dimensional. When the effects of
viscosity and diffusion are included, however, it may be possible that the weaker mode is
initially three-dimensional. Detailed analysis of the energy transfer in the linear regime
indicates that when the background flow loses its symmetry, it is the mode with the larger
growth rate that is primarily responsible for the extraction of energy from the mean flow. Two-dimensional numerical simulations are used to examine the nonlinear development of “non-symmetric” Holmboe instabilities. We start by perturbing the flow with
the stronger mode predicted by linear theory. We then examine the response of the flow
to weaker mode. Finally, we impose both modes. By comparing the development of these
three flows we are able to study the interaction between the two modes and the effect of
initial conditions on the development of instabilities. Although the initial development
of instabilities depends on the initial conditions, this dependence weakens as the density
interface offset is increased. Also, preliminary results indicate that long term behaviour
of the perturbations are independent of initial conditions.
Results of the numerical simulations are compared to both symmetric and nonsymmetric
Holmboe instabilities that have been observed in laboratory experiments.
Since we are unable to compute the flow at the high Prandtl number of the salt stratified
experimental flows, we run the simulation for a range of increasing Prandtl numbers to
determine how instabilities of thermally stratified flows (with low Prandtl number) and
salt stratified flows differ. Results of these simulations indicate that differences between
the experimental and numerical flows can be attributed to the thicker density interface
and lower Prandtl number used in the numerical simulations. When the density interface
is thicker and the Prandtl number lower, the waves or billows formed by the instabilities
are not as sharply defined as in the experiments.
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