diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..4f93e6f --- /dev/null +++ b/.gitignore @@ -0,0 +1,8 @@ +*.aux +*.blg +*.bbl +*.out +*.gz +*.log +*.toc +*.ipynb_checkpoints* \ No newline at end of file diff --git a/figures/remappings_atmos.pdf b/figures/remappings_atmos.pdf new file mode 100644 index 0000000..6c7c0df Binary files /dev/null and b/figures/remappings_atmos.pdf differ diff --git a/figures/remappings_bottom.pdf b/figures/remappings_bottom.pdf new file mode 100644 index 0000000..07775fc Binary files /dev/null and b/figures/remappings_bottom.pdf differ diff --git a/figures/remappings_intersection.pdf b/figures/remappings_intersection.pdf new file mode 100644 index 0000000..594eb0c Binary files /dev/null and b/figures/remappings_intersection.pdf differ diff --git a/report.pdf b/report.pdf index 60b093d..4880be9 100644 Binary files a/report.pdf and b/report.pdf differ diff --git a/report.tex b/report.tex index d9c95cb..dbbf7a3 100644 --- a/report.tex +++ b/report.tex @@ -7,7 +7,7 @@ \usepackage{amssymb} \usepackage{graphicx} \usepackage{fourier} -\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} +\usepackage[left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm]{geometry} %\usepackage{graphicx} \usepackage{float} \usepackage{caption} @@ -493,6 +493,104 @@ In other words, the vertical profile for a station $\zeta$ and seasons $S$ is gi \\ The described analysis is automatically done within developed framework that is driving the coupled \esm, as it will be published elsewhere with technical details. +\section{Impact of the chosen exchange grid} + +In order to investigate the impact of the described exchange grid approach, see \Sec{sec:exchange_grid_and_flux_calculator}, two alternative exchange grid setups are considered for comparison. +% +For the sake of clarity within this discussion, we first have to generalize the exchange grid term as it was introduced above. + +In \Sec{sec:exchange_grid_and_flux_calculator}, the term exchange grid was synonymously used for the grid that is formed from the intersections of the involved models grids. +% +In the following we want to refer to this (in fact) \textit{special case} of an exchange grid, as the \textit{intersection grid}. +% +The two alternative exchange grid cases that are apparent, can then be constructed from either the \textit{atmospheric model grid} or the \textit{ocean model grid} themselves. + +Since for a typical coupled model setup, the atmospheric grid has the lower resolution than the ocean model, the two resulting alternative exchange grids can differ quite substantially. +% +In any case both alternatives will have (by construction) a lower resolution than the intersection-type exchange grid, see again \Sec{sec:exchange_grid_and_flux_calculator}. + +With these more general formulation for the exchange grid we are now able to consider three different kind of coupling approaches on equal footing. +% +First, we employ the approach introduced in this manuscript and calculate fluxes by the flux calculator with state variables locally resolved on the intersection grid and subsequently communicate the fluxes to the models. +% +Second and third, we employ one of the model grids as the exchange grid and calculate fluxes with spatially averaged fields and communicate then the fluxes to the involved models. +% +These cases include also the ``standard'' coupling approach, i.e. using a conservative mapping of state variables from the ocean to the atmospheric model accompanied by the flux calculation via the latter and the communication back to the former~\open{\cite{citation-needed}}. + +Importantly, the flux calculator enables to investigate all three approaches with the very same infrastructure. +% +The only differences are in the exchange grid and the resulting mapping matrices to and from the model grids. +% +The different mapping matrices are visualized in \Fig{fig:remappings}. +% +\begin{figure*} + \centering + \begin{subfigure}[t]{0.49\textwidth} + \includegraphics[width=\linewidth]{"./figures/remappings_intersection.pdf"} + \caption{\label{fig:remappings-intersection} Intersection grid is used as the exchange grid.} + \end{subfigure} + \hfill + \begin{subfigure}[t]{0.49\textwidth} + \includegraphics[width=\linewidth]{"./figures/remappings_bottom.pdf"} + \caption{\label{fig:remappings-bottom} Ocean model grid is used as the exchange grid.} + \end{subfigure} + \\ + \begin{subfigure}[t]{0.49\textwidth} + \includegraphics[width=\linewidth]{"./figures/remappings_atmos.pdf"} + \caption{\label{fig:remappings-atmos} Atmospheric grid is used as the exchange grid.} + \end{subfigure} + \captionsetup{width=\linewidth} + \caption{\label{fig:remappings}\textbf{Mapping weights for the different kinds of exchange grids.} + % + The atmospheric grid is depicted by the grey lines, the ocean model's grid corresponds to the dark blue lines and the exchange grid is shown with the thin orange lines. + % + Exchange grid cells are additionally filled with transparent orange color. + % + The opaque background colors (corresponding to the color bar green-blue-white) refer to the mean mapping weight contributing to a particular cell on the destination grid. + % + The white numbers show how many cells contribute to that mean. + % + If there is no number given, then only one cell contributes to the particular cell. + % + The columns of each figure correspond to different phases during one coupling time step (from left to right). + % + Left panels depict the sending of state variables from the models to the exchange grid, right panels the communication of fluxes back to the models. + % + The rows account for the two involved models. + % + For further description see text. + } +\end{figure*} +% +As it can be seen therein, the different mapping matrices for different exchange grids can be distinguished at which phase of the coupling spatial averaging is performed. + +For instance in case of the intersection grid, \Fig{fig:remappings-intersection}, the weights are all equal to one when the model's state variables are mapped to the exchange grid, see white grid cell areas in the left panels therein and note the color bar and figure caption. +% +After the fluxes are calculated from the state variables, the fluxes have to be communicated back to the model grids. +% +This mapping naturally involves averaging over several cells and cannot be avoided since the models do eventually feature different grids. +% +This is illustrated by first the background color of the cells accounting for the mean mapping weight and second by the white numbers that count how many cells contribute to the particular destination grid cell. +% +The higher this counter (and, thus, the lower the average mapping weight is) the more information is lost during the conservative mapping from one grid to the other. +% +Importantly, when using the intersection grid as the exchange grid, the state variables are communicated to the flux calculator without any loos of information. +% +Thus, there is no local inconsistency due to any non-linearity of the flux formulas and no errors stemming from the mapping procedure can be amplified by strongly non-linear dependencies of the fluxes. +% +Averaging over several cells only happens when the fluxes are finally communicated to the models. + +In contrast, for the other two cases, there is a loss of information, when the state variables are communicated to the exchange grid for the flux calculation, see \Figs{\ref{fig:remappings-bottom} and \ref{fig:remappings-atmos}}. +% +The case when the ocean model's grid is used, see \Fig{fig:remappings-bottom}, seems to be quite similar to the intersection grid case. +% +However, some local information is lost when mapping the atmospheric state variables to the exchange grid before the flux calculation. +% +One can suppose from \Fig{fig:remappings-atmos}, that largest local inconsistencies will occur when the standard approach is employed, i.e. the ocean's state variables are first communicated to the atmospheric grid, fluxes are calculated by the atmospheric model and finally the fluxes are communicated back to the ocean. +% +The impact of these inconsistencies is quantitatively discussed in \open{\Sec{???}}. + + \section{Results of the uncorrected model} \subsection{Atmospheric model output from \cclm}