chap_DYN.tex

\section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})}
\label{sec:DYN_nxt}

Options are defined through the \nam{dom}{dom} namelist variables.
The general framework for dynamics time stepping is a leap-frog scheme,
\ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}).
The scheme is applied to the velocity, except when
using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux})
in the variable volume case (\np[=.false.]{ln_linssh}{ln\_linssh}),
where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum})

$\bullet$ vector invariant form or linear free surface
(\np[=.true.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} or \np[=.true.]{ln_linssh}{ln\_linssh}):
\[
  % \label{eq:DYN_nxt_vec}
  \left\{
    \begin{aligned}
      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t  	\\
      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
    \end{aligned}
  \right.
\]

$\bullet$ flux form and nonlinear free surface
(\np[=.false.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} and \np[=.false.]{ln_linssh}{ln\_linssh}):
\[
  % \label{eq:DYN_nxt_flux}
  \left\{
    \begin{aligned}
      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t  	\\
      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right]
    \end{aligned}
  \right.
\]
where RHS is the right hand side of the momentum equation,
the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
$\gamma$ is initialized as \np{nn_atfp}{nn\_atfp} (namelist parameter).
Its default value is \np[=10.e-3]{nn_atfp}{nn\_atfp}.
In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for
the momentum equations.

\subinc{\input{../../global/epilogue}}

\end{document}