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May 07, 2013 at 3:30 pm.
Lecture 1.
We will show that the Euler equations for potential flow of
incompressible fluid with free surface is a Hamiltonian system.
The canonical variables are the surface elevation and the
potential evaluated on the surface. Instead of these ”natural
variables” we can introduce normal canonical variables,
which are classical analogs of annihilation and creation operators
in the quantum field theory. This language is very convenient.
Then one can perform the canonical transformation excluding
the cubic terms in the Hamiltonian. The resulting effective
Hamiltonian describes scattering of the ”riplons”
and conserves the total wave action. The coupling coefficient
for nontrivial scattering of riplons in 2D geometry happens
to be identically zero. This fact makes possible the further
simplification of the Hamiltonian. The resulting ”compact
Hamiltonian” describes interaction of riplons moving
in one direction. From this Hamiltonian, the Nonlinear Schrodinger
equation and the Dysthe equation can be obtained in the case
of narrow frequency band. The ”compact” four-wave
equation is very suitable for numerical simulation. It admits
solitonic solution that is very similar to the rogue waves
observed in the ocean.
May 09, 2013 at 3:30 pm.
Lecture 2 on Hasselmann kinetic equation and its solutions
Using the four-riplon Hamiltonian one can easily derive the
Hasselmann kinetic equation for the wave action spectrum.
This equation is almost identical to the standard kinetic
equations for phonons in the condensed matter physics. The
Hasselmann kinetic equation has remarkable powerlike solutions
of Kolmogorov type that describe the direct cascade of energy
and the inverse cascade of wave action. These solutions are
observed in laboratory and field experiments as well as in
numerical simulation of primordial deterministic equations.
The Hasselmann kinetic equation admits very broad class of
self-similar solutions that describe the dependence of major
characteristics of wind-driven sea: the average energy and
the mean frequency on duration and fetch. The use of selfsimilar
solutions makes possible to explain the results of numerous
field and laboratory experiments. They show the domination
of four-riplon scattering over other processes: interaction
with the wind and dissipation due to wave breaking. This fact
is supported by the direct numerical solution of the kinetic
equation.
May 10, 2013 at 3:30 pm.
Lecture 3. About the nature of Phillips spectrum
Numerous experiments show that the KZ spectrum $I(\omega)\simeq
\omega^{-4}$ is realized in a limited (one decade) range of
wave numbers right behind the spectral peak. More short waves
obey to more steep Phillips law $I(\omega)\simeq g^2 \,\omega^{-5}$.
This spectrum can be obtained from dimensional consideration
(O. Phillips 1957). From the beginning, it was clear that
existence of this spectrum is connected with wave breaking
event but its physical nature was understood just recently.
Each breaking event is defined by a characteristic length,
while the Phillips spectrum includes one dimension quantity
only, the gravity acceleration $g$. This contradiction is
resolved as follow. Existence of Phillips spectrum presumes
coexistence of plethora of wave breaking events with different
characterizing scales. Moreover, there is equipartition of
inverted wave breaking scales. Simultaneous presence of the
weak turbulent KZ spectrum $\omega^{-4}$ and the Phillips
spectrum $\omega^{-5}$ means that in the real sea coexist
both the weak and the strong wave turbulence. Long waves weakly
interact due to four-wave resonant processes. This interaction
forms the direct cascade of energy that feeds the wave breaking
mechanism. Also, the four-wave interaction creates the inverse
cascade, the frequency downshift. Weakly interacting long
waves are covered by the net of microscopic wave-breakings.
Including into kinetic equation an additional term of very
sophisticated form, we can take into account the wave-breaking
event.
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