Real
ocean density is not constant and varies with the depth (due to change in
water temperature and salinity). In most oceans, density jumps from a
lighter upper layer fluid density to a heavier lower layer fluid one over a
surface called Thermocline. Waves - similar to surface waves - can
propagate over Thermocline and are called internal (interfacial) waves.
Internal waves, when they propagate over topography, may exchange energy
(via nonlinear resonance) with surface waves. If bottom is periodically
modulated, the phenomenon is called Bragg Resonance and very well
understood for a homogeneous fluid. Bragg resonance of internal waves is
important in explaining the generation of internal gravity waves
particularly in littoral zones, dead water phenomenon (high resistance on
ships in stratified waters), and has many applications in navigation and
design of off-shore structures.
Depth of isotherms as a function of time
(From
NOAA, USDC).
Density change in the ocean is mostly due to temperature and sality. A
similar distribution exists for salinity. Therefore a two-layer density
stratified fluid is a good approximation for ocean stratification.
We have studied Bragg resonance of interfacial waves theoretically and
numerically. Starting from perturbation analysis, and proceeding to higher
order nonlinearities, closed-form solutions for the resonance condition and
the growth rate of resonant waves are obtained. Part II of this work is
devoted to the development of a very efficient spectral-based numerical
scheme. Theoretical results are cross-validated with our direct simulation
and effect of higher nonlinearity and complicated cases - which are beyond
the capacity of analytical tools - are studied. It is shown for example,
that by multiple resonances between a chosen set of waves, water surface
may undergo chaotic oscillations. We also show that Bragg resonance in
two-layer fluids can significantly contribute to the development of the
ocean spectrum, and, is a potential mechanism for the mysterious phenomenon
of generation of high-frequency interfacial waves in lakes and on
continental shelves.
Schematic of the problem that we studied:
Nonlinear (second and third order in perturbation series) of interaction of
surface waves, internal waves and bottom topography in a two-layer density
stratified fluid. Specifically we focused on triad and quartet resonance and
near resonance interactions.
Artist rendering
of Bragg resonance in a 2-layer density stratified fluid
References:
- Alam, M.R., Liu, Y. and Yue,
D.K.P., "Bragg resonance of waves in a two-layer fluid propagating
over bottom ripples. PartI. Perturbation analysis.",J. Fluid Mechanics,
Volume 624 (2009), pages 191-224.
(PDF)
- Alam, M.R., Liu, Y. and Yue,
D.K.P., "Bragg resonance of waves in a two-layer fluid propagating
over bottom ripples. Part II. Numerical simulation.",J. Fluid Mechanics,
Volume 624 (2009), pages 225-253.
(PDF)
- Alam, M.R., Liu, Y. and Yue, D.K.P., ``Chaotic Internal Wave Motion due to
Multiple Bragg Resonances'', American Geophysical Union Fall Meeting. San Francisco,
CA,
15-19 December 2008.
