Deterministic and Stochastic Modeling of Surface Gravity
Waves in Finite Water Depth
In this thesis several aspects regarding modelling of surface
gravity waves by deterministic and stochastic evolution equations
are considered.
The deterministic Zakharov equation and the stochastic Boltzmann
integral of Hasselmann are re-derived and extended to govern the
case of waves propagating at a finite water depth on a current, and
also to govern evolution of inhomogeneous seas in deep water where
the wave-induced current and the wave-induced change in the mean
water level vanish. It is shown that the Zakharov equation includes
amplitude dispersion, that bound waves have been taken into account
in the approach and it is partly shown that these nonlinear effects
are consistent with a Stokes wave expansion. The Boltzmann integral
does not, however, include amplitude dispersion, and a possibility
to include amplitude dispersion is discussed. Furthermore it is
shown that the approach breaks down when shallow waters are
entered.
Based on the wave-action balance equation a one-dimensional model
is implemented, and it is shown that the Discrete Interaction
Approximation (DIA) most often used in third generation wind wave
models simulates the evolution of the peak-frequency and the
significant wave height in agreement with the Boltzmann integral of
Hasselmann, whereas DIA underestimates the mean frequency, cannot
model the evolution towards a f^(-4) form of the spectral tail,
nor the transfer towards frequencies in the infragravity wave
regime.
Different approaches which account for wave-wave interactions in
shallow waters are discussed. In this thesis a one-dimensional
stochastic two-equation model based on Boussinesq type equations
with improved linear dispersion characteristics for
uni-directional waves is proposed. This model is able to simulate
transformation of wave spectra in shallow water including
near-resonant triad wave interaction, generation of bound sub- and
superharmonics and wave breaking. This model is evaluated in a
number of cases with acceptable agreement with measurements.
Furthermore a stochastic one-equation model is presented and
evaluated in a few cases with acceptable agreement with
measurements, especially on the upslope part of the experimental
bathymetry. The latter result suggests that the Lumped Triad
Approximation (LTA) approach is acceptable on bottom upslopes. It
is, however, shown that neither the LTA approach nor the
one-equation model can simulate wave evolution on horizontal or
nearly horizontal bottoms very well.
