Changeset 213 for altifloat

Timestamp:

12/11/15 10:32:02 (8 years ago)

Author:

leila_ocean

Message:

Dec 11, some divergence discussion added

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (3 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -363 +363 @@
 
 We perform sequences of optimizations, where we minimise the following objective function with respect to the time independent correction $\delta \bo{u},$ in a specific time window $[0,T_w]$
-\begin{equation}
+\begin{equation}\label{Optim}
 \mathcal{J}(\delta \bo{u})=  \sum _{i=1}^{N_f} \sum_{m=1}^{\left \lfloor{T_w/\Delta t}\right \rfloor} \vectornorm{\bo{r}^{\,b}_{i}(\bo{u^b})+\delta \bo{r}_i(\delta \bo{u}) -\bo{r}_i^{\,obs}(m\Delta t) }^2 +\alpha_1 \vectornorm{ \bo{\delta u} }^2_{\bo{B}} +\alpha_2 \,\sum_{i,j} (\nabla \cdot \bo{\delta u})^2. 
 \end{equation}
@@ -373 +373 @@
 Here the $B$-norm is defined as $\vectornorm{\psi}^2_{\bo{B}} \equiv \psi^T \mathbf{B}^{-1} \psi,$ where $\bo{B}$ is the error covariance matrix.  This term serves the dual purpose of regularisation and information spreading or smoothing. To obtain $\bo{B}$, we use the diffusion filter method of \citet{weaver2001correlation}, where a priori information on the typical length scale $R$ of the Eulerian velocity can be inserted.
 The parameter $\alpha_1$ represents the relative weight of this regularisation term with respect to the other terms.
-The last component is a constraint on the geostrophic part of the velocity, required to stay divergence free. We note here that the total velocity may have a divergent component due to the wind. \textcolor{red}{This term is added to ensure a physical correction, avoiding artefacts especially near the coasts}. 
+The last component is a constraint on the geostrophic part of the velocity, required to stay divergence free. We note here that the total velocity may have a divergent component due to the wind. \textcolor{red}{This term is added to ensure a physical correction, avoiding artefacts especially near the coasts. It promotes the emergence of eddies and forces the field to go along the coast not perpendicular to it.} 
 
 
@@ -462 +462 @@
 \subsection{Sensitivity to the effect of the divergence constraint}
 
-\textcolor{red}{blabla }
+\textcolor{red}{The role of the divergence constraint in the optimization is determined by a delicate balance between the various terms. This term should be non negligible because as mentioned earlier, it forces the correction to be in the direction tangent to the coast, making the component perpendicular to the coast small. However, it cannot be too strong as to interfere with the regularization term, because that would make the optimization ill-conditionned. To show its effect on the correction, we conduct a sensitivity experiment where we compare the results (in the same setting as the previous experiments) with and without this term. As seen from Fig.~\ref{fig:div}, we obtain an improvement of about $10\%$ in the overall error if we have this term. This is expected because we are correcting the velocity in a region close to the coast.}
+
+
 \begin{figure}[htbp]
 \begin{center}
 \includegraphics[scale=0.4]{./fig/Div_win24_dt1_f14_tf72.pdf}
 %\vspace{-30mm}
-\caption{The effect of divergence constraint. Here $T_w=24$ h, and $N_f=14.$ }
+\caption{The effect of divergence constraint. The curve in $\scriptstyle -*-$ is obtained without the divergence constraint ($\alpha_2=0$ in Eq.~\ref{Optim}) whereas the one in $\scriptstyle -+-$ is obtained by adding the divergence constraint. An improvement of about $10\%$ in the error is observed in this coastal setting. Here $T_w=24$ h, $\Delta t=6$ h,  $\sigma=6$ h, and $N_f=14.$ }
 \label{fig:div}
 \end{center}