Skip to content
Snippets Groups Projects
Commit 76725a18 authored by Andrew Coward's avatar Andrew Coward
Browse files

Revert "Update file chap_DYN.tex"

This reverts commit 2d52af2e.
which temporarily deleted a section for debugging purposes
parent 2d52af2e
No related branches found
No related tags found
1 merge request!31Resolve "Review/update DYN chapter"
......@@ -392,6 +392,163 @@ the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
\right.
\]
%% =================================================================================================
\section{Coriolis and advection: flux form}
\label{sec:DYN_adv_cor_flux}
Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables.
In the flux form (as in the vector invariant form),
the Coriolis and momentum advection terms are evaluated using either a leapfrog scheme or a RK3 scheme.
In the leapfrog case the velocity appearing in these expressions is centred in time (\textit{now} velocity).
In the RK3 case the velocity appearing in these expressions is forward in time (\textit{before} velocity) at stage 1,
it is is centred in time (\textit{now} velocity) at stage 2 and 3.
At the lateral boundaries either free slip,
no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
%% =================================================================================================
\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
\label{subsec:DYN_cor_flux}
In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
This altered Coriolis parameter is thus discretised at $f$-points.
It is given by:
\begin{aligned*}
% \label{eq:DYN_cor_metric}
f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right)
\equiv f + \frac{1}{e_{1f} e_{2f} } \left( \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]
- \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right] \right)
\end{aligned*}
% energy conserving scheme at T-point
%% =================================================================================================
\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_enT})]{Energy conserving scheme (\protect\np{ln_dynvor_enT}{ln\_dynvor\_enT})}
\label{subsec:DYN_vor_enT}
The kinetic energy conserving scheme at T-point (ENT scheme) conserves the global kinetic energy but not the global enstrophy.
It is given by:
\begin{equation}
\label{eq:DYN_vor_enT}
\left\{
\begin{aligned}
&+\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
e_{1t}\,e_{2t}\,e_{3t} \, \overline{v}^{\,j}}^{\,i+1/2} \\
&-\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
e_{1t}\,e_{2t}\,e_{3t} \, \overline{u}^{\,i}}^{\,j+1/2}
\end{aligned}
\right.
\end{equation}
\noindent Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}), (\autoref{eq:DYN_vor_enT}) and (\autoref{eq:DYN_vor_een})
schemes can be used to
compute the product of the Coriolis parameter and the vorticity.
However, the energy-conserving schemes (\autoref{eq:DYN_vor_een} and \autoref{eq:DYN_vor_enT})
have exclusively been used to date.
\vskip 0.5cm
\noindent This term is evaluated using either a leapfrog scheme or a RK3 scheme.
In the leapfrog case it is centred in time (\textit{now} velocity).
In the RK3 case it is forward in time (\textit{before} velocity) at stage 1,
it is is centred in time (\textit{now} velocity) at stage 2 and 3.
%% =================================================================================================
\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
\label{subsec:DYN_adv_flux}
The discrete expression of the advection term is given by:
\[
% \label{eq:DYN_adv}
\left\{
\begin{aligned}
\frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
\left( \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i} \ u_t \right]
+ & \delta_{j} \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2} \ u_f \right] \right. \\
\left. + & \delta_{k} \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2} \ u_{uw} \right] \right) \\[10pt]
\frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
\left( \delta_{i} \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f \right]
+ & \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i} \ v_t \right] \right. \\
\left. + & \delta_{k} \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw} \right] \right) \\
\end{aligned}
\right.
\]
Two advection schemes are available:
a $2^{nd}$ order centered finite difference scheme, CEN2,
or a $3^{rd}$ order upstream biased scheme, UP3.
The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_up3}{ln\_dynadv\_up3}.
In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
$u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
% 2nd order centred scheme
%% =================================================================================================
\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln_dynadv_cen2}{ln\_dynadv\_cen2})}
\label{subsec:DYN_adv_cen2}
In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
\begin{equation}
\label{eq:DYN_adv_cen2}
\left\{
\begin{aligned}
u_T^{cen2} &=\overline u^{i } \quad & u_F^{cen2} &=\overline u^{j+1/2} \quad & u_{uw}^{cen2} &=\overline u^{k+1/2} \\
v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j \quad & v_{vw}^{cen2} &=\overline v ^{k+1/2} \\
\end{aligned}
\right.
\end{equation}
The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
produce a sensible solution.
% UP3 scheme
%% =================================================================================================
\subsubsection[UP3: Upstream Biased Scheme (\forcode{ln_dynadv_up3})]{UP3: Upstream Biased Scheme (\protect\np{ln_dynadv_up3}{ln\_dynadv\_up3})}
\label{subsec:DYN_adv_up3}
The UP3 advection scheme is an upstream biased third order scheme based on
an upstream-biased parabolic interpolation.
For example, the evaluation of $u_T^{up3} $ is done as follows:
\begin{equation}
\label{eq:DYN_adv_up3}
u_T^{up3} =\overline u ^i-\;\frac{1}{6}
\begin{cases}
u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\
u"_{i+1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i < 0$ }
\end{cases}
\end{equation}
where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
\citep{shchepetkin.mcwilliams_OM05}.
The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
It is a relatively good compromise between accuracy and smoothness.
It is not a \emph{positive} scheme, meaning that false extrema are permitted.
But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
(\ie\ \np[=.true.]{ln_dynldf_OFF}{ln\_dynldf\_OFF}),
and it is recommended to do so.
The UP3 scheme is used in all directions.
UP3 is diffusive and is associated with vertical mixing of momentum. \cmtgm{ gm pursue the
sentence:Since vertical mixing of momentum is a source term of the TKE equation... }
In a leapfrog environment, for stability reasons, the first term in (\autoref{eq:DYN_adv_up3}),
which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
while the second term, which is the diffusion part of the scheme,
is evaluated using the \textit{before} velocity (forward in time).
In an RK3 environment, the first term in (\autoref{eq:DYN_adv_up3}),
which corresponds to a second order centred scheme, is evaluated using the \textit{before} velocity at stage 1
and using the \textit{before} velocity (centred in time) at stage 2 and 3,
while the second term, which is the diffusion part of the scheme,
is evaluated using the \textit{before} velocity (forward in time).
This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme.
Note that the UP3 and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
one coefficient.
Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_up3}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_up3} module and obtain a QUICK scheme.
%% =================================================================================================
\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}
......
0% Loading or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment