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revision 1.3 by adcroft, Fri Sep 28 03:28:32 2001 UTC revision 1.4 by cnh, Thu Oct 25 00:55:28 2001 UTC
# Line 106  discretized: Line 106  discretized:
106  \end{eqnarray}  \end{eqnarray}
107  where the Coriolis parameters $f$ and $f'$ are defined:  where the Coriolis parameters $f$ and $f'$ are defined:
108  \begin{eqnarray}  \begin{eqnarray}
109  f & = & 2 \Omega \sin{\phi} \\  f & = & 2 \Omega \sin{\varphi} \\
110  f' & = & 2 \Omega \cos{\phi}  f' & = & 2 \Omega \cos{\varphi}
111  \end{eqnarray}  \end{eqnarray}
112  where $\phi$ is geographic latitude when using spherical geometry,  where $\varphi$ is geographic latitude when using spherical geometry,
113  otherwise the $\beta$-plane definition is used:  otherwise the $\beta$-plane definition is used:
114  \begin{eqnarray}  \begin{eqnarray}
115  f & = & f_o + \beta y \\  f & = & f_o + \beta y \\
# Line 152  $G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (loca Line 152  $G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (loca
152  \subsection{Curvature metric terms}  \subsection{Curvature metric terms}
153    
154  The most commonly used coordinate system on the sphere is the  The most commonly used coordinate system on the sphere is the
155  geographic system $(\lambda,\phi)$. The curvilinear nature of these  geographic system $(\lambda,\varphi)$. The curvilinear nature of these
156  coordinates on the sphere lead to some ``metric'' terms in the  coordinates on the sphere lead to some ``metric'' terms in the
157  component momentum equations. Under the thin-atmosphere and  component momentum equations. Under the thin-atmosphere and
158  hydrostatic approximations these terms are discretized:  hydrostatic approximations these terms are discretized:
159  \begin{eqnarray}  \begin{eqnarray}
160  {\cal A}_w \Delta r_f h_w G_u^{metric} & = &  {\cal A}_w \Delta r_f h_w G_u^{metric} & = &
161    \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\    \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\
162  {\cal A}_s \Delta r_f h_s G_v^{metric} & = &  {\cal A}_s \Delta r_f h_s G_v^{metric} & = &
163  - \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\  - \overline{ \frac{ \overline{u}^i }{a} \tan{\varphi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\
164  G_w^{metric} & = & 0  G_w^{metric} & = & 0
165  \end{eqnarray}  \end{eqnarray}
166  where $a$ is the radius of the planet (sphericity is assumed) or the  where $a$ is the radius of the planet (sphericity is assumed) or the
167  radial distance of the particle (i.e. a function of height).  It is  radial distance of the particle (i.e. a function of height).  It is
168  easy to see that this discretization satisfies all the properties of  easy to see that this discretization satisfies all the properties of
169  the discrete Coriolis terms since the metric factor $\frac{u}{a}  the discrete Coriolis terms since the metric factor $\frac{u}{a}
170  \tan{\phi}$ can be viewed as a modification of the vertical Coriolis  \tan{\varphi}$ can be viewed as a modification of the vertical Coriolis
171  parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$.  parameter: $f \rightarrow f+\frac{u}{a} \tan{\varphi}$.
172    
173  However, as for the Coriolis terms, a non-energy conserving form has  However, as for the Coriolis terms, a non-energy conserving form has
174  exclusively been used to date:  exclusively been used to date:
175  \begin{eqnarray}  \begin{eqnarray}
176  G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\phi} \\  G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\varphi} \\
177  G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\phi}  G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\varphi}
178  \end{eqnarray}  \end{eqnarray}
179  where $\tan{\phi}$ is evaluated at the $u$ and $v$ points  where $\tan{\varphi}$ is evaluated at the $u$ and $v$ points
180  respectively.  respectively.
181    
182  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
# Line 240  tensor such as symmetry. Line 240  tensor such as symmetry.
240    
241  The lateral viscous stresses are discretized:  The lateral viscous stresses are discretized:
242  \begin{eqnarray}  \begin{eqnarray}
243  \tau_{11} & = & A_h c_{11\Delta}(\phi) \frac{1}{\Delta x_f} \delta_i u  \tau_{11} & = & A_h c_{11\Delta}(\varphi) \frac{1}{\Delta x_f} \delta_i u
244                 -A_4 c_{11\Delta^2}(\phi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\                 -A_4 c_{11\Delta^2}(\varphi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\
245  \tau_{12} & = & A_h c_{12\Delta}(\phi) \frac{1}{\Delta y_u} \delta_j u  \tau_{12} & = & A_h c_{12\Delta}(\varphi) \frac{1}{\Delta y_u} \delta_j u
246                 -A_4 c_{12\Delta^2}(\phi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\                 -A_4 c_{12\Delta^2}(\varphi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\
247  \tau_{21} & = & A_h c_{21\Delta}(\phi) \frac{1}{\Delta x_v} \delta_i v  \tau_{21} & = & A_h c_{21\Delta}(\varphi) \frac{1}{\Delta x_v} \delta_i v
248                 -A_4 c_{21\Delta^2}(\phi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\                 -A_4 c_{21\Delta^2}(\varphi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\
249  \tau_{22} & = & A_h c_{22\Delta}(\phi) \frac{1}{\Delta y_f} \delta_j v  \tau_{22} & = & A_h c_{22\Delta}(\varphi) \frac{1}{\Delta y_f} \delta_j v
250                 -A_4 c_{22\Delta^2}(\phi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v                 -A_4 c_{22\Delta^2}(\varphi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v
251  \end{eqnarray}  \end{eqnarray}
252  where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in  where the non-dimensional factors $c_{lm\Delta^n}(\varphi), \{l,m,n\} \in
253  \{1,2\}$ define the ``cosine'' scaling with latitude which can be  \{1,2\}$ define the ``cosine'' scaling with latitude which can be
254  applied in various ad-hoc ways. For instance, $c_{11\Delta} =  applied in various ad-hoc ways. For instance, $c_{11\Delta} =
255  c_{21\Delta} = (\cos{\phi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would  c_{21\Delta} = (\cos{\varphi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would
256  represent the an-isotropic cosine scaling typically used on the  represent the an-isotropic cosine scaling typically used on the
257  ``lat-lon'' grid for Laplacian viscosity.  ``lat-lon'' grid for Laplacian viscosity.
258  \marginpar{Need to tidy up method for controlling this in code}  \marginpar{Need to tidy up method for controlling this in code}
# Line 298  conditions. The ``body'' force takes the Line 298  conditions. The ``body'' force takes the
298  \begin{eqnarray}  \begin{eqnarray}
299  G_u^{side-drag} & = &  G_u^{side-drag} & = &
300  \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j  \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j
301  \left( A_h c_{12\Delta}(\phi) u - A_4 c_{12\Delta^2}(\phi) \nabla^2 u \right)  \left( A_h c_{12\Delta}(\varphi) u - A_4 c_{12\Delta^2}(\varphi) \nabla^2 u \right)
302  \\  \\
303  G_v^{side-drag} & = &  G_v^{side-drag} & = &
304  \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i  \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i
305  \left( A_h c_{21\Delta}(\phi) v - A_4 c_{21\Delta^2}(\phi) \nabla^2 v \right)  \left( A_h c_{21\Delta}(\varphi) v - A_4 c_{21\Delta^2}(\varphi) \nabla^2 v \right)
306  \end{eqnarray}  \end{eqnarray}
307    
308  In fact, the above discretization is not quite complete because it  In fact, the above discretization is not quite complete because it
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