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 at C-points in the center of the grid
cell. Discretization requires only averaging of $\zeta$ and $\eta$ to
vorticity or Z-points at the bottom left corner of the cell to give
$\overline{\zeta}^{Z}$ and $\overline{\eta}^{Z}$. 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 (which follow
the model code documentation at \url{http://mitgcm.org} except that
vorticity or $\zeta$-points have been renamed to Z-points in order to
avoid confusion with the bulk viscosity $\zeta$), the strain
rates are discretized as:
\begin{linenomath*}\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}\end{linenomath*}
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{linenomath*}\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}\end{linenomath*}

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{linenomath*}\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}\end{linenomath*}

Similarly, we have for the $v$-equation ($\alpha=2$):
\begin{linenomath*}\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}\end{linenomath*}


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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	mlosch	1.5	naturally defined at C-points in the center of the grid
7	mlosch	1.4	cell. Discretization requires only averaging of $\zeta$ and $\eta$ to
8			vorticity or Z-points at the bottom left corner of the cell to give
9			$\overline{\zeta}^{Z}$ and $\overline{\eta}^{Z}$. In the following,
10			the superscripts indicate location at Z or C points, distance across
11			the cell (F), along the cell edge (G), between $u$-points (U),
12			$v$-points (V), and C-points (C). The control volumes of the $u$- and
13			$v$-equations in the grid cell at indices $(i,j)$ are $A_{i,j}^{w}$
14			and $A_{i,j}^{s}$, respectively. With these definitions (which follow
15			the model code documentation at \url{http://mitgcm.org} except that
16	mlosch	1.6	vorticity or $\zeta$-points have been renamed to Z-points in order to
17			avoid confusion with the bulk viscosity $\zeta$), the strain
18	mlosch	1.5	rates are discretized as:
19	mlosch	1.4	\begin{linenomath*}\begin{align}
20	mlosch	1.3	\dot{\epsilon}_{11} &= \partial_{1}{u}_{1} + k_{2}u_{2} \\ \notag
21			=> (\epsilon_{11})_{i,j}^C &= \frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}}
22			+ k_{2,i,j}^{C}\frac{v_{i,j+1}+v_{i,j}}{2} \\
23			\dot{\epsilon}_{22} &= \partial_{2}{u}_{2} + k_{1}u_{1} \\\notag
24			=> (\epsilon_{22})_{i,j}^C &= \frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}}
25			+ k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\
26			\dot{\epsilon}_{12} = \dot{\epsilon}_{21} &= \frac{1}{2}\biggl(
27			\partial_{1}{u}_{2} + \partial_{2}{u}_{1} - k_{1}u_{2} - k_{2}u_{1}
28			\biggr) \\ \notag
29			=> (\epsilon_{12})_{i,j}^Z &= \frac{1}{2}
30			\biggl( \frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^V}
31			+ \frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^U} \\\notag
32			&\phantom{=\frac{1}{2}\biggl(}
33			- k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2}
34			- k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2}
35			\biggr),
36	mlosch	1.4	\end{align}\end{linenomath*}
37	mlosch	1.3	so that the diagonal terms of the strain rate tensor are naturally
38			defined at C-points and the symmetric off-diagonal term at
39			Z-points. No-slip boundary conditions ($u_{i,j-1}+u_{i,j}=0$ and
40			$v_{i-1,j}+v_{i,j}=0$ across boundaries) are implemented via
41			``ghost-points''; for free slip boundary conditions
42			$(\epsilon_{12})^Z=0$ on boundaries.
43
44			For a spherical polar grid, the coefficients of the metric terms are
45			$k_{1}=0$ and $k_{2}=-\tan\phi/a$, with the spherical radius $a$ and
46			the latitude $\phi$; $\Delta{x}_1 = \Delta{x} = a\cos\phi
47			\Delta\lambda$, and $\Delta{x}_2 = \Delta{y}=a\Delta\phi$. For a
48			general orthogonal curvilinear grid as used in this paper, $k_{1}$ and
49			$k_{2}$ can be approximated by finite differences of the cell widths:
50			%\enlargethispage{1cm}
51	mlosch	1.4	\begin{linenomath*}\begin{align}
52	mlosch	1.3	k_{1,i,j}^{C} &= \frac{1}{\Delta{y}_{i,j}^{F}}
53			\frac{\Delta{y}_{i+1,j}^{G}-\Delta{y}_{i,j}^{G}}{\Delta{x}_{i,j}^{F}} \\
54			k_{2,i,j}^{C} &= \frac{1}{\Delta{x}_{i,j}^{F}}
55			\frac{\Delta{x}_{i,j+1}^{G}-\Delta{x}_{i,j}^{G}}{\Delta{y}_{i,j}^{F}} \\
56			k_{1,i,j}^{Z} &= \frac{1}{\Delta{y}_{i,j}^{U}}
57			\frac{\Delta{y}_{i,j}^{C}-\Delta{y}_{i-1,j}^{C}}{\Delta{x}_{i,j}^{V}} \\
58			k_{2,i,j}^{Z} &= \frac{1}{\Delta{x}_{i,j}^{V}}
59			\frac{\Delta{x}_{i,j}^{C}-\Delta{x}_{i,j-1}^{C}}{\Delta{y}_{i,j}^{U}}
60	mlosch	1.4	\end{align}\end{linenomath*}
61	heimbach	1.1
62	mlosch	1.3	The stress tensor is given by the constitutive viscous-plastic
63			relation $\sigma_{\alpha\beta} = 2\eta\dot{\epsilon}_{\alpha\beta} +
64			[(\zeta-\eta)\dot{\epsilon}_{\gamma\gamma} - P/2
65			]\delta_{\alpha\beta}$ \citep{hibler79}. The stress tensor divergence
66			$(\nabla\sigma)_{\alpha} = \partial_\beta\sigma_{\beta\alpha}$, is
67			discretized in finite volumes. This conveniently avoids dealing with
68			further metric terms, as these are ``hidden'' in the differential cell
69			widths. For the $u$-equation ($\alpha=1$) we have:
70	mlosch	1.4	\begin{linenomath*}\begin{align}
71	mlosch	1.3	(\nabla\sigma)_{1}: \phantom{=}&
72			\frac{1}{A_{i,j}^w}
73			\int_{\mathrm{cell}}(\partial_1\sigma_{11}+\partial_2\sigma_{21})\,dx_1\,dx_2
74			\\\notag
75			=& \frac{1}{A_{i,j}^w} \biggl\{
76			\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{11}dx_2\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
77			+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{21}dx_1\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
78			\biggr\} \\ \notag
79			\approx& \frac{1}{A_{i,j}^w} \biggl\{
80			\Delta{x}_2\sigma_{11}\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
81			+ \Delta{x}_1\sigma_{21}\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
82			\biggr\} \\ \notag
83			=& \frac{1}{A_{i,j}^w} \biggl\{
84			(\Delta{x}_2\sigma_{11})_{i,j}^C - (\Delta{x}_2\sigma_{11})_{i-1,j}^C \\\notag
85			\phantom{=}& \phantom{\frac{1}{A_{i,j}^w} \biggl\{}
86			+ (\Delta{x}_1\sigma_{21})_{i,j+1}^Z - (\Delta{x}_1\sigma_{21})_{i,j}^Z
87			\biggr\}
88			\intertext{with}
89			(\Delta{x}_2\sigma_{11})_{i,j}^C =& \phantom{+}
90			\Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
91			\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag
92			&+ \Delta{y}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
93			k_{2,i,j}^C \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag
94			\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
95			\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag
96			\phantom{=}& + \Delta{y}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
97			k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag
98			\phantom{=}& - \Delta{y}_{i,j}^{F} \frac{P}{2} \\
99			%
100			(\Delta{x}_1\sigma_{21})_{i,j}^Z =& \phantom{+}
101			\Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
102			\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\ \notag
103			& + \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
104			\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag
105			& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
106			k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag
107			& - \Delta{x}_{i,j}^{V}\overline{\eta}^{Z}_{i,j}
108			k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2}
109	mlosch	1.4	\end{align}\end{linenomath*}
110	heimbach	1.1
111	mlosch	1.3	Similarly, we have for the $v$-equation ($\alpha=2$):
112	mlosch	1.4	\begin{linenomath*}\begin{align}
113	mlosch	1.3	(\nabla\sigma)_{2}: \phantom{=}&
114			\frac{1}{A_{i,j}^s}
115			\int_{\mathrm{cell}}(\partial_1\sigma_{12}+\partial_2\sigma_{22})\,dx_1\,dx_2
116			\\\notag
117			=& \frac{1}{A_{i,j}^s} \biggl\{
118			\int_{x_2}^{x_2+\Delta{x}_2}\sigma_{12}dx_2\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
119			+ \int_{x_1}^{x_1+\Delta{x}_1}\sigma_{22}dx_1\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
120			\biggr\} \\ \notag
121			\approx& \frac{1}{A_{i,j}^s} \biggl\{
122			\Delta{x}_2\sigma_{12}\biggl\|_{x_{1}}^{x_{1}+\Delta{x}_{1}}
123			+ \Delta{x}_1\sigma_{22}\biggl\|_{x_{2}}^{x_{2}+\Delta{x}_{2}}
124			\biggr\} \\ \notag
125			=& \frac{1}{A_{i,j}^s} \biggl\{
126			(\Delta{x}_2\sigma_{12})_{i+1,j}^Z - (\Delta{x}_2\sigma_{12})_{i,j}^Z
127			\\ \notag
128			\phantom{=}& \phantom{\frac{1}{A_{i,j}^s} \biggl\{}
129			+ (\Delta{x}_1\sigma_{22})_{i,j}^C - (\Delta{x}_1\sigma_{22})_{i,j-1}^C
130			\biggr\}
131			\intertext{with}
132			(\Delta{x}_1\sigma_{12})_{i,j}^Z =& \phantom{+}
133			\Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
134			\frac{u_{i,j}-u_{i,j-1}}{\Delta{y}_{i,j}^{U}} \\\notag
135			&+ \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
136			\frac{v_{i,j}-v_{i-1,j}}{\Delta{x}_{i,j}^{V}} \\ \notag
137			&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
138			k_{2,i,j}^{Z}\frac{u_{i,j}+u_{i,j-1}}{2} \\ \notag
139			&- \Delta{y}_{i,j}^{U}\overline{\eta}^{Z}_{i,j}
140			k_{1,i,j}^{Z}\frac{v_{i,j}+v_{i-1,j}}{2} \\ \notag
141			%
142			(\Delta{x}_2\sigma_{22})_{i,j}^C =& \phantom{+}
143			\Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
144			\frac{u_{i+1,j}-u_{i,j}}{\Delta{x}_{i,j}^{F}} \\ \notag
145			&+ \Delta{x}_{i,j}^{F}(\zeta - \eta)^{C}_{i,j}
146			k_{2,i,j}^{C} \frac{v_{i,j+1}+v_{i,j}}{2} \\ \notag
147			& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
148			\frac{v_{i,j+1}-v_{i,j}}{\Delta{y}_{i,j}^{F}} \\ \notag
149			& + \Delta{x}_{i,j}^{F}(\zeta + \eta)^{C}_{i,j}
150			k_{1,i,j}^{C}\frac{u_{i+1,j}+u_{i,j}}{2} \\ \notag
151			& -\Delta{x}_{i,j}^{F} \frac{P}{2}
152	mlosch	1.4	\end{align}\end{linenomath*}
153	heimbach	1.1
154
155			%%% Local Variables:
156			%%% mode: latex
157	mlosch	1.3	%%% TeX-master: "ceaice_part1"
158	heimbach	1.1	%%% End: