Direct Numerical Simulations of the Degeneration and Shear Instability of Large and Small Amplitude Basin Scale Internal Waves at Varied Aspect Ratios
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Date
2018-08-14
Authors
Grace, Andrew
Journal Title
Journal ISSN
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Publisher
University of Waterloo
Abstract
This thesis presents high resolution simulations of the degeneration and shear instability
of standing waves, or seiches, of varying amplitudes and aspect ratios in a continuously
stratified fluid. It is well known that such waves evolve to form non–linear, dispersive
wave trains under certain conditions. When the initial amplitude scaled by the upper
layer depth (the dimensionless amplitude) is sufficiently large, it is possible that stratified
shear instability develops, possibly at the same time as the formation of wave trains early
in the evolution of the flow. While both of these physical phenomena serve to move
energy from large to small scales, they are fundamentally different. The development
into wave trains is non-dissipative in nature, and in the asymptotic limit of small, but
finite amplitude seiches may be described by variants of the Korteweg–de–Vries (KdV)
equation. Shear instability, on the other hand yields Kelvin-Helmholtz billows which in
turn provide one of the basic archetypes of transition to turbulence, with greatly increased
rates of mixing and viscous dissipation. Discussed is how the two phenomena vary as the
aspect ratio of the tank and the height of the interface between lighter and denser fluid
are changed, finding examples of cases where the two phenomena co-exist. Beginning with
an expository set of examples of small amplitude seiches, the process by which a seiche
changes from a traditional standing wave to a more complicated small scale set of dynamics
is discussed. The results demonstrate that when the initial dimensionless amplitude is
small, the seiche takes more than one oscillation period for non–linear effects to become
obviously present in the flow. The small amplitude results put into context the cases where
the dimensionless amplitude becomes large enough such that non–linear process occur at
much earlier times and there is a competition between the formation of wave trains and
stratified shear instability. A quantitative accounting for the evolution of the horizontal
modewise decomposition of the kinetic energy of the system is presented along with a
semi-analytical model of the evolution of the fundamental mode of the seiche. Using two
well known methodologies from the literature, the evolution of the mixing dynamics of
the seiche is compared from an energetic perspective and a density variability perspective
which illustrates a fundamental transition that occurs as the aspect ratio is decreased.
Finally, the seiche degeneration and the mixing dynamics are summarized and the most
likely future directions of study are highlighted.
Description
Keywords
Geophysical fluid dynamics, Internal waves, Seiche