diff --git a/accronyms.tex b/accronyms.tex
index a91443f..180664e 100644
--- a/accronyms.tex
+++ b/accronyms.tex
@@ -134,4 +134,11 @@
short=LW,
long=Long-Wave,
}
-\newcommand{\lw}{\ac{lw}}
\ No newline at end of file
+\newcommand{\lw}{\ac{lw}}
+
+\DeclareAcronym{dof}{
+ short=DOF,
+ long=Degree of Freedom,
+}
+\newcommand{\dof}{\ac{dof}}
+\newcommand{\dofs}{\acp{dof}}
\ No newline at end of file
diff --git a/report.pdf b/report.pdf
index 89cbe0f..b793fda 100644
Binary files a/report.pdf and b/report.pdf differ
diff --git a/report.tex b/report.tex
index d7eb75d..5425338 100644
--- a/report.tex
+++ b/report.tex
@@ -252,13 +252,30 @@ Importantly, as mentioned in \Sec{sec:exchange_grid_and_flux_calculator} the \es
\subsubsection{Coupling cycle}
\label{sec:coupling_cycle}
-\open{See \Fig{fig:coupling_cycle}...
-\begin{itemize}
-\item describe which quantities are exchanged
-\item detailed description of flux calculation is written \Sec{sec:flux_formulas}
-\end{itemize}
-}
+In the current implementation the \mom\ ocean model and the \cclm\ atmospheric model exchange the following quantities during one coupling time step via the flux calculator, see also \Fig{fig:coupling_cycle}.
+%
+First, the ocean model sends its state variables, i.e. surface temperature, the albedo and the ice coverage, for each surface type (water and different ice classes; depicted by double arrows in \Fig{fig:coupling_cycle}) and each grid cell to the flux calculator.
+%
+While the data is transferred it is mapped from the ocean model's grid to the exchange grid.
+%
+With this input, the flux calculator may compute the \bbr, see \Sec{sec:flux_formulas}, that is emitted by the ocean.
+%
+This quantity is then sent to both models (and mapped to their grids) where it is added to the atmospheric thermal radiation budget and subtracted from the ocean's one.
+%
+Note that the thermal radiation that is emitted by the atmosphere is entirely computed in the atmospheric model since it is not simply given as \bbr.
+%
+Since the atmospheric model also requires the three ocean's state variables mentioned above (for computing transfer coefficients, radiation fluxes and precipitation), they are passed through the flux calculator to the atmospheric model as well.
+Second, the atmospheric model calculates its own state variables that are then send to the flux calculator.
+%
+Using this information the remaining fluxes, i.e. evaporation, latent and sensible heat as well as momentum fluxes, may be computed on the exchange grid and send to both models, see \Sec{sec:flux_formulas}.
+%
+Note that for the ocean model these fluxes may be calculated depending on the surface type (double arrows in \Fig{fig:coupling_cycle}).
+%
+Radiation and precipitation fluxes that are not computed by the flux calculator, are simply passed through to the ocean model.
+%
+Additionally the ocean model requires a few atmospheric state variables, i.e. atmospheric pressure and ten-meter wind-speed components, for \open{...} and the implemented wave model, respectively.
+%
\begin{figure}
\centering
\captionsetup{width=\linewidth}
@@ -269,17 +286,18 @@ Importantly, as mentioned in \Sec{sec:exchange_grid_and_flux_calculator} the \es
}
\end{figure}
+
\subsubsection{Flux formulas}
\label{sec:flux_formulas}
-In the current implementation of the \esm, flux formulas are used, as they are implemented in the \cclm~\citesqr{cclmManualII2011}.
+In the current implementation of the \esm, the formulas for the exchanged fluxes are used, that are implemented in the \cclm~\citesqr{cclmManualII2011}.
%
The formulas are based on the work in~\citesqr{louis1979} and will be briefly presented in the following.
-Using the surface's air pressure $p_s(x,y,t)$ and the specific water vapor content $q^v_{s}(x,y,t)$ can be calculated via
+Using the air pressure $p_a(x,y,t)$ directly above the sea surface and the specific water vapor content $q^v_{s}(x,y,t)$ can be calculated via
%
\begin{align}
-q^v_{s}(x,y,t) =\frac{R_d/R_v p_{\mathrm{sat}}(x,y,t)}{p_s(x,y,t) - (1-R_d/R_v) p_{\mathrm{sat}}(x,y,t)},
+q^v_{s}(x,y,t) =\frac{R_d/R_v p_{\mathrm{sat}}(x,y,t)}{p_a(x,y,t) - (1-R_d/R_v) p_{\mathrm{sat}}(x,y,t)},
\end{align}
%
with the gas constants $R_d$ for dry air and $R_v$ for water vapor.
@@ -299,7 +317,7 @@ Having the water vapor content $q^s_{v}(x,y,t)$ at hand, one may then calculate
%
This temperature is related to the air's density by the ideal gas law (valid for dry air)
\begin{align}
-\rho(x,y,t) = \frac{p_s(x,y,t)}{R_d\tilde{T}(x,y,t)}
+\rho(x,y,t) = \frac{p_a(x,y,t)}{R_d\tilde{T}(x,y,t)}
\end{align}
%
With the density and the coefficients $c_h(x,y,t)$ for turbulent moisture and heat transfer as well as $c_m(x,y,t)$ for the turbulent momentum transfer, one can compute the underlying fluxes according to the following formulas.
@@ -327,7 +345,7 @@ The \sh\ flux is determined by the difference between the temperatures of the lo
\phi_{\mathrm{SH}}(x,y,t) = c_h C_p \rho | \vec{u} | (T_s - \theta_s).
\end{align}
%
-The appearing $\theta_s(x,y,t)=(p_s/p_a)^{R_d/C_p} T_a$ is the atmospheric potential directly at the surface.
+The appearing $\theta_a(x,y,t)$ is the atmospheric potential temperature directly at the surface.
The momentum fluxes (i.e. the shear stress at the components interface) depend non-linearly on the wind velocity $\vec{u}(x,y,t)$ at the lowest atmospheric layer and are calculated as
%
@@ -355,6 +373,7 @@ The downward radiation fluxes (i.e. \sw\ and \lw radiation) as computed by the a
%
Thus, these fluxes are still calculated by the atmospheric model and then passed through the flux calculator to the ocean model.
+
\section{Simulation setup for the validation}
In order to validate the quality of the developed coupled \esm, a setup is used as described in the following section and is illustrated in \Fig{fig:simulation-setup}.
@@ -408,14 +427,14 @@ This comparison has been implemented into an automatic post-processing procedure
\centering
\includegraphics[width=0.8\textwidth]{"./figures/flow-diagram.pdf"}
\captionsetup{width=\linewidth}
- \caption{\label{fig:flow-diagram}\textbf{Diagram for the data flow during automatized validation procedure.} The main data flow is from left to right. The arrows depict the particular flow of the data from one validation task to the next. The raw model/reference data (left column) is first pre-processed into a generic format (right column) and subsequently analyzed by \cdo\ operations (upper row). The figures are then plotted by Python tools (lower row) and compiled into a validation report (bottom).}
+ \caption{\label{fig:flow-diagram}\textbf{Diagram for the data flow during automatized validation procedure.} The main data flow is from left to right and is exemplified for the ocean model's variable \texttt{SST} (sea surface temperature). The arrows depict the particular flow of the data from one validation task to the next. The raw model/reference data (left column) is first pre-processed into a generic format (right column) and subsequently analyzed by \cdo\ operations (upper row). The figures are then plotted by Python tools (lower row) and compiled into a validation report (bottom).}
\end{figure}
The reasoning of of the procedure is as follows.
%
First, the raw data of one model within the \esm\ as well as the chosen reference data are pre-processed such that all considered model variables are stored in files with the name of the model variable.
%
-For instance the MOM variable \texttt{SST} (sea-surface temperature) is stored in file \texttt{SST.nc} that is the broadly used NetCDF~\cite{rew1990netcdf} format.
+For instance the MOM variable \texttt{SST} (sea surface temperature) is stored in file \texttt{SST.nc} that is the broadly used NetCDF~\cite{rew1990netcdf} format.
%
The same step is applied to the reference data, where usually the variable is renamed such that it coincides with model's variable name.
%
@@ -467,10 +486,14 @@ An example of two validation reports in the PDF format can be found in \Sec{sec:
\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.
+
+\subsection{Two alternative exchange grids}
+
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}.
@@ -560,10 +583,11 @@ However, some local information is lost when mapping the atmospheric state varia
%
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{???}}.
+The impact of these inconsistencies is more quantitatively discussed in \Secs{\ref{sec:instability-atmos-grid} and \ref{sec:ocean-model-grid}}.
-\subsection{Instability with atmospheric exchange}
+\subsection{Instability with atmospheric exchange grid}
+\label{sec:instability-atmos-grid}
\begin{figure*}
\centering
@@ -584,8 +608,27 @@ The impact of these inconsistencies is quantitatively discussed in \open{\Sec{??
\caption{\label{fig:crash}\textbf{Ocean variables directly before instability.}
}
\end{figure*}
+%
+When using the atmospheric grid as the exchange grid it was not possible to integrate the coupled model over the whole time period \FullTime.
+%
+Instead the model becomes instable after 12 months featuring an unrealistic low \sst\ at specific points on the ocean's grid, see \Fig{fig:SST-crash}, where a snapshot directly before the model stops is depicted, i.e. January 9, 1961.
+%
+For instance the cell located at 21.08$^\circ$ east and 59.2$^\circ$ north exhibits a temperature that falls rapidly down by 40$^\circ$C within approximately 6 hours, see \Fig{fig:time-series-crash} blue curve.
+%
+Before this event occurred, the evaporation increases, see orange curve therein.
+%
+Due to the low exchange grid (given by the atmospheric model, orange boxes in \Figs{\ref{fig:SST-crash} and \ref{fig:EVAP-crash}}) the magnitude of the evaporation is mainly given by the surrounding ice-free cells, see white boxes in \Fig{fig:SST-crash}.
+%
+Thus, the ice covered cell is cooled down by a rate that is determined by the liquid water contained in the ice-free cells.
+%
+Hence, the instability is a direct consequence of the inconsistency when calculating fluxes on the low-resolution atmospheric grid.
+
+
+\subsection{Intersection grid vs. ocean model grid}
+\label{sec:ocean-model-grid}
-\section{Results of the uncorrected model}
+
+\section{Coupled model vs. uncoupled models}
\subsection{Atmospheric model output from \cclm}
\label{sec:CCLM}
@@ -757,6 +800,145 @@ See \Fig{fig:FI_anomalies}.
\newpage
\section{Appendix}
\label{sec:appendix}
+
+\subsection{Simple model for the cold-bias in the coupled model}
+
+
+In order to explain the strong cold bias during summer time in the coupled model simulation, a simplistic model is considered.
+
+
+\subsubsection{The simplistic model}
+
+The atmospheric temperature directly above the water surface $T_a$ and the water-surface temperature $T_w$ are written as functions of a set of quantities, i.e.
+%
+\begin{align}
+T_a = T_a(T_w, \{A_i\}), \qquad T_w = T_w(T_a, \phi_{\sw}, \phi_{\lh}, \{W_i\}),
+\end{align}
+%
+where $\{A_i\}$ and $\{W_i\}$ are all \dofs\ in the atmosphere and the water, respectively, that are not of interest now.
+%
+The downward shortwave radiation flux that heats up the ocean is explicitly taken into account via $\phi_{\sw}$ as well as the latent heat flux $\phi_{\lh}$ that mainly cools the ocean.
+%
+The total change of these quantities can then be written down as
+\begin{align}
+\d T_a = \del{T_a}{T_w} \d T_w + \sum_{i} \del{T_a}{A_i} \d A_i \\
+\d T_w = \del{T_w}{T_a} \d T_a + \del{T_w}{\phi_{\sw}} \d \phi_{\sw} + \del{T_w}{\phi_{\lh}} \d \phi_{\lh} + \sum_{i} \del{T_w}{W_i} \d W_i.
+\end{align}
+
+Now we want to construct a simplistic scenario, where all the unconsidered \dofs\ are fixed, i.e.
+%
+\begin{align}
+\d W_i = \d A_i = 0
+\end{align}
+%
+We then get
+%
+\begin{align}
+\d T_a = \del{T_a}{T_w} \d T_w \\
+\d T_w = \del{T_w}{T_a} \d T_a + \del{T_w}{\phi_{\sw}} \d \phi_{\sw} + \del{T_w}{\phi_{\lh}} \d \phi_{\lh}.
+\end{align}
+
+We may now make some intuitive assumptions.
+%
+First, we assume that
+\begin{align}
+\del{T_a}{T_w} =: \alpha \geq 0 \\
+\del{T_w}{T_a} =: \beta \geq 0.
+\end{align}
+which basically states that if the atmosphere becomes warmer the ocean will likely become warmer as well and vice versa.
+%
+Practically, this mutual heating is simply mediated by sensible heat and thermal radiation fluxes.
+%
+In a similar fashion we assume for the shortwave radiation that it is mainly heating the ocean, i.e.
+%
+\begin{align}
+\del{T_w}{\phi_{\sw}} =: \gamma \geq 0.
+\end{align}
+%
+and that the latent heat flux is cooling the ocean
+%
+\begin{align}
+\del{T_w}{\phi_{\lh}} =: \delta \leq 0,
+\end{align}
+%
+since we have mainly evaporation from the ocean to the atmosphere.
+%
+Plugging these assumptions into the above equations we get
+%
+\begin{align}
+\d T_a = \alpha \d T_w \\
+\d T_w = \beta \d T_a + \gamma \d \phi_{\sw} + \delta \d \phi_{\lh}.
+\end{align}
+%
+Now we insert the upper into the lower equation to obtain a closed formula for the water's temperature change, i.e.
+%
+\begin{align}
+\d T_w = \alpha \beta \d T_w + \gamma \d \phi_{\sw} + \delta \d \phi_{\lh}.
+\end{align}
+%
+Obvious rearranging yields
+%
+\begin{align}
+\d T_w = \frac{1}{1- \alpha \beta } (\gamma \d \phi_{\sw} + \delta \d \phi_{\lh})
+\end{align}
+%
+which states that, in our simplistic scenario, the change of the water's temperature is proportional to the change in shortwave radiation flux.
+
+
+\subsubsection{Comparing coupled and uncoupled cases}
+
+In the following we want to compare the cases when the ocean and the atmosphere are coupled or uncoupled in our simplistic model.
+
+First we consider the uncoupled case.
+%
+Here we simply have
+\begin{align}
+\alpha = \del{T_a}{T_w} \equiv 0
+\end{align}
+%
+(and thus $\alpha \beta = 0$), which means that there is no feedback from the ocean to the atmosphere.
+%
+In that case the change of the water's temperature is simply given by
+%
+\begin{align}
+\d T_w = \gamma \d \phi_{\sw} + \delta \d \phi_{\lh}
+\end{align}
+%
+
+Second, we consider the coupled case.
+%
+{\color{red} Here we can assume that
+\begin{align}
+0 < \alpha < 1 \\
+0 < \beta < 1
+\end{align}
+}
+In that case we have
+%
+\begin{align}
+\d T_w =\tilde{\gamma} \d \phi_{\sw} + \tilde{\delta} \d \phi_{\lh}
+\end{align}
+%
+with
+%
+\begin{align}
+\tilde{\gamma} := \frac{\gamma}{1- \alpha \beta }, \qquad \tilde{\delta} := \frac{\delta}{1- \alpha \beta }.
+\end{align}
+%
+{\color{red} Since $0<\alpha<1$ and $0<\beta<1$} we have
+%
+\begin{align}
+\tilde{\gamma} > \gamma, \qquad \tilde{\delta} < \delta.
+\end{align}
+%
+The different signs are due to the assumption $\gamma > 0$ and $\delta < 0$.
+%
+Consequently, in the coupled case the impact of a changing shortwave radiation and latent heat flux on the water's temperature is \textit{amplified} with respect to the uncoupled case.
+%
+Thus, reducing the incoming shortwave radiation ($\d \phi_{\sw} < 0$) and increasing the latent heat flux ($\d \phi_{\lh} > 0$) we decrease the water's temperature more strongly in the coupled case.
+%
+Hence, a deficit in the radiation flux and an overestimated latent heat (as it is present for CCLM) leads to a stronger cold bias for the coupled model.
+
\subsection{Validation of the ocean model}
\includepdf[pages=-]{./appendix/validation_report_MOM5_Baltic.pdf}
diff --git a/useful_commands.tex b/useful_commands.tex
index e5a911d..a228247 100644
--- a/useful_commands.tex
+++ b/useful_commands.tex
@@ -19,4 +19,6 @@
\bibliographystyle{unsrtnat}
\newcommand{\citesqr}[1]{\cite{#1}}
+\renewcommand{\d}{\mathrm{d}}
+\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}}