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revision 1.2 by adcroft, Wed Aug 8 22:19:02 2001 UTC revision 1.14 by afe, Tue Mar 23 16:47:04 2004 UTC
# Line 8  method. This amounts to a grid-point met Line 8  method. This amounts to a grid-point met
8  centered finite difference) in the fluid interior but allows  centered finite difference) in the fluid interior but allows
9  boundaries to intersect a regular grid allowing a more accurate  boundaries to intersect a regular grid allowing a more accurate
10  representation of the position of the boundary. We treat the  representation of the position of the boundary. We treat the
11  horizontal and veritical directions as seperable and thus slightly  horizontal and vertical directions as separable and differently.
 differently.  
12    
13  Initialization of grid data is controlled by subroutine {\em  \input{part2/notation}
 INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the  
 vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em  
 INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to  
 initialize the horizontal grid for cartesian, spherical-polar or  
 curvilinear coordinates respectively.  
   
 The reciprocals of all grid quantities are pre-calculated and this is  
 done in subroutine {\em INI\_MASKS\_ETC} which is called later by  
 subroutine {\em INITIALIZE\_FIXED}.  
   
 All grid descriptors are global arrays and stored in common blocks in  
 {\em GRID.h} and a generally declared as {\em \_RS}.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})  
14    
 {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})  
15    
16  grid data: ({\em model/inc/GRID.h})  \subsection{The finite volume method: finite volumes versus finite difference}
17  \end{minipage} }  \begin{rawhtml}
18    <!-- CMIREDIR:finite_volume: -->
19    \end{rawhtml}
20    
21    
 \subsection{The finite volume method: finite volumes versus finite difference}  
22    
23  The finite volume method is used to discretize the equations in  The finite volume method is used to discretize the equations in
24  space. The expression ``finite volume'' actually has two meanings; one  space. The expression ``finite volume'' actually has two meanings; one
25  involves invocation of the weak formulation (e.g. integral  is the method of embedded or intersecting boundaries (shaved or lopped
26  formulation) and the other involves non-linear expressions for  cells in our terminology) and the other is non-linear interpolation
27  interpolation of data (e.g. flux limiters). We use both but they can  methods that can deal with non-smooth solutions such as shocks
28  and will be ddiscussed seperately. The finite volume method discretizes by invoking the weak formulation of the equations or integral form. For example, the 1-D advection-diffusion equation:  (i.e. flux limiters for advection). Both make use of the integral form
29    of the conservation laws to which the {\it weak solution} is a
30    solution on each finite volume of (sub-domain). The weak solution can
31    be constructed out of piece-wise constant elements or be
32    differentiable. The differentiable equations can not be satisfied by
33    piece-wise constant functions.
34    
35    As an example, the 1-D constant coefficient advection-diffusion
36    equation:
37  \begin{displaymath}  \begin{displaymath}
38  \partial_t \theta + \partial_x ( u \theta - \kappa \partial_x \theta ) = 0  \partial_t \theta + \partial_x ( u \theta - \kappa \partial_x \theta ) = 0
39  \end{displaymath}  \end{displaymath}
40  can be discretized by integrating of finite lengths $\Delta x$:  can be discretized by integrating over finite sub-domains, i.e.
41    the lengths $\Delta x_i$:
42  \begin{displaymath}  \begin{displaymath}
43  \Delta x \partial_t \theta + \delta_i ( F ) = 0  \Delta x \partial_t \theta + \delta_i ( F ) = 0
44  \end{displaymath}  \end{displaymath}
45  is exact but where the flux  is exact if $\theta(x)$ is piece-wise constant over the interval
46    $\Delta x_i$ or more generally if $\theta_i$ is defined as the average
47    over the interval $\Delta x_i$.
48    
49    The flux, $F_{i-1/2}$, must be approximated:
50  \begin{displaymath}  \begin{displaymath}
51  F = u \overline{\theta} - \frac{\kappa}{\Delta x_c} \partial_i \theta  F = u \overline{\theta} - \frac{\kappa}{\Delta x_c} \partial_i \theta
52  \end{displaymath}  \end{displaymath}
53  is approximate. The method for obtained $\overline{\theta}$ is  and this is where truncation errors can enter the solution. The
54  unspecified and non-lienar finite volume methods can be invoked.  method for obtaining $\overline{\theta}$ is unspecified and a wide
55    range of possibilities exist including centered and upwind
56  ....  INCOMPLETE  interpolation, polynomial fits based on the the volume average
57    definitions of quantities and non-linear interpolation such as
58    flux-limiters.
59    
60    Choosing simple centered second-order interpolation and differencing
61    recovers the same ODE's resulting from finite differencing for the
62    interior of a fluid. Differences arise at boundaries where a boundary
63    is not positioned on a regular or smoothly varying grid. This method
64    is used to represent the topography using lopped cell, see
65    \cite{adcroft:97}. Subtle difference also appear in more than one
66    dimension away from boundaries. This happens because the each
67    direction is discretized independently in the finite difference method
68    while the integrating over finite volume implicitly treats all
69    directions simultaneously. Illustration of this is given in
70    \cite{ac:02}.
71    
72  \subsection{C grid staggering of variables}  \subsection{C grid staggering of variables}
73    
74  \begin{figure}  \begin{figure}
75  \centerline{ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} }  \begin{center}
76    \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}}
77    \end{center}
78  \caption{Three dimensional staggering of velocity components. This  \caption{Three dimensional staggering of velocity components. This
79  facilitates the natural discretization of the continuity and tracer  facilitates the natural discretization of the continuity and tracer
80  equations. }  equations. }
# Line 71  equations. } Line 84  equations. }
84  The basic algorithm employed for stepping forward the momentum  The basic algorithm employed for stepping forward the momentum
85  equations is based on retaining non-divergence of the flow at all  equations is based on retaining non-divergence of the flow at all
86  times. This is most naturally done if the components of flow are  times. This is most naturally done if the components of flow are
87  staggered in space in the form of an Arakawa C grid \cite{Arakawa70}.  staggered in space in the form of an Arakawa C grid \cite{arakawa:77}.
88    
89  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)  Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)
90  staggered in space such that the zonal component falls on the  staggered in space such that the zonal component falls on the
91  interface between continiuty cells in the zonal direction. Similarly  interface between continuity cells in the zonal direction. Similarly
92  for the meridional and vertical directions.  The continiuty cell is  for the meridional and vertical directions.  The continuity cell is
93  synonymous with tracer cells (they are one and the same).  synonymous with tracer cells (they are one and the same).
94    
95    
96    
97    \subsection{Grid initialization and data}
98    
99    Initialization of grid data is controlled by subroutine {\em
100    INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the
101    vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em
102    INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to
103    initialize the horizontal grid for cartesian, spherical-polar or
104    curvilinear coordinates respectively.
105    
106    The reciprocals of all grid quantities are pre-calculated and this is
107    done in subroutine {\em INI\_MASKS\_ETC} which is called later by
108    subroutine {\em INITIALIZE\_FIXED}.
109    
110    All grid descriptors are global arrays and stored in common blocks in
111    {\em GRID.h} and a generally declared as {\em \_RS}.
112    
113    \fbox{ \begin{minipage}{4.75in}
114    {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})
115    
116    {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
117    
118    grid data: ({\em model/inc/GRID.h})
119    \end{minipage} }
120    
121    
122  \subsection{Horizontal grid}  \subsection{Horizontal grid}
123    \label{sec:spatial_discrete_horizontal_grid}
124    
125  \begin{figure}  \begin{figure}
126  \centerline{ \begin{tabular}{cc}  \begin{center}
127    \begin{tabular}{cc}
128    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}    \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
129  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}  & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
130  \\  \\
131    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}    \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
132  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}  & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
133  \end{tabular} }  \end{tabular}
134    \end{center}
135  \caption{  \caption{
136  Staggering of horizontal grid descriptors (lengths and areas). The  Staggering of horizontal grid descriptors (lengths and areas). The
137  grid lines indicate the tracer cell boundaries and are the reference  grid lines indicate the tracer cell boundaries and are the reference
# Line 106  is bordered by the lengths $\Delta x_f$ Line 146  is bordered by the lengths $\Delta x_f$
146    
147  The model domain is decomposed into tiles and within each tile a  The model domain is decomposed into tiles and within each tile a
148  quasi-regular grid is used. A tile is the basic unit of domain  quasi-regular grid is used. A tile is the basic unit of domain
149  decomposition for parallelization but may be used whether parallized  decomposition for parallelization but may be used whether parallelized
150  or not; see section \ref{sect:tiles} for more details. Although the  or not; see section \ref{sect:tiles} for more details. Although the
151  tiles may be patched together in an unstructured manner  tiles may be patched together in an unstructured manner
152  (i.e. irregular or non-tessilating pattern), the interior of tiles is  (i.e. irregular or non-tessilating pattern), the interior of tiles is
153  a structered grid of quadrilateral cells. The horizontal coordinate  a structured grid of quadrilateral cells. The horizontal coordinate
154  system is orthogonal curvilinear meaning we can not necessarily treat  system is orthogonal curvilinear meaning we can not necessarily treat
155  the two horizontal directions as seperable. Instead, each cell in the  the two horizontal directions as separable. Instead, each cell in the
156  horizontal grid is described by the length of it's sides and it's  horizontal grid is described by the length of it's sides and it's
157  area.  area.
158    
159  The grid information is quite general and describes any of the  The grid information is quite general and describes any of the
160  available coordinates systems, cartesian, spherical-polar or  available coordinates systems, cartesian, spherical-polar or
161  curvilinear. All that is necessary to distinguish between the  curvilinear. All that is necessary to distinguish between the
162  coordinate systems is to initialize the grid data (discriptors)  coordinate systems is to initialize the grid data (descriptors)
163  appropriately.  appropriately.
164    
 \marginpar{Caution!}  
165  In the following, we refer to the orientation of quantities on the  In the following, we refer to the orientation of quantities on the
166  computational grid using geographic terminology such as points of the  computational grid using geographic terminology such as points of the
167  compass. This is purely for convenience but should note be confused  compass.
168    \marginpar{Caution!}
169    This is purely for convenience but should note be confused
170  with the actual geographic orientation of model quantities.  with the actual geographic orientation of model quantities.
171    
 \marginpar{$A_c$: {\bf rAc}}  
 \marginpar{$\Delta x_g$: {\bf DXg}}  
 \marginpar{$\Delta y_g$: {\bf DYg}}  
172  Fig.~\ref{fig:hgrid}a shows the tracer cell (synonymous with the  Fig.~\ref{fig:hgrid}a shows the tracer cell (synonymous with the
173  continuity cell). The length of the southern edge, $\Delta x_g$,  continuity cell). The length of the southern edge, $\Delta x_g$,
174  western edge, $\Delta y_g$ and surface area, $A_c$, presented in the  western edge, $\Delta y_g$ and surface area, $A_c$, presented in the
175  vertical are stored in arrays {\bf DXg}, {\bf DYg} and {\bf rAc}. The  vertical are stored in arrays {\bf DXg}, {\bf DYg} and {\bf rAc}.
176  ``g'' suffix indicates that the lengths are along the defining grid  \marginpar{$A_c$: {\bf rAc}}
177  boundaries. The ``c'' suffix associates the quantity with the cell  \marginpar{$\Delta x_g$: {\bf DXg}}
178  centers. The quantities are staggered in space and the indexing is  \marginpar{$\Delta y_g$: {\bf DYg}}
179  such that {\bf DXg(i,j)} is positioned to the south of {\bf rAc(i,j)}  The ``g'' suffix indicates that the lengths are along the defining
180  and {\bf DYg(i,j)} positioned to the west.  grid boundaries. The ``c'' suffix associates the quantity with the
181    cell centers. The quantities are staggered in space and the indexing
182    is such that {\bf DXg(i,j)} is positioned to the south of {\bf
183    rAc(i,j)} and {\bf DYg(i,j)} positioned to the west.
184    
 \marginpar{$A_\zeta$: {\bf rAz}}  
 \marginpar{$\Delta x_c$: {\bf DXc}}  
 \marginpar{$\Delta y_c$: {\bf DYc}}  
185  Fig.~\ref{fig:hgrid}b shows the vorticity cell. The length of the  Fig.~\ref{fig:hgrid}b shows the vorticity cell. The length of the
186  southern edge, $\Delta x_c$, western edge, $\Delta y_c$ and surface  southern edge, $\Delta x_c$, western edge, $\Delta y_c$ and surface
187  area, $A_\zeta$, presented in the vertical are stored in arrays {\bf  area, $A_\zeta$, presented in the vertical are stored in arrays {\bf
188  DXg}, {\bf DYg} and {\bf rAz}.  The ``z'' suffix indicates that the  DXg}, {\bf DYg} and {\bf rAz}.
189  lengths are measured between the cell centers and the ``$\zeta$'' suffix  \marginpar{$A_\zeta$: {\bf rAz}}
190  associates points with the vorticity points. The quantities are  \marginpar{$\Delta x_c$: {\bf DXc}}
191  staggered in space and the indexing is such that {\bf DXc(i,j)} is  \marginpar{$\Delta y_c$: {\bf DYc}}
192  positioned to the north of {\bf rAc(i,j)} and {\bf DYc(i,j)} positioned  The ``z'' suffix indicates that the lengths are measured between the
193  to the east.  cell centers and the ``$\zeta$'' suffix associates points with the
194    vorticity points. The quantities are staggered in space and the
195    indexing is such that {\bf DXc(i,j)} is positioned to the north of
196    {\bf rAc(i,j)} and {\bf DYc(i,j)} positioned to the east.
197    
 \marginpar{$A_w$: {\bf rAw}}  
 \marginpar{$\Delta x_v$: {\bf DXv}}  
 \marginpar{$\Delta y_f$: {\bf DYf}}  
198  Fig.~\ref{fig:hgrid}c shows the ``u'' or western (w) cell. The length of  Fig.~\ref{fig:hgrid}c shows the ``u'' or western (w) cell. The length of
199  the southern edge, $\Delta x_v$, eastern edge, $\Delta y_f$ and  the southern edge, $\Delta x_v$, eastern edge, $\Delta y_f$ and
200  surface area, $A_w$, presented in the vertical are stored in arrays  surface area, $A_w$, presented in the vertical are stored in arrays
201  {\bf DXv}, {\bf DYf} and {\bf rAw}.  The ``v'' suffix indicates that  {\bf DXv}, {\bf DYf} and {\bf rAw}.
202  the length is measured between the v-points, the ``f'' suffix  \marginpar{$A_w$: {\bf rAw}}
203  indicates that the length is measured between the (tracer) cell faces  \marginpar{$\Delta x_v$: {\bf DXv}}
204  and the ``w'' suffix associates points with the u-points (w stands for  \marginpar{$\Delta y_f$: {\bf DYf}}
205  west). The quantities are staggered in space and the indexing is such  The ``v'' suffix indicates that the length is measured between the
206  that {\bf DXv(i,j)} is positioned to the south of {\bf rAw(i,j)} and  v-points, the ``f'' suffix indicates that the length is measured
207  {\bf DYf(i,j)} positioned to the east.  between the (tracer) cell faces and the ``w'' suffix associates points
208    with the u-points (w stands for west). The quantities are staggered in
209    space and the indexing is such that {\bf DXv(i,j)} is positioned to
210    the south of {\bf rAw(i,j)} and {\bf DYf(i,j)} positioned to the east.
211    
 \marginpar{$A_s$: {\bf rAs}}  
 \marginpar{$\Delta x_f$: {\bf DXf}}  
 \marginpar{$\Delta y_u$: {\bf DYu}}  
212  Fig.~\ref{fig:hgrid}d shows the ``v'' or southern (s) cell. The length of  Fig.~\ref{fig:hgrid}d shows the ``v'' or southern (s) cell. The length of
213  the northern edge, $\Delta x_f$, western edge, $\Delta y_u$ and  the northern edge, $\Delta x_f$, western edge, $\Delta y_u$ and
214  surface area, $A_s$, presented in the vertical are stored in arrays  surface area, $A_s$, presented in the vertical are stored in arrays
215  {\bf DXf}, {\bf DYu} and {\bf rAs}.  The ``u'' suffix indicates that  {\bf DXf}, {\bf DYu} and {\bf rAs}.
216  the length is measured between the u-points, the ``f'' suffix  \marginpar{$A_s$: {\bf rAs}}
217  indicates that the length is measured between the (tracer) cell faces  \marginpar{$\Delta x_f$: {\bf DXf}}
218  and the ``s'' suffix associates points with the v-points (s stands for  \marginpar{$\Delta y_u$: {\bf DYu}}
219  south). The quantities are staggered in space and the indexing is such  The ``u'' suffix indicates that the length is measured between the
220  that {\bf DXf(i,j)} is positioned to the north of {\bf rAs(i,j)} and  u-points, the ``f'' suffix indicates that the length is measured
221  {\bf DYu(i,j)} positioned to the west.  between the (tracer) cell faces and the ``s'' suffix associates points
222    with the v-points (s stands for south). The quantities are staggered
223    in space and the indexing is such that {\bf DXf(i,j)} is positioned to
224    the north of {\bf rAs(i,j)} and {\bf DYu(i,j)} positioned to the west.
225    
226  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
227  {\em S/R INI\_CARTESIAN\_GRID} ({\em  {\em S/R INI\_CARTESIAN\_GRID} ({\em
# Line 257  using\-Cartes\-ianGrid} in namelist {\em Line 298  using\-Cartes\-ianGrid} in namelist {\em
298  spacing can be set to uniform via scalars {\bf dXspacing} and {\bf  spacing can be set to uniform via scalars {\bf dXspacing} and {\bf
299  dYspacing} in namelist {\em PARM04} or to variable resolution by the  dYspacing} in namelist {\em PARM04} or to variable resolution by the
300  vectors {\bf DELX} and {\bf DELY}. Units are normally  vectors {\bf DELX} and {\bf DELY}. Units are normally
301  meters. Non-dimensional coordinates can be used by interpretting the  meters. Non-dimensional coordinates can be used by interpreting the
302  gravitational constant as the Rayleigh number.  gravitational constant as the Rayleigh number.
303    
304  \subsubsection{Spherical-polar coordinates}  \subsubsection{Spherical-polar coordinates}
# Line 282  other grids, the horizontal grid descrip Line 323  other grids, the horizontal grid descrip
323  \subsection{Vertical grid}  \subsection{Vertical grid}
324    
325  \begin{figure}  \begin{figure}
326  \centerline{ \begin{tabular}{cc}  \begin{center}
327      \begin{tabular}{cc}
328    \raisebox{4in}{a)} \resizebox{!}{4in}{    \raisebox{4in}{a)} \resizebox{!}{4in}{
329    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}    \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}
330    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}    \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
331  \end{tabular} }  \end{tabular}
332    \end{center}
333  \caption{Two versions of the vertical grid. a) The cell centered  \caption{Two versions of the vertical grid. a) The cell centered
334  approach where the interface depths are specified and the tracer  approach where the interface depths are specified and the tracer
335  points centered in between the interfaces. b) The interface centered  points centered in between the interfaces. b) The interface centered
# Line 309  The vertical grid is calculated in subro Line 352  The vertical grid is calculated in subro
352  INI\_VERTICAL\_GRID} and specified via the vector {\bf DELR} in  INI\_VERTICAL\_GRID} and specified via the vector {\bf DELR} in
353  namelist {\em PARM04}. The units of ``r'' are either meters or Pascals  namelist {\em PARM04}. The units of ``r'' are either meters or Pascals
354  depending on the isomorphism being used which in turn is dependent  depending on the isomorphism being used which in turn is dependent
355  only on the choise of equation of state.  only on the choice of equation of state.
356    
357  There are alternative namelist vectors {\bf DELZ} and {\bf DELP} which  There are alternative namelist vectors {\bf DELZ} and {\bf DELP} which
358  dictate whether z- or  dictate whether z- or
# Line 348  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e Line 391  $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\e
391  \end{minipage} }  \end{minipage} }
392    
393    
394  \subsection{Continuity and horizontal pressure gradient terms}  \subsection{Topography: partially filled cells}
395    
396    \begin{figure}
397    \begin{center}
398    \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}
399    \end{center}
400    \caption{
401    A schematic of the x-r plane showing the location of the
402    non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a
403    tracer cell is given by $h_c(i,j,k) \Delta r_f(k)$ and the physical
404    thickness of the open side is given by $h_w(i,j,k) \Delta r_f(k)$.}
405    \label{fig:hfacs}
406    \end{figure}
407    
408    \cite{adcroft:97} presented two alternatives to the step-wise finite
409    difference representation of topography. The method is known to the
410    engineering community as {\em intersecting boundary method}. It
411    involves allowing the boundary to intersect a grid of cells thereby
412    modifying the shape of those cells intersected. We suggested allowing
413    the topography to take on a piece-wise linear representation (shaved
414    cells) or a simpler piecewise constant representation (partial step).
415    Both show dramatic improvements in solution compared to the
416    traditional full step representation, the piece-wise linear being the
417    best. However, the storage requirements are excessive so the simpler
418    piece-wise constant or partial-step method is all that is currently
419    supported.
420    
421    Fig.~\ref{fig:hfacs} shows a schematic of the x-r plane indicating how
422    the thickness of a level is determined at tracer and u points.
423    \marginpar{$h_c$: {\bf hFacC}}
424    \marginpar{$h_w$: {\bf hFacW}}
425    \marginpar{$h_s$: {\bf hFacS}}
426    The physical thickness of a tracer cell is given by $h_c(i,j,k) \Delta
427    r_f(k)$ and the physical thickness of the open side is given by
428    $h_w(i,j,k) \Delta r_f(k)$. Three 3-D descriptors $h_c$, $h_w$ and
429    $h_s$ are used to describe the geometry: {\bf hFacC}, {\bf hFacW} and
430    {\bf hFacS} respectively. These are calculated in subroutine {\em
431    INI\_MASKS\_ETC} along with there reciprocals {\bf RECIP\_hFacC}, {\bf
432    RECIP\_hFacW} and {\bf RECIP\_hFacS}.
433    
434    The non-dimensional fractions (or h-facs as we call them) are
435    calculated from the model depth array and then processed to avoid tiny
436    volumes. The rule is that if a fraction is less than {\bf hFacMin}
437    then it is rounded to the nearer of $0$ or {\bf hFacMin} or if the
438    physical thickness is less than {\bf hFacMinDr} then it is similarly
439    rounded. The larger of the two methods is used when there is a
440    conflict. By setting {\bf hFacMinDr} equal to or larger than the
441    thinnest nominal layers, $\min{(\Delta z_f)}$, but setting {\bf
442    hFacMin} to some small fraction then the model will only lop thick
443    layers but retain stability based on the thinnest unlopped thickness;
444    $\min{(\Delta z_f,\mbox{\bf hFacMinDr})}$.
445    
446    \fbox{ \begin{minipage}{4.75in}
447    {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
448    
449    $h_c$: {\bf hFacC} ({\em GRID.h})
450    
451    $h_w$: {\bf hFacW} ({\em GRID.h})
452    
453    $h_s$: {\bf hFacS} ({\em GRID.h})
454    
455    $h_c^{-1}$: {\bf RECIP\_hFacC} ({\em GRID.h})
456    
457    $h_w^{-1}$: {\bf RECIP\_hFacW} ({\em GRID.h})
458    
459    $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRID.h})
460    
461    \end{minipage} }
462    
463    
464    \section{Continuity and horizontal pressure gradient terms}
465    
466  The core algorithm is based on the ``C grid'' discretization of the  The core algorithm is based on the ``C grid'' discretization of the
467  continuity equation which can be summarized as:  continuity equation which can be summarized as:
468  \begin{eqnarray}  \begin{eqnarray}
469  \partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}' & = & G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h' \\  \partial_t u + \frac{1}{\Delta x_c} \delta_i \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta x_c} \delta_i \Phi_{nh}' & = & G_u - \frac{1}{\Delta x_c} \delta_i \Phi_h' \label{eq:discrete-momu} \\
470  \partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}' & = & G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h' \\  \partial_t v + \frac{1}{\Delta y_c} \delta_j \left. \frac{ \partial \Phi}{\partial r}\right|_{s} \eta + \frac{\epsilon_{nh}}{\Delta y_c} \delta_j \Phi_{nh}' & = & G_v - \frac{1}{\Delta y_c} \delta_j \Phi_h' \label{eq:discrete-momv} \\
471  \epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right) & = & \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}' \\  \epsilon_{nh} \left( \partial_t w + \frac{1}{\Delta r_c} \delta_k \Phi_{nh}' \right) & = & \epsilon_{nh} G_w + \overline{b}^k - \frac{1}{\Delta r_c} \delta_k \Phi_{h}' \label{eq:discrete-momw} \\
472  \delta_i \Delta y_g \Delta r_f h_w u +  \delta_i \Delta y_g \Delta r_f h_w u +
473  \delta_j \Delta x_g \Delta r_f h_s v +  \delta_j \Delta x_g \Delta r_f h_s v +
474  \delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0}  \delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0}
# Line 363  continuity equation which can be summari Line 476  continuity equation which can be summari
476  \end{eqnarray}  \end{eqnarray}
477  where the continuity equation has been most naturally discretized by  where the continuity equation has been most naturally discretized by
478  staggering the three components of velocity as shown in  staggering the three components of velocity as shown in
479  Fig.~\ref{fig-cgrid3d}. The grid lengths $\Delta x_c$ and $\Delta y_c$  Fig.~\ref{fig:cgrid3d}. The grid lengths $\Delta x_c$ and $\Delta y_c$
480  are the lengths between tracer points (cell centers). The grid lengths  are the lengths between tracer points (cell centers). The grid lengths
481  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell  $\Delta x_g$, $\Delta y_g$ are the grid lengths between cell
482  corners. $\Delta r_f$ and $\Delta r_c$ are the distance (in units of  corners. $\Delta r_f$ and $\Delta r_c$ are the distance (in units of
# Line 376  A}_c$.  The factors $h_w$ and $h_s$ are Line 489  A}_c$.  The factors $h_w$ and $h_s$ are
489  \marginpar{$h_s$: {\bf hFacS}}  \marginpar{$h_s$: {\bf hFacS}}
490    
491  The last equation, the discrete continuity equation, can be summed in  The last equation, the discrete continuity equation, can be summed in
492  the vertical to yeild the free-surface equation:  the vertical to yield the free-surface equation:
493  \begin{equation}  \begin{equation}
494  {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v =  {\cal A}_c \partial_t \eta + \delta_i \sum_k \Delta y_g \Delta r_f h_w
495  {\cal A}_c(P-E)_{r=0}  u + \delta_j \sum_k \Delta x_g \Delta r_f h_s v = {\cal
496    A}_c(P-E)_{r=0} \label{eq:discrete-freesurface}
497  \end{equation}  \end{equation}
498  The source term $P-E$ on the rhs of continuity accounts for the local  The source term $P-E$ on the rhs of continuity accounts for the local
499  addition of volume due to excess precipitation and run-off over  addition of volume due to excess precipitation and run-off over
500  evaporation and only enters the top-level of the {\em ocean} model.  evaporation and only enters the top-level of the {\em ocean} model.
501    
502  \subsection{Hydrostatic balance}  \section{Hydrostatic balance}
503    
504  The vertical momentum equation has the hydrostatic or  The vertical momentum equation has the hydrostatic or
505  quasi-hydrostatic balance on the right hand side. This discretization  quasi-hydrostatic balance on the right hand side. This discretization
# Line 394  derived from the buoyancy equation exact Line 508  derived from the buoyancy equation exact
508  from the pressure gradient terms when forming the kinetic energy  from the pressure gradient terms when forming the kinetic energy
509  equation.  equation.
510    
511  In the ocean, using z-ccordinates, the hydrostatic balance terms are  In the ocean, using z-coordinates, the hydrostatic balance terms are
512  discretized:  discretized:
513  \begin{equation}  \begin{equation}
514  \epsilon_{nh} \partial_t w  \epsilon_{nh} \partial_t w
515  + g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots  + g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots
516    \label{eq:discrete_hydro_ocean}
517  \end{equation}  \end{equation}
518    
519  In the atmosphere, using p-coordinates, hydrostatic balance is  In the atmosphere, using p-coordinates, hydrostatic balance is
520  discretized:  discretized:
521  \begin{equation}  \begin{equation}
522  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0  \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0
523    \label{eq:discrete_hydro_atmos}
524  \end{equation}  \end{equation}
525  where $\Delta \Pi$ is the difference in Exner function between the  where $\Delta \Pi$ is the difference in Exner function between the
526  pressure points. The non-hydrostatic equations are not available in  pressure points. The non-hydrostatic equations are not available in
# Line 412  the atmosphere. Line 528  the atmosphere.
528    
529  The difference in approach between ocean and atmosphere occurs because  The difference in approach between ocean and atmosphere occurs because
530  of the direct use of the ideal gas equation in forming the potential  of the direct use of the ideal gas equation in forming the potential
531  energy conversion term $\alpha \omega$. The form of these consversion  energy conversion term $\alpha \omega$. The form of these conversion
532  terms is discussed at length in \cite{Adcroft01}.  terms is discussed at length in \cite{adcroft:02}.
533    
534  Because of the different representation of hydrostatic balance between  Because of the different representation of hydrostatic balance between
535  ocean and atmosphere there is no elegant way to represent both systems  ocean and atmosphere there is no elegant way to represent both systems
# Line 428  CALC\_PHI\_HYD}. Inside this routine, on Line 544  CALC\_PHI\_HYD}. Inside this routine, on
544  atmospheric/oceanic form is selected based on the string variable {\bf  atmospheric/oceanic form is selected based on the string variable {\bf
545  buoyancyRelation}.  buoyancyRelation}.
546    
 \subsection{Flux-form momentum equations}  
   
 The original finite volume model was based on the Eulerian flux form  
 momentum equations. This is the default though the vector invariant  
 form is optionally available (and recommended in some cases).  
   
 The ``G's'' (our colloquial name for all terms on rhs!) are broken  
 into the various advective, Coriolis, horizontal dissipation, vertical  
 dissipation and metric forces:  
 \marginpar{$G_u$: {\bf Gu} }  
 \marginpar{$G_v$: {\bf Gv} }  
 \marginpar{$G_w$: {\bf Gw} }  
 \begin{eqnarray}  
 G_u & = & G_u^{adv} + G_u^{cor} + G_u^{h-diss} + G_u^{v-diss} +  
 G_u^{metric} + G_u^{nh-metric} \\  
 G_v & = & G_v^{adv} + G_v^{cor} + G_v^{h-diss} + G_v^{v-diss} +  
 G_v^{metric} + G_v^{nh-metric} \\  
 G_w & = & G_w^{adv} + G_w^{cor} + G_w^{h-diss} + G_w^{v-diss} +  
 G_w^{metric} + G_w^{nh-metric}  
 \end{eqnarray}  
 In the hydrostatic limit, $G_w=0$ and $\epsilon_{nh}=0$, reducing the  
 vertical momentum to hydrostatic balance.  
   
 These terms are calculated in routines called from subroutine {\em  
 CALC\_MOM\_RHS} a collected into the global arrays {\bf Gu}, {\bf Gv},  
 and {\bf Gw}.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_fluxform/calc\_mom\_rhs.F})  
   
 $G_u$: {\bf Gu} ({\em DYNVARS.h})  
   
 $G_v$: {\bf Gv} ({\em DYNVARS.h})  
   
 $G_w$: {\bf Gw} ({\em DYNVARS.h})  
 \end{minipage} }  
   
   
 \subsubsection{Advection of momentum}  
   
 The advective operator is second order accurate in space:  
 \begin{eqnarray}  
 {\cal A}_w \Delta r_f h_w G_u^{adv} & = &  
   \delta_i \overline{ U }^i \overline{ u }^i  
 + \delta_j \overline{ V }^i \overline{ u }^j  
 + \delta_k \overline{ W }^i \overline{ u }^k \\  
 {\cal A}_s \Delta r_f h_s G_v^{adv} & = &  
   \delta_i \overline{ U }^j \overline{ v }^i  
 + \delta_j \overline{ V }^j \overline{ v }^j  
 + \delta_k \overline{ W }^j \overline{ v }^k \\  
 {\cal A}_c \Delta r_c G_w^{adv} & = &  
   \delta_i \overline{ U }^k \overline{ w }^i  
 + \delta_j \overline{ V }^k \overline{ w }^j  
 + \delta_k \overline{ W }^k \overline{ w }^k \\  
 \end{eqnarray}  
 and because of the flux form does not contribute to the global budget  
 of linear momentum. The quantities $U$, $V$ and $W$ are volume fluxes  
 defined:  
 \marginpar{$U$: {\bf uTrans} }  
 \marginpar{$V$: {\bf vTrans} }  
 \marginpar{$W$: {\bf rTrans} }  
 \begin{eqnarray}  
 U & = & \Delta y_g \Delta r_f h_w u \\  
 V & = & \Delta x_g \Delta r_f h_s v \\  
 W & = & {\cal A}_c w  
 \end{eqnarray}  
 The advection of momentum takes the same form as the advection of  
 tracers but by a translated advective flow. Consequently, the  
 conservation of second moments, derived for tracers later, applies to  
 $u^2$ and $v^2$ and $w^2$ so that advection of momentum correctly  
 conserves kinetic energy.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_ADV\_UU} ({\em mom\_u\_adv\_uu.F})  
   
 {\em S/R MOM\_U\_ADV\_VU} ({\em mom\_u\_adv\_vu.F})  
   
 {\em S/R MOM\_U\_ADV\_WU} ({\em mom\_u\_adv\_wu.F})  
   
 {\em S/R MOM\_U\_ADV\_UV} ({\em mom\_u\_adv\_uv.F})  
   
 {\em S/R MOM\_U\_ADV\_VV} ({\em mom\_u\_adv\_vv.F})  
   
 {\em S/R MOM\_U\_ADV\_WV} ({\em mom\_u\_adv\_wv.F})  
   
 $uu$, $uv$, $vu$, $vv$: {\bf aF} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
   
 \subsubsection{Coriolis terms}  
   
 The ``pure C grid'' Coriolis terms (i.e. in absence of C-D scheme) are  
 discretized:  
 \begin{eqnarray}  
 {\cal A}_w \Delta r_f h_w G_u^{Cor} & = &  
   \overline{ f {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i  
 - \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ w }^k }^i \\  
 {\cal A}_s \Delta r_f h_s G_v^{Cor} & = &  
 - \overline{ f {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\  
 {\cal A}_c \Delta r_c G_w^{Cor} & = &  
  \epsilon_{nh} \overline{ f' {\cal A}_c \Delta r_f h_c \overline{ u }^i }^k  
 \end{eqnarray}  
 where the Coriolis parameters $f$ and $f'$ are defined:  
 \begin{eqnarray}  
 f & = & 2 \Omega \sin{\phi} \\  
 f' & = & 2 \Omega \cos{\phi}  
 \end{eqnarray}  
 when using spherical geometry, otherwise the $\beta$-plane definition is used:  
 \begin{eqnarray}  
 f & = & f_o + \beta y \\  
 f' & = & 0  
 \end{eqnarray}  
   
 This discretization globally conserves kinetic energy. It should be  
 noted that despite the use of this discretization in former  
 publications, all calculations to date have used the following  
 different discretization:  
 \begin{eqnarray}  
 G_u^{Cor} & = &  
   f_u \overline{ v }^{ji}  
 - \epsilon_{nh} f_u' \overline{ w }^{ik} \\  
 G_v^{Cor} & = &  
 - f_v \overline{ u }^{ij} \\  
 G_w^{Cor} & = &  
  \epsilon_{nh} f_w' \overline{ u }^{ik}  
 \end{eqnarray}  
 \marginpar{Need to change the default in code to match this}  
 where the subscripts on $f$ and $f'$ indicate evaluation of the  
 Coriolis parameters at the appropriate points in space. The above  
 discretization does {\em not} conserve anything, especially energy. An  
 option to recover this discretization has been retained for backward  
 compatibility testing (set run-time logical {\bf  
 useNonconservingCoriolis} to {\em true} which otherwise defaults to  
 {\em false}).  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F})  
   
 {\em S/R MOM\_U\_CORIOLIS} ({\em mom\_u\_coriolis.F})  
   
 {\em S/R MOM\_V\_CORIOLIS} ({\em mom\_v\_coriolis.F})  
   
 $G_u^{Cor}$, $G_v^{Cor}$: {\bf cF} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Curvature metric terms}  
   
 The most commonly used coordinate system on the sphere is the  
 geographic system $(\lambda,\phi)$. The curvilinear nature of these  
 coordinates on the sphere lead to some ``metric'' terms in the  
 component momentum equations. Under the thin-atmosphere and  
 hydrostatic approximations these terms are discretized:  
 \begin{eqnarray}  
 {\cal A}_w \Delta r_f h_w G_u^{metric} & = &  
   \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ v }^j }^i \\  
 {\cal A}_s \Delta r_f h_s G_v^{metric} & = &  
 - \overline{ \frac{ \overline{u}^i }{a} \tan{\phi} {\cal A}_c \Delta r_f h_c \overline{ u }^i }^j \\  
 G_w^{metric} & = & 0  
 \end{eqnarray}  
 where $a$ is the radius of the planet (sphericity is assumed) or the  
 radial distance of the particle (i.e. a function of height).  It is  
 easy to see that this discretization satisfies all the properties of  
 the discrete Coriolis terms since the metric factor $\frac{u}{a}  
 \tan{\phi}$ can be viewed as a modification of the vertical Coriolis  
 parameter: $f \rightarrow f+\frac{u}{a} \tan{\phi}$.  
   
 However, as for the Coriolis terms, a non-energy conserving form has  
 exclusively been used to date:  
 \begin{eqnarray}  
 G_u^{metric} & = & \frac{u \overline{v}^{ij} }{a} \tan{\phi} \\  
 G_v^{metric} & = & \frac{ \overline{u}^{ij} \overline{u}^{ij}}{a} \tan{\phi}  
 \end{eqnarray}  
 where $\tan{\phi}$ is evaluated at the $u$ and $v$ points  
 respectively.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_metric\_sphere.F})  
   
 {\em S/R MOM\_V\_METRIC\_SPHERE} ({\em mom\_v\_metric\_sphere.F})  
   
 $G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
   
 \subsubsection{Non-hydrostatic metric terms}  
   
 For the non-hydrostatic equations, dropping the thin-atmosphere  
 approximation re-introduces metric terms involving $w$ and are  
 required to conserve anglular momentum:  
 \begin{eqnarray}  
 {\cal A}_w \Delta r_f h_w G_u^{metric} & = &  
 - \overline{ \frac{ \overline{u}^i \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c }^i \\  
 {\cal A}_s \Delta r_f h_s G_v^{metric} & = &  
 - \overline{ \frac{ \overline{v}^j \overline{w}^k }{a} {\cal A}_c \Delta r_f h_c}^j \\  
 {\cal A}_c \Delta r_c G_w^{metric} & = &  
   \overline{ \frac{ {\overline{u}^i}^2 + {\overline{v}^j}^2}{a} {\cal A}_c \Delta r_f h_c }^k  
 \end{eqnarray}  
   
 Because we are always consistent, even if consistently wrong, we have,  
 in the past, used a different discretization in the model which is:  
 \begin{eqnarray}  
 G_u^{metric} & = &  
 - \frac{u}{a} \overline{w}^{ik} \\  
 G_v^{metric} & = &  
 - \frac{v}{a} \overline{w}^{jk} \\  
 G_w^{metric} & = &  
   \frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )  
 \end{eqnarray}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metric\_nh.F})  
   
 {\em S/R MOM\_V\_METRIC\_NH} ({\em mom\_v\_metric\_nh.F})  
   
 $G_u^{metric}$, $G_v^{metric}$: {\bf mT} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Lateral dissipation}  
   
 Historically, we have represented the SGS Reynolds stresses as simply  
 down gradient momentum fluxes, ignoring constraints on the stress  
 tensor such as symmetry.  
 \begin{eqnarray}  
 {\cal A}_w \Delta r_f h_w G_u^{h-diss} & = &  
   \delta_i  \Delta y_f \Delta r_f h_c \tau_{11}  
 + \delta_j  \Delta x_v \Delta r_f h_\zeta \tau_{12} \\  
 {\cal A}_s \Delta r_f h_s G_v^{h-diss} & = &  
   \delta_i  \Delta y_u \Delta r_f h_\zeta \tau_{21}  
 + \delta_j  \Delta x_f \Delta r_f h_c \tau_{22}  
 \end{eqnarray}  
 \marginpar{Check signs of stress definitions}  
   
 The lateral viscous stresses are discretized:  
 \begin{eqnarray}  
 \tau_{11} & = & A_h c_{11\Delta}(\phi) \frac{1}{\Delta x_f} \delta_i u  
                -A_4 c_{11\Delta^2}(\phi) \frac{1}{\Delta x_f} \delta_i \nabla^2 u \\  
 \tau_{12} & = & A_h c_{12\Delta}(\phi) \frac{1}{\Delta y_u} \delta_j u  
                -A_4 c_{12\Delta^2}(\phi)\frac{1}{\Delta y_u} \delta_j \nabla^2 u \\  
 \tau_{21} & = & A_h c_{21\Delta}(\phi) \frac{1}{\Delta x_v} \delta_i v  
                -A_4 c_{21\Delta^2}(\phi) \frac{1}{\Delta x_v} \delta_i \nabla^2 v \\  
 \tau_{22} & = & A_h c_{22\Delta}(\phi) \frac{1}{\Delta y_f} \delta_j v  
                -A_4 c_{22\Delta^2}(\phi) \frac{1}{\Delta y_f} \delta_j \nabla^2 v  
 \end{eqnarray}  
 where the non-dimensional factors $c_{lm\Delta^n}(\phi), \{l,m,n\} \in  
 \{1,2\}$ define the ``cosine'' scaling with latitude which can be  
 applied in various ad-hoc ways. For instance, $c_{11\Delta} =  
 c_{21\Delta} = (\cos{\phi})^{3/2}$, $c_{12\Delta}=c_{22\Delta}=0$ would  
 represent the an-isotropic cosine scaling typically used on the  
 ``lat-lon'' grid for Laplacian viscosity.  
 \marginpar{Need to tidy up method for controlling this in code}  
   
 It should be noted that dispite the ad-hoc nature of the scaling, some  
 scaling must be done since on a lat-lon grid the converging meridians  
 make it very unlikely that a stable viscosity parameter exists across  
 the entire model domain.  
   
 The Laplacian viscosity coefficient, $A_h$ ({\bf viscAh}), has units  
 of $m^2 s^{-1}$. The bi-harmonic viscosity coefficient, $A_4$ ({\bf  
 viscA4}), has units of $m^4 s^{-1}$.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_XVISCFLUX} ({\em mom\_u\_xviscflux.F})  
   
 {\em S/R MOM\_U\_YVISCFLUX} ({\em mom\_u\_yviscflux.F})  
   
 {\em S/R MOM\_V\_XVISCFLUX} ({\em mom\_v\_xviscflux.F})  
   
 {\em S/R MOM\_V\_YVISCFLUX} ({\em mom\_v\_yviscflux.F})  
   
 $\tau_{11}$, $\tau_{12}$, $\tau_{22}$, $\tau_{22}$: {\bf vF}, {\bf  
 v4F} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
 Two types of lateral boundary condition exist for the lateral viscous  
 terms, no-slip and free-slip.  
   
 The free-slip condition is most convenient to code since it is  
 equivalent to zero-stress on boundaries. Simple masking of the stress  
 components sets them to zero. The fractional open stress is properly  
 handled using the lopped cells.  
   
 The no-slip condition defines the normal gradient of a tangential flow  
 such that the flow is zero on the boundary. Rather than modify the  
 stresses by using complicated functions of the masks and ``ghost''  
 points (see \cite{Adcroft+Marshall98}) we add the boundary stresses as  
 an additional source term in cells next to solid boundaries. This has  
 the advantage of being able to cope with ``thin walls'' and also makes  
 the interior stress calculation (code) independent of the boundary  
 conditions. The ``body'' force takes the form:  
 \begin{eqnarray}  
 G_u^{side-drag} & = &  
 \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta x_v}{\Delta y_u} }^j  
 \left( A_h c_{12\Delta}(\phi) u - A_4 c_{12\Delta^2}(\phi) \nabla^2 u \right)  
 \\  
 G_v^{side-drag} & = &  
 \frac{4}{\Delta z_f} \overline{ (1-h_\zeta) \frac{\Delta y_u}{\Delta x_v} }^i  
 \left( A_h c_{21\Delta}(\phi) v - A_4 c_{21\Delta^2}(\phi) \nabla^2 v \right)  
 \end{eqnarray}  
   
 In fact, the above discretization is not quite complete because it  
 assumes that the bathymetry at velocity points is deeper than at  
 neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F})  
   
 {\em S/R MOM\_V\_SIDEDRAG} ({\em mom\_v\_sidedrag.F})  
   
 $G_u^{side-drag}$, $G_v^{side-drag}$: {\bf vF} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Vertical dissipation}  
   
 Vertical viscosity terms are discretized with only partial adherence  
 to the variable grid lengths introduced by the finite volume  
 formulation. This reduces the formal accuracy of these terms to just  
 first order but only next to boundaries; exactly where other terms  
 appear such as linar and quadratic bottom drag.  
 \begin{eqnarray}  
 G_u^{v-diss} & = &  
 \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\  
 G_v^{v-diss} & = &  
 \frac{1}{\Delta r_f h_s} \delta_k \tau_{23} \\  
 G_w^{v-diss} & = & \epsilon_{nh}  
 \frac{1}{\Delta r_f h_d} \delta_k \tau_{33}  
 \end{eqnarray}  
 represents the general discrete form of the vertical dissipation terms.  
   
 In the interior the vertical stresses are discretized:  
 \begin{eqnarray}  
 \tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\  
 \tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v \\  
 \tau_{33} & = & A_v \frac{1}{\Delta r_f} \delta_k w  
 \end{eqnarray}  
 It should be noted that in the non-hydrostatic form, the stress tensor  
 is even less consistent than for the hydrostatic (see Wazjowicz  
 \cite{Waojz}). It is well known how to do this properly (see Griffies  
 \cite{Griffies}) and is on the list of to-do's.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})  
   
 {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})  
   
 $\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})  
   
 $\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 As for the lateral viscous terms, the free-slip condition is  
 equivalent to simply setting the stress to zero on boundaries.  The  
 no-slip condition is implemented as an additional term acting on top  
 of the interior and free-slip stresses. Bottom drag represents  
 additional friction, in addition to that imposed by the no-slip  
 condition at the bottom. The drag is cast as a stress expressed as a  
 linear or quadratic function of the mean flow in the layer above the  
 topography:  
 \begin{eqnarray}  
 \tau_{13}^{bottom-drag} & = &  
 \left(  
 2 A_v \frac{1}{\Delta r_c}  
 + r_b  
 + C_d \sqrt{ \overline{2 KE}^i }  
 \right) u \\  
 \tau_{23}^{bottom-drag} & = &  
 \left(  
 2 A_v \frac{1}{\Delta r_c}  
 + r_b  
 + C_d \sqrt{ \overline{2 KE}^j }  
 \right) v  
 \end{eqnarray}  
 where these terms are only evaluated immediately above topography.  
 $r_b$ ({\bf bottomDragLinear}) has units of $m s^{-1}$ and a typical value  
 of the order 0.0002 $m s^{-1}$. $C_d$ ({\bf bottomDragQuadratic}) is  
 dimensionless with typical values in the range 0.001--0.003.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_BOTTOMDRAG} ({\em mom\_u\_bottomdrag.F})  
   
 {\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F})  
   
 $\tau_{13}^{bottom-drag}$, $\tau_{23}^{bottom-drag}$: {\bf vf} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
   
   
   
 \subsection{Tracer equations}  
   
 The tracer equations are discretized consistantly with the continuity  
 equation to facilitate conservation properties analogous to the  
 continuum:  
 \begin{equation}  
 {\cal A}_c \Delta r_f h_c \partial_\theta  
 + \delta_i U \overline{ \theta }^i  
 + \delta_j V \overline{ \theta }^j  
 + \delta_k W \overline{ \theta }^k  
 = {\cal A}_c \Delta r_f h_c {\cal S}_\theta + \theta {\cal A}_c \delta_k (P-E)_{r=0}  
 \end{equation}  
 The quantities $U$, $V$ and $W$ are volume fluxes defined:  
 \marginpar{$U$: {\bf uTrans} }  
 \marginpar{$V$: {\bf vTrans} }  
 \marginpar{$W$: {\bf rTrans} }  
 \begin{eqnarray}  
 U & = & \Delta y_g \Delta r_f h_w u \\  
 V & = & \Delta x_g \Delta r_f h_s v \\  
 W & = & {\cal A}_c w  
 \end{eqnarray}  
 ${\cal S}$ represents the ``parameterized'' SGS processes and  
 physics associated with the tracer. For instance, potential  
 temperature equation in the ocean has is forced by surface and  
 partially penetrating heat fluxes:  
 \begin{equation}  
 {\cal A}_c \Delta r_f h_c {\cal S}_\theta = \frac{1}{c_p \rho_o} \delta_k {\cal A}_c {\cal Q}  
 \end{equation}  
 while the salt equation has no real sources, ${\cal S}=0$, which  
 leaves just the $P-E$ term.  
   
 The continuity equation can be recovered by setting ${\cal Q}=0$ and  
 $\theta=1$. The term $\theta (P-E)_{r=0}$ is required to retain local  
 conservation of $\theta$. Global conservation is not possible using  
 the flux-form (as here) and a linearized free-surface  
 (\cite{Griffies00,Campin02}).  
   
   
   
   
 \subsection{Derivation of discrete energy conservation}  
   
 These discrete equations conserve kinetic plus potential energy using the  
 following definitions:  
 \begin{equation}  
 KE = \frac{1}{2} \left( \overline{ u^2 }^i + \overline{ v^2 }^j +  
 \epsilon_{nh} \overline{ w^2 }^k \right)  
 \end{equation}  
   
   
 \subsection{Vector invariant momentum equations}  
   
 The finite volume method lends itself to describing the continuity and  
 tracer equations in curvilinear coordinate systems but the appearance  
 of new metric terms in the flux-form momentum equations makes  
 generalizing them far from elegant. The vector invariant form of the  
 momentum equations are exactly that; invariant under coordinate  
 transformations.  
   
 The non-hydrostatic vector invariant equations read:  
 \begin{equation}  
 \partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v}  
 - b \hat{r}  
 + \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau}  
 \end{equation}  
 which describe motions in any orthogonal curvilinear coordinate  
 system. Here, $B$ is the Bernoulli function and $\vec{\zeta}=\nabla  
 \wedge \vec{v}$ is the vorticity vector. We can take advantage of the  
 elegance of these equations when discretizing them and use the  
 discrete definitions of the grad, curl and divergence operators to  
 satisfy constraints. We can also consider the analogy to forming  
 derived equations, such as the vorticity equation, and examine how the  
 discretization can be adjusted to give suitable vorticity advection  
 among other things.  
   
 The underlying algorithm is the same as for the flux form  
 equations. All that has changed is the contents of the ``G's''. For  
 the time-being, only the hydrostatic terms have been coded but we will  
 indicate the points where non-hydrostatic contributions will enter:  
 \begin{eqnarray}  
 G_u & = & G_u^{fv} + G_u^{\zeta_3 v} + G_u^{\zeta_2 w} + G_u^{\partial_x B}  
 + G_u^{\partial_z \tau^x} + G_u^{h-dissip} + G_u^{v-dissip} \\  
 G_v & = & G_v^{fu} + G_v^{\zeta_3 u} + G_v^{\zeta_1 w} + G_v^{\partial_y B}  
 + G_v^{\partial_z \tau^y} + G_v^{h-dissip} + G_v^{v-dissip} \\  
 G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G_w^{\zeta_2 u} + G_w^{\partial_z B}  
 + G_w^{h-dissip} + G_w^{v-dissip}  
 \end{eqnarray}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/calc\_mom\_rhs.F})  
   
 $G_u$: {\bf Gu} ({\em DYNVARS.h})  
   
 $G_v$: {\bf Gv} ({\em DYNVARS.h})  
   
 $G_w$: {\bf Gw} ({\em DYNVARS.h})  
 \end{minipage} }  
   
 \subsubsection{Relative vorticity}  
   
 The vertical component of relative vorticity is explicitly calculated  
 and use in the discretization. The particular form is crucial for  
 numerical stablility; alternative definitions break the conservation  
 properties of the discrete equations.  
   
 Relative vorticity is defined:  
 \begin{equation}  
 \zeta_3 = \frac{\Gamma}{A_\zeta}  
 = \frac{1}{{\cal A}_\zeta} ( \delta_i \Delta y_c v - \delta_j \Delta x_c u )  
 \end{equation}  
 where ${\cal A}_\zeta$ is the area of the vorticity cell presented in  
 the vertical and $\Gamma$ is the circulation about that cell.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})  
   
 $\zeta_3$: {\bf vort3} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Kinetic energy}  
   
 The kinetic energy, denoted $KE$, is defined:  
 \begin{equation}  
 KE = \frac{1}{2} ( \overline{ u^2 }^i + \overline{ v^2 }^j  
 + \epsilon_{nh} \overline{ w^2 }^k )  
 \end{equation}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})  
   
 $KE$: {\bf KE} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Coriolis terms}  
   
 The potential enstrophy conserving form of the linear Coriolis terms  
 are written:  
 \begin{eqnarray}  
 G_u^{fv} & = &  
 \frac{1}{\Delta x_c}  
 \overline{ \frac{f}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\  
 G_v^{fu} & = & -  
 \frac{1}{\Delta y_c}  
 \overline{ \frac{f}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j  
 \end{eqnarray}  
 Here, the Coriolis parameter $f$ is defined at vorticity (corner)  
 points.  
 \marginpar{$f$: {\bf fCoriG}}  
 \marginpar{$h_\zeta$: {\bf hFacZ}}  
   
 The potential enstrophy conserving form of the non-linear Coriolis  
 terms are written:  
 \begin{eqnarray}  
 G_u^{\zeta_3 v} & = &  
 \frac{1}{\Delta x_c}  
 \overline{ \frac{\zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\  
 G_v^{\zeta_3 u} & = & -  
 \frac{1}{\Delta y_c}  
 \overline{ \frac{\zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j  
 \end{eqnarray}  
 \marginpar{$\zeta_3$: {\bf vort3}}  
   
 The Coriolis terms can also be evaluated together and expressed in  
 terms of absolute vorticity $f+\zeta_3$. The potential enstrophy  
 conserving form using the absolute vorticity is written:  
 \begin{eqnarray}  
 G_u^{fv} + G_u^{\zeta_3 v} & = &  
 \frac{1}{\Delta x_c}  
 \overline{ \frac{f + \zeta_3}{h_\zeta} }^j \overline{ \overline{ \Delta x_g h_s v }^j }^i \\  
 G_v^{fu} + G_v^{\zeta_3 u} & = & -  
 \frac{1}{\Delta y_c}  
 \overline{ \frac{f + \zeta_3}{h_\zeta} }^i \overline{ \overline{ \Delta y_g h_w u }^i }^j  
 \end{eqnarray}  
   
 \marginpar{Run-time control needs to be added for these options} The  
 disctinction between using absolute vorticity or relative vorticity is  
 useful when constructing higher order advection schemes; monotone  
 advection of relative vorticity behaves differently to monotone  
 advection of absolute vorticity. Currently the choice of  
 relative/absolute vorticity, centered/upwind/high order advection is  
 available only through commented subroutine calls.  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_CORIOLIS} ({\em mom\_vi\_coriolis.F})  
   
 {\em S/R MOM\_VI\_U\_CORIOLIS} ({\em mom\_vi\_u\_coriolis.F})  
   
 {\em S/R MOM\_VI\_V\_CORIOLIS} ({\em mom\_vi\_v\_coriolis.F})  
   
 $G_u^{fv}$, $G_u^{\zeta_3 v}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})  
   
 $G_v^{fu}$, $G_v^{\zeta_3 u}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Shear terms}  
   
 The shear terms ($\zeta_2w$ and $\zeta_1w$) are are discretized to  
 guarantee that no spurious generation of kinetic energy is possible;  
 the horizontal gradient of Bernoulli function has to be consistent  
 with the vertical advection of shear:  
 \marginpar{N-H terms have not been tried!}  
 \begin{eqnarray}  
 G_u^{\zeta_2 w} & = &  
 \frac{1}{ {\cal A}_w \Delta r_f h_w } \overline{  
 \overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )  
 }^k \\  
 G_v^{\zeta_1 w} & = &  
 \frac{1}{ {\cal A}_s \Delta r_f h_s } \overline{  
 \overline{ {\cal A}_c w }^i ( \delta_k u - \epsilon_{nh} \delta_j w )  
 }^k  
 \end{eqnarray}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_U\_VERTSHEAR} ({\em mom\_vi\_u\_vertshear.F})  
   
 {\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})  
   
 $G_u^{\zeta_2 w}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})  
   
 $G_v^{\zeta_1 w}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
   
 \subsubsection{Gradient of Bernoulli function}  
   
 \begin{eqnarray}  
 G_u^{\partial_x B} & = &  
 \frac{1}{\Delta x_c} \delta_i ( \phi' + KE ) \\  
 G_v^{\partial_y B} & = &  
 \frac{1}{\Delta x_y} \delta_j ( \phi' + KE )  
 %G_w^{\partial_z B} & = &  
 %\frac{1}{\Delta r_c} h_c \delta_k ( \phi' + KE )  
 \end{eqnarray}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_U\_GRAD\_KE} ({\em mom\_vi\_u\_grad\_ke.F})  
   
 {\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})  
   
 $G_u^{\partial_x KE}$: {\bf uCf} (local to {\em calc\_mom\_rhs.F})  
   
 $G_v^{\partial_y KE}$: {\bf vCf} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
   
 \subsubsection{Horizontal dissipation}  
   
 The horizontal divergence, a complimentary quantity to relative  
 vorticity, is used in parameterizing the Reynolds stresses and is  
 discretized:  
 \begin{equation}  
 D = \frac{1}{{\cal A}_c h_c} (  
   \delta_i \Delta y_g h_w u  
 + \delta_j \Delta x_g h_s v )  
 \end{equation}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})  
   
 $D$: {\bf hDiv} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Horizontal dissipation}  
   
 The following discretization of horizontal dissipation conserves  
 potential vorticity (thickness weighted relative vorticity) and  
 divergence and dissipates energy, enstrophy and divergence squared:  
 \begin{eqnarray}  
 G_u^{h-dissip} & = &  
   \frac{1}{\Delta x_c} \delta_i ( A_D D - A_{D4} D^*)  
 - \frac{1}{\Delta y_u h_w} \delta_j h_\zeta ( A_\zeta \zeta - A_{\zeta4} \zeta^* )  
 \\  
 G_v^{h-dissip} & = &  
   \frac{1}{\Delta x_v h_s} \delta_i h_\zeta ( A_\zeta \zeta - A_\zeta \zeta^* )  
 + \frac{1}{\Delta y_c} \delta_j ( A_D D - A_{D4} D^* )  
 \end{eqnarray}  
 where  
 \begin{eqnarray}  
 D^* & = & \frac{1}{{\cal A}_c h_c} (  
   \delta_i \Delta y_g h_w \nabla^2 u  
 + \delta_j \Delta x_g h_s \nabla^2 v ) \\  
 \zeta^* & = & \frac{1}{{\cal A}_\zeta} (  
   \delta_i \Delta y_c \nabla^2 v  
 - \delta_j \Delta x_c \nabla^2 u )  
 \end{eqnarray}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})  
   
 $G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F})  
   
 $G_v^{h-dissip}$: {\bf vDiss} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
   
   
 \subsubsection{Vertical dissipation}  
   
 Currently, this is exactly the same code as the flux form equations.  
 \begin{eqnarray}  
 G_u^{v-diss} & = &  
 \frac{1}{\Delta r_f h_w} \delta_k \tau_{13} \\  
 G_v^{v-diss} & = &  
 \frac{1}{\Delta r_f h_s} \delta_k \tau_{23}  
 \end{eqnarray}  
 represents the general discrete form of the vertical dissipation terms.  
   
 In the interior the vertical stresses are discretized:  
 \begin{eqnarray}  
 \tau_{13} & = & A_v \frac{1}{\Delta r_c} \delta_k u \\  
 \tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v  
 \end{eqnarray}  
   
 \fbox{ \begin{minipage}{4.75in}  
 {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})  
   
 {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})  
   
 $\tau_{13}$: {\bf urf} (local to {\em calc\_mom\_rhs.F})  
   
 $\tau_{23}$: {\bf vrf} (local to {\em calc\_mom\_rhs.F})  
 \end{minipage} }  
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