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Hence for any $\bk$ one quadruplet selects $\bk_{2,+}$ and $\bk_{3,+}$, and the other quadruplet selects its mirror image
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$\bk_{2,-}$, $\bk_{2,-}$. Because there are 3 different components interacting in the two DIA-selected quadruplets, any discrete spectral component $(f_r,\theta)$ is actually involved in 6 quadruplets and directly exchanges energy with 12 other components $(f_r',\theta')$. Because the values of $f'_r$ and $\theta'$ do not fall exacly on other discrete components, the spectral density is interpolated using a bilinear interpolation, so that each source term value
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$S_{nl}(\bk)$ contains the direct exchange of energy with 48 other discrete components.
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we compute the three contributions that correspond to the situation in which $\bk$ takes the role of $\bk$,$\bk_{2,+}$, $\bk_{2,-}$, $\bk_{3,+}$ and $\bk_{3,-}$ in the quadruplet, namely the full source term is, without making explicit that bilinear interpolation,
where the geometry of the quadruplet $(\bk_4,\bk_4,\bk,\bk_5)$ is obtained from that of $(\bk,\bk,\bk_{2,+},\bk_{3,+})$ by a dilation by a factor $(1+\lambda)^2$ and rotation by the angle $\delta_{\theta,2}$; $(\bk_6,\bk_6,\bk,\bk_7)$ has the same dilation but the opposite rotation; $(\bk_8,\bk_8,\bk_9,\bk)$ is dilated by a factor $(1-\lambda)^2$ and rotated by the angle $-\delta_{\theta,3}$: and $(\bk_{10},\bk_{10},\bk_{11},\bk)$ is dilated by the same factor and rotated by the opposite angle.
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The elementary contributions $\delta S_{\mathrm{nl}}(\bk_l,\bk_m,\bk_n)$ are given by
where the spectral densities are $F_l = F(f_{r,l} ,\theta_l)$, etc.
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$C$ is a proportionality constant that was tuned to reproduce the inverse energy cascade. Default values for different source term packages are presented in Table~\ref{tab:snl_par}.
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% tab:snl_par
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\begin{table} \begin{center}
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\begin{tabular}{|l|c|c|} \hline
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& $\lambda_{nl}$ & $C$ \\ \hline
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& $\lambda$ & $C$ \\ \hline
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ST6 & 0.25 & $3.00\;10^7$ \\ \hline
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\wam-3 & 0.25 & $2.78\;10^7$ \\ \hline
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ST4 (Ardhuin et al.)& 0.25 & $2.50\;10^7$ \\ \hline
\opthead{NL1 , but with a negative IQTYPE}{TOMAWAC model, M. Benoit}{adaptation to WW3 by S. Siadatmousavi \& F. Ardhuin}
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\noindent
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Changing the namelist parameter IQTYPE to a negative value replaces the
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DIA parameterization with the possibility to perform an exact but fast cal-
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culation of $S_{\mathrm{nl}}$ using the Gaussian Quadrature Method of \cite{Lavrenov2001}.
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More details can be found in \cite{Gagnaire-Renou2009}.
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The quadruplet configurations that are used correspond to the three integrals over $f_1$, $f_2$ and $\theta_1$, with all other frequencies and directions given by the resonance conditions (\ref{eq:resonance}) with only one ambiguity on the angle $\theta_2$ which can be defined by a sign index $s$, as in the DIA. Starting from eq. (A4) in \cite{Lavrenov2001} as writen in (2.25) of \cite{Gagnaire-Renou2009}, the source term is
where $ D(\bk,\bk_1,\bk_2,\bk_3)$ is given by \cite{Webb1978} in his eq. (A1).
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This triple integral is performed using quadrature functions to best resolve the effect of the singularities in the denominator. It is thus replaced with weighted sums over the 3 dimensions.
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Compared to the DIA, there is no bilinear interpolation and the nearest neighbor is used in frequency and direction. Also,
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the source term is computed by a loop over the quadruplet configuration, which allows for filtering based on
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both the value of the coupling coefficient and the energy level at the frequency corresponding to $\bk$. Within
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that loop, the source term contribution is computed for all 4 interacting components, so that any filtering still
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conserves energy, action, momentum ... (One may argue that this multiplies by 4 the number of calculations, but it may have the benefit of properly dealing with the high frequency boundary... this is to be verified. The same question arises for the DIA: why have the wavenumber $\bk$ play the role of the other members of the quadruplets when this will also be computed as we loop on the spectral components?).
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If a very aggressive filtering is performed, the source may need to be rescaled.
in \para\ref{sec:ww3shel}. That input file also provides a list of flags
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indicating if output parameters are available in different field
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output file types (ASCII, grib, igrads, NetCDF).
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For any details on how these parameters are computed, the user may read the code of the {\code w3iogo} routine, in the {\code w3iogomd.ftn} module.
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For any details on how these parameters are computed, the user may read the code of the {\code w3iogo} routine, in the {\code w3iogomd.F90} module.
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Selection of field outputs in {\code ww3\_shel.inp} is most easily performed by providing a list of the
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Selection of field outputs in {\code ww3\_shel.nml} or {\code ww3\_shel.inp} is most easily performed by providing a list of the
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requested parameters, for example, {\textbf HS DIR SPR} will request the calculation of significant wave height, mean direction and directional spread. These will thus be stored in the {\code out\_grd.XX} file and can be post-processed, for example in NetCDF using {\code ww3\_ouf}. Examples are given in \para\ref{sec:ww3multi} and
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\para\ref{sec:ww3ounf}. The names for these namelists are the bold names below, for
file extensions, NetCDF variable names and namelist-based selection (see
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also \para\ref{sec:ww3ounf}), and the long parameter name/definition.
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When the result is not overly sensitive to the contribution of the unresolved part of the spectrum (for $f<f_{NK}$), the contribution of the tail is parameterized assuming a power law decay of the spectrum, by default $E(f,\theta) = E(f_{NK},\theta) (f_{NK}/f)^{-5}$, for some parameters this is either unnecessary or misleading, and the integrals are computed only up to $f_{NK}$.
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Finally we note that in all definitions the frequency is the \emph{relative} frequency. Thus, in the presence of currents, these can only be compared to drifting measurement data or data obtained in wavenumber and converted to frequency. Comparison to fixed instrument data requires the use of the full spectrum and proper conversion to the fixed reference frame.
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