High-resolution wave climate analysis in the Helgoland area - GKSS

2.1 Methods, data and tools 7
For the wave simulation in this study the K-model is used, which is a third generation
spectral wave model adapted to small-scale shallow water applications. Third generation
models are characterized by the transition to full two-dimensional discretization of the
wave spectrum instead of artificial separation of the wind and swell parts of spectrum. The
development of third generation models started in mid 1980s by WAMDI Group [1988]. This
approach has been successfully used for the major part of spectral wave models constructed
for both open ocean applications (e.g. WAM-cycle4 (Günther et al. [1992])) and smallscale
and shallow water applications (e.g. high-resolution version of WAM (Monbaliu et al.
[2000]) or SWAN (Ris et al. [1999])). The K-model was developed on the basis of the
WAM-cycle4 model with some changes aiming mainly at the adaptation of the model to
small-scale coastal zone applications. With the increasing spatial and time resolutions and
consideration of areas with shallow water and complex topography, the model faces the
problems of inhomogeneity and non-stationarity of external fields. The main part of the
wave theory was developed for idealized conditions which can be accepted for the most of
medium-scale deep-water applications but need a strong revision and adaptation to the nearshore
small-scale environment. The changes between WAM and K-model include transitions
from the frequency-direction (f, θ) to the wavenumber-direction (k, θ) domain. This allows
avoiding additional multipliers with partial time derivatives in the balance equation, which
is crucial in non-stationary cases (Schneggenburger [1998]). Another difference is related
to the form of the source function. Non-linear wave-wave interactions have been neglected
following the reasoning of Schneggenburger [1998], who argued that in shallow water the
assumptions of homogeneity for the application of this theory are violated. Instead, a nonlinear
dissipation source function (Günther and Rosenthal [1997] or Schneggenburger et al.
[1997]), accounting for the dissipation by wave turbulence, is used. In the K-model energy
input by wind is parameterized by a modified Philips linear function (Cavaleri and Rizzoli
[1981]) and a modified Snyder exponential function (WAMDI Group [1988]) similar to the
WAM model. Bottom dissipation is taken into account according to Hasselmann et al.
[1973].
The modeled and measured wave fields were originally given in the form of a wave
spectrum but for practical reasons different representations of spectrum or integrated parameters
are used. This provides the opportunity to relate model results to observed data in
a sense that integrated parameters can be easily statistically interpreted as characteristics
of the sea state. The most usable integrated parameters are the moments mn:
�
mn = F (f, θ)f n dfdθ, n = −1, 0, 1, 2
and quantities related to them. The zeroth order moment m0 is the total variance of
the spectrum which proportional to the total energy and defines another frequently used
integrated parameter the significant wave height (SWH):
Hs = 4 √ m0
which corresponds to the mean of the highest third of the observed waves (Longuett-Higgins
et al. [1963]). Different integrated periods can also be useful. The peak period Tp is defined