ceaice_split_version/ceaice_part1/ceaice_appendix.tex

\section{Finite-volume discretization of the stress tensor
  divergence}
\label{app:discretization}
On an Arakawa C~grid, ice thickness and concentration and thus ice
strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are
naturally defined a C-points in the center of the grid
cell. Discretization requires only averaging of $\eta$ to vorticity or
Z-points at the bottom left corner of the cell to give
$\overline{\eta}^{Z}$. In the following, the superscripts indicate
location at Z or C points, distance across the cell (F), along the
cell edge (G), between $u$-points (U), $v$-points (V), and C-points
(C). The control volumes of the $u$- and $v$-equations in the grid
cell at indices $(i,j)$ are $A_{i,j}^{w}$ and $A_{i,j}^{s}$,
respectively. With these definitions, the strain rates are discretized as:
\begin{align}
  \dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag
  => (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} 
   + k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\ 
  \dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag
  => (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} 
   + k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ 
   \dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl(
   \partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1}
   \biggr) \\ \notag
  => (\epsilon_{12})_{i,j}^Z &= \frac{1}{2}
  \biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V} 
   + \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag
  &\phantom{=\frac{1}{2}\biggl(}
   - k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2}
   - k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2}
   \biggr),
\end{align}
so that the diagonal terms of the strain rate tensor are naturally
defined at C-points and the symmetric off-diagonal term at
Z-points. No-slip boundary conditions ($u_{i,j-1}+u_{i,j}=0$ and
$v_{i-1,j}+v_{i,j}=0$ across boundaries) are implemented via
``ghost-points''; for free slip boundary conditions
$(\epsilon_{12})^Z=0$ on boundaries.

For a spherical polar grid, the coefficients of the metric terms are
$k_{1}=0$ and $k_{2}=-\tan\phi/a$, with the spherical radius $a$ and
the latitude $\phi$; $\Delta{x}_1 = \Delta{x} = a\cos\phi
\Delta\lambda$, and $\Delta{x}_2 = \Delta{y}=a\Delta\phi$. For a
general orthogonal curvilinear grid as used in this paper, $k_{1}$ and
$k_{2}$ can be approximated by finite differences of the cell widths:
%\enlargethispage{1cm}
\begin{align}
  k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}}
  \frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\
  k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}}
  \frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\
  k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}}
  \frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\
  k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}}
  \frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}}
\end{align}

The stress tensor is given by the constitutive viscous-plastic
relation $\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} +
[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2
]\delta_{\alpha\beta}$ \citep{hibler79}. The stress tensor divergence
$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is
discretized in finite volumes. This conveniently avoids dealing with
further metric terms, as these are ``hidden'' in the differential cell
widths. For the $u$-equation ($\alpha=1$) we have:
\begin{align}
  (\nabla\sigma)_{1}: \phantom{=}&
  \frac{1}{A_{i,j}^w}
  \int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21})\,dx_1\,dx_2
  \\\notag
  =& \frac{1}{A_{i,j}^w} \biggl\{
  \int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
  + \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
  \biggr\} \\ \notag
  \approx& \frac{1}{A_{i,j}^w} \biggl\{
  \Delta{x}_2\sigma_{11}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
  + \Delta{x}_1\sigma_{21}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
  \biggr\} \\ \notag
  =& \frac{1}{A_{i,j}^w} \biggl\{
  (\Delta{x}_2\sigma_{11})_{i,j}^C - (\Delta{x}_2\sigma_{11})_{i-1,j}^C \\\notag
  \phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{}
  + (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z
  \biggr\}
  \intertext{with}
  (\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+}
  \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
  \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag
  &+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
  k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag
  \phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
  \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag
  \phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
  k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag
  \phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\
  % 
  (\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+}
  \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
  \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag
  & + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
  \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag
  & - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} 
  k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag
  & - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j} 
  k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2}
\end{align}

Similarly, we have for the $v$-equation ($\alpha=2$):
\begin{align}
  (\nabla\sigma)_{2}: \phantom{=}&
  \frac{1}{A_{i,j}^s}
  \int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22})\,dx_1\,dx_2 
  \\\notag
  =& \frac{1}{A_{i,j}^s} \biggl\{
  \int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
  + \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
  \biggr\} \\ \notag
  \approx& \frac{1}{A_{i,j}^s} \biggl\{
  \Delta{x}_2\sigma_{12}\biggl|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
  + \Delta{x}_1\sigma_{22}\biggl|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
  \biggr\} \\ \notag
  =& \frac{1}{A_{i,j}^s} \biggl\{
  (\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z
  \\ \notag
  \phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{}
  + (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C
  \biggr\} 
  \intertext{with}
  (\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+}
  \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
  \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\\notag
  &+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
  \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag
  &- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
  k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag
  &- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
  k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag
  % 
  (\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+}
  \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
  \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag
  &+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
  k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag
  & + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
  \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag
  & + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
  k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag
  & -\Delta{x}_{i,j}^{F} \frac{P}{2}
\end{align}


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "ceaice_part1"
%%% End: 
1	mlosch	1.3	\section{Finite-volume discretization of the stress tensor
2			divergence}
3			\label{app:discretization}
4			On an Arakawa C~grid, ice thickness and concentration and thus ice
5			strength $P$ and bulk and shear viscosities $\zeta$ and $\eta$ are
6			naturally defined a C-points in the center of the grid
7			cell. Discretization requires only averaging of $\eta$ to vorticity or
8			Z-points at the bottom left corner of the cell to give
9			$\overline{\eta}^{Z}$. In the following, the superscripts indicate
10			location at Z or C points, distance across the cell (F), along the
11			cell edge (G), between $u$-points (U), $v$-points (V), and C-points
12			(C). The control volumes of the $u$- and $v$-equations in the grid
13			cell at indices $(i,j)$ are $A_{i,j}^{w}$ and $A_{i,j}^{s}$,
14			respectively. With these definitions, the strain rates are discretized as:
15			\begin{align}
16			\dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag
17			=> (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}}
18			+ k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\
19			\dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag
20			=> (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}}
21			+ k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\
22			\dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl(
23			\partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1}
24			\biggr) \\ \notag
25			=> (\epsilon_{12})_{i,j}^Z &= \frac{1}{2}
26			\biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V}
27			+ \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag
28			&\phantom{=\frac{1}{2}\biggl(}
29			- k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2}
30			- k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2}
31			\biggr),
32			\end{align}
33			so that the diagonal terms of the strain rate tensor are naturally
34			defined at C-points and the symmetric off-diagonal term at
35			Z-points. No-slip boundary conditions ($u_{i,j-1}+u_{i,j}=0$ and
36			$v_{i-1,j}+v_{i,j}=0$ across boundaries) are implemented via
37			``ghost-points''; for free slip boundary conditions
38			$(\epsilon_{12})^Z=0$ on boundaries.
39
40			For a spherical polar grid, the coefficients of the metric terms are
41			$k_{1}=0$ and $k_{2}=-\tan\phi/a$, with the spherical radius $a$ and
42			the latitude $\phi$; $\Delta{x}_1 = \Delta{x} = a\cos\phi
43			\Delta\lambda$, and $\Delta{x}_2 = \Delta{y}=a\Delta\phi$. For a
44			general orthogonal curvilinear grid as used in this paper, $k_{1}$ and
45			$k_{2}$ can be approximated by finite differences of the cell widths:
46			%\enlargethispage{1cm}
47			\begin{align}
48			k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}}
49			\frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\
50			k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}}
51			\frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\
52			k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}}
53			\frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\
54			k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}}
55			\frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}}
56			\end{align}
57	heimbach	1.1
58	mlosch	1.3	The stress tensor is given by the constitutive viscous-plastic
59			relation $\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} +
60			[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2
61			]\delta_{\alpha\beta}$ \citep{hibler79}. The stress tensor divergence
62			$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is
63			discretized in finite volumes. This conveniently avoids dealing with
64			further metric terms, as these are ``hidden'' in the differential cell
65			widths. For the $u$-equation ($\alpha=1$) we have:
66			\begin{align}
67			(\nabla\sigma)_{1}: \phantom{=}&
68			\frac{1}{A_{i,j}^w}
69			\int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21})\,dx_1\,dx_2
70			\\\notag
71			=& \frac{1}{A_{i,j}^w} \biggl\{
72			\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
73			+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
74			\biggr\} \\ \notag
75			\approx& \frac{1}{A_{i,j}^w} \biggl\{
76			\Delta{x}_2\sigma_{11}\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
77			+ \Delta{x}_1\sigma_{21}\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
78			\biggr\} \\ \notag
79			=& \frac{1}{A_{i,j}^w} \biggl\{
80			(\Delta{x}_2\sigma_{11})_{i,j}^C - (\Delta{x}_2\sigma_{11})_{i-1,j}^C \\\notag
81			\phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{}
82			+ (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z
83			\biggr\}
84			\intertext{with}
85			(\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+}
86			\Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
87			\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag
88			&+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
89			k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag
90			\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
91			\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag
92			\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
93			k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag
94			\phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\
95			%
96			(\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+}
97			\Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
98			\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag
99			& + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
100			\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag
101			& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
102			k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag
103			& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
104			k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2}
105			\end{align}
106	heimbach	1.1
107	mlosch	1.3	Similarly, we have for the $v$-equation ($\alpha=2$):
108	heimbach	1.1	\begin{align}
109	mlosch	1.3	(\nabla\sigma)_{2}: \phantom{=}&
110			\frac{1}{A_{i,j}^s}
111			\int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22})\,dx_1\,dx_2
112			\\\notag
113			=& \frac{1}{A_{i,j}^s} \biggl\{
114			\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
115			+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
116			\biggr\} \\ \notag
117			\approx& \frac{1}{A_{i,j}^s} \biggl\{
118			\Delta{x}_2\sigma_{12}\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
119			+ \Delta{x}_1\sigma_{22}\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
120			\biggr\} \\ \notag
121			=& \frac{1}{A_{i,j}^s} \biggl\{
122			(\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z
123			\\ \notag
124			\phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{}
125			+ (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C
126			\biggr\}
127			\intertext{with}
128			(\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+}
129			\Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
130			\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\\notag
131			&+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
132			\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag
133			&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
134			k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag
135			&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
136			k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag
137			%
138			(\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+}
139			\Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
140			\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag
141			&+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
142			k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag
143			& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
144			\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag
145			& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
146			k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag
147			& -\Delta{x}_{i,j}^{F} \frac{P}{2}
148	heimbach	1.1	\end{align}
149
150
151			%%% Local Variables:
152			%%% mode: latex
153	mlosch	1.3	%%% TeX-master: "ceaice_part1"
154	heimbach	1.1	%%% End: