We consider a strongly nonlinear long wave model for large amplitude internal waves in a three-layer flow between two rigid boundaries. The model extends the two-layer Miyata–Choi–Camassa (MCC) model (Miyata, Proceedings of the IUTAM Symposium on Nonlinear Water Waves , eds. H. Horikawa & H. Maruo, 1988, pp. 399–406; Choi & Camassa, J. Fluid Mech. , vol. 396, 1999, pp. 1–36) and is able to describe the propagation of long internal waves of both the first and second baroclinic modes. Solitary-wave solutions of the model are shown to be governed by a Hamiltonian system with two degrees of freedom. Emphasis is given to the solitary waves of the second baroclinic mode (mode 2) and their strongly nonlinear characteristics that fail to be captured by weakly nonlinear models. In certain asymptotic limits relevant to oceanic applications and previous laboratory experiments, it is shown that large amplitude mode-2 waves with single-hump profiles can be described by the solitary-wave solutions of the MCC model, originally developed for mode-1 waves in a two-layer system. In other cases, however, e.g. when the density stratification is weak and the density transition layer is thin, the richness of the dynamical system with two degrees of freedom becomes apparent and new classesmore »
Onset of energy equipartition among surface and body waves
We derive a radiative transfer equation that accounts
for coupling from surface waves to body waves and
the other way around. The model is the acoustic
wave equation in a two-dimensional waveguide
with reflecting boundary. The waveguide has a thin,
weakly randomly heterogeneous layer near the top
surface, and a thick homogeneous layer beneath it.
There are two types of modes that propagate along the
axis of the waveguide: those that are almost trapped
in the thin layer, and thus model surface waves,
and those that penetrate deep in the waveguide,
and thus model body waves. The remaining modes
are evanescent waves. We introduce a mathematical
theory of mode coupling induced by scattering in the
thin layer, and derive a radiative transfer equation
which quantifies the mean mode power exchange.We
study the solution of this equation in the asymptotic
limit of infinite width of the waveguide. The main
result is a quantification of the rate of convergence of
the mean mode powers toward equipartition.
- Award ID(s):
- 2010046
- Publication Date:
- NSF-PAR ID:
- 10281522
- Journal Name:
- Proceedings of the Royal Society of London
- Volume:
- 477
- ISSN:
- 2053-9150
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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- https://doi.org/doi.org/10.1098/rspa.2020.0775
