Changeset 1223 for trunk/DOC/TexFiles/Chapters/Annex_E.tex

Timestamp:

2008-11-26T13:12:16+01:00 (15 years ago)

Author:

gm

Message:

minor corrections in the Appendix from Steven see ticket #282

File:

• trunk/DOC/TexFiles/Chapters/Annex_E.tex (modified) (7 diffs)

trunk/DOC/TexFiles/Chapters/Annex_E.tex

@@ -29 +29 @@
 - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
 \end{equation}
-where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with a constant i-grid spacing ($\Delta i=1$). 
-
-Alternative choice: introduce the scale factors:  $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$.
+where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and 
+$\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. 
+By choosing this expression for $\tau "$ we consider a fourth order approximation 
+of $\partial_i^2$ with a constant i-grid spacing ($\Delta i=1$). 
+
+Alternative choice: introduce the scale factors:  
+$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$.
 
 
@@ -48 +52 @@
 scheme when \np{ln\_traadv\_ubs}=T.
 
-For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term 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 diffusive part of the scheme, is 
-evaluated using the \textit{before} velocity (forward in time. This is discussed by \citet{Webb1998} in the context of the Quick advection scheme. UBS and QUICK 
+For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term 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 diffusive part of the scheme, 
+is evaluated using the \textit{before} velocity (forward in time. This is discussed 
+by \citet{Webb1998} in the context of the Quick advection scheme. UBS and QUICK 
 schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 
-(\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. This 
-option is not available through a namelist parameter, since the 1/6 
+(\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. 
+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 \mdl{traadv\_ubs} module and obtain a QUICK scheme
@@ -67 +72 @@
 NB 2 : In a forthcoming release four options will be proposed for the 
 vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be 
-evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , or  \textit{(b)} a TVD 
-scheme, or  \textit{(c)} an interpolation based on conservative parabolic splines 
-following \citet{Sacha2005} implementation of UBS in ROMS, 
+evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , 
+or  \textit{(b)} a TVD scheme, or  \textit{(c)} an interpolation based on conservative 
+parabolic splines following \citet{Sacha2005} implementation of UBS in ROMS, 
 or  \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an 
 eight-order accurate conventional scheme.
@@ -88 +93 @@
 \end{split}
 \end{equation}
-\eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that the UBS scheme is based on the fourth order scheme to which is added an upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order part as stated above using \eqref{Eq_tra_adv_ubs}. Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient with is simply proportional to the velocity.
-
+\eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that 
+the UBS scheme is based on the fourth order scheme to which is added an 
+upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order 
+part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order 
+part as stated above using \eqref{Eq_tra_adv_ubs}. Third, the diffusive term is 
+in fact a biharmonic operator with a eddy coefficient with is simply proportional 
+to the velocity.
 
 laplacian diffusion:
@@ -179 +189 @@
 \end{align}
 \end{subequations}
-As for space operator, the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\Delta t/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$, respectively. 
+As for space operator, the adjoint of the derivation and averaging time operators are 
+$\delta_t^*=\delta_{t+\Delta t/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$
+, respectively. 
 
 The Leap-frog time stepping given by \eqref{Eq_DOM_nxt} can be defined as:
@@ -187 +199 @@
       =         \frac{q^{t+\Delta t}-q^{t-\Delta t}}{2\Delta t}
 \end{equation} 
-Note that \eqref{LF} shows that the leapfrog time step is $\Delta t$, not $2\Delta t$ as it can be found sometime in literature. 
-The leap-Frog time stepping is a second order centered scheme. As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property,
+Note that \eqref{LF} shows that the leapfrog time step is $\Delta t$, not $2\Delta t$ 
+as it can be found sometime in literature. 
+The leap-Frog time stepping is a second order centered scheme. As such it respects 
+the quadratic invariant in integral forms, $i.e.$ the following continuous property,
 \begin{equation} \label{Energy}
 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
@@ -205 +219 @@
       \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right)
 \end{split} \end{equation}
-NB here pb of boundary condition when applying the adjoin! In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition (equivalently of the boundary value of the integration by part). In time this boundary condition is not physical and \textbf{add something here!!!}
-
-
-
+NB here pb of boundary condition when applying the adjoin! In space, setting to 0 
+the quantity in land area is sufficient to get rid of the boundary condition 
+(equivalently of the boundary value of the integration by part). In time this boundary 
+condition is not physical and \textbf{add something here!!!}
+
+
+