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4f2617d475 Jeff*0001
0002 Vertical grid
0003 -------------
0004
0005 .. figure:: figs/vgrid-accur-center.*
0006 :width: 60%
0007 :align: center
0008 :alt: vgrid-accur-center
0009 :name: vgrid-accur-center
0010
9c8516d9da Jeff*0011 Two versions of the vertical grid. a) The cell centered approach where the interface depths are specified
0012 and the tracer points centered in between the interfaces. b) The interface centered approach where tracer
0013 levels are specified and the w-interfaces are centered in between.
4f2617d475 Jeff*0014
0015 As for the horizontal grid, we use the suffixes “c” and “f” to indicates
0016 faces and centers. :numref:`vgrid-accur-center` (a) shows the default vertical grid
0017 used by the model. :math:`\Delta r_f` is the difference in :math:`r`
0018 (vertical coordinate) between the faces (i.e. :math:`\Delta r_f \equiv -
0019 \delta_k r` where the minus sign appears due to the convention that the
0020 surface layer has index :math:`k=1`.).
0021
0022 The vertical grid is calculated in subroutine :filelink:`INI_VERTICAL_GRID <model/src/ini_vertical_grid.F>` and
0023 specified via the vector :varlink:`delR` in namelist ``PARM04``. The units of “r”
0024 are either meters or Pascals depending on the isomorphism being used
0025 which in turn is dependent only on the choice of equation of state.
0026
0027 There are alternative namelist vectors :varlink:`delZ` and :varlink:`delP` which
0028 dictate whether z- or p- coordinates are to be used but we intend to
0029 phase this out since they are redundant.
0030
0031 The reciprocals :math:`\Delta r_f^{-1}` and :math:`\Delta r_c^{-1}` are
0032 pre-calculated (also in subroutine :filelink:`INI_VERTICAL_GRID <model/src/ini_vertical_grid.F>`). All vertical
0033 grid descriptors are stored in common blocks in :filelink:`GRID.h <model/inc/GRID.h>`.
0034
0035 The above grid :numref:`vgrid-accur-center` (a) is known as the cell centered
0036 approach because the tracer points are at cell centers; the cell centers
0037 are mid-way between the cell interfaces. This discretization is selected
0038 when the thickness of the levels are provided (:varlink:`delR`, parameter file
0039 ``data``, namelist ``PARM04``) An alternative, the vertex or interface
0040 centered approach, is shown in :numref:`vgrid-accur-center` (b). Here, the interior
0041 interfaces are positioned mid-way between the tracer nodes (no longer
0042 cell centers). This approach is formally more accurate for evaluation of
0043 hydrostatic pressure and vertical advection but historically the cell
0044 centered approach has been used. An alternative form of subroutine
0045 :filelink:`INI_VERTICAL_GRID <model/src/ini_vertical_grid.F>` is used to select the interface centered approach
0046 This form requires to specify :math:`Nr+1` vertical distances :varlink:`delRc`
0047 (parameter file ``data``, namelist ``PARM04``, e.g.
0048 :filelink:`ideal\_2D\_oce/input/data <verification/ideal_2D_oce/input/data>`) corresponding to surface to
0049 center, :math:`Nr-1` center to center, and center to bottom distances.
0050
0051 .. admonition:: S/R :filelink:`INI_VERTICAL_GRID <model/src/ini_vertical_grid.F>`
0052 :class: note
0053
0054 | :math:`\Delta r_f , \Delta r_c` : :varlink:`drF`, :varlink:`drC` ( :filelink:`GRID.h <model/inc/GRID.h>` )
0055 | :math:`\Delta r_f^{-1} , \Delta r_c^{-1}` : :varlink:`recip_drF`, :varlink:`recip_drC` ( :filelink:`GRID.h <model/inc/GRID.h>` )
0056
9c8516d9da Jeff*0057 .. _sec_topo_partial_cells:
4f2617d475 Jeff*0058
0059 Topography: partially filled cells
0060 ----------------------------------
0061
0062 Adcroft et al. (1997) :cite:`adcroft:97` presented two alternatives to the
0063 step-wise finite difference representation of topography. The method is
0064 known to the engineering community as *intersecting boundary method*. It
0065 involves allowing the boundary to intersect a grid of cells thereby
0066 modifying the shape of those cells intersected. We suggested allowing
0067 the topography to take on a piece-wise linear representation (shaved
0068 cells) or a simpler piecewise constant representation (partial step).
0069 Both show dramatic improvements in solution compared to the traditional
0070 full step representation, the piece-wise linear being the best. However,
0071 the storage requirements are excessive so the simpler piece-wise
0072 constant or partial-step method is all that is currently supported.
0073
0074 .. figure:: figs/vgrid-xz.*
0075 :width: 60%
0076 :align: center
0077 :alt: vgrid-xz
0078 :name: vgrid-xz
0079
0080 A schematic of the x-r plane showing the location of the non-dimensional fractions :math:`h_c` and :math:`h_w` . The physical thickness of a tracer cell is given by :math:`h_c(i,j,k) \Delta r_f(k)` and the physical thickness of the open side is given by :math:`h_w(i,j,k) \Delta r_f(k)` .
0081
0082 :numref:`vgrid-xz` shows a schematic of the x-r plane indicating how the
0083 thickness of a level is determined at tracer and u points. The physical
0084 thickness of a tracer cell is given by :math:`h_c(i,j,k) \Delta
0085 r_f(k)` and the physical thickness of the open side is given by
0086 :math:`h_w(i,j,k) \Delta r_f(k)`. Three 3-D descriptors :math:`h_c`,
0087 :math:`h_w` and :math:`h_s` are used to describe the geometry:
0088 :varlink:`hFacC`, :varlink:`hFacW` and :varlink:`hFacS` respectively. These are calculated in
0089 subroutine :filelink:`INI_MASKS_ETC <model/src/ini_masks_etc.F>` along with there reciprocals
0090 :varlink:`recip_hFacC`, :varlink:`recip_hFacW` and :varlink:`recip_hFacS`.
0091
0092 The non-dimensional fractions (or h-facs as we call them) are calculated
0093 from the model depth array and then processed to avoid tiny volumes. The
0094 rule is that if a fraction is less than :varlink:`hFacMin` then it is rounded
0095 to the nearer of :math:`0` or :varlink:`hFacMin` or if the physical thickness
0096 is less than :varlink:`hFacMinDr` then it is similarly rounded. The larger of
0097 the two methods is used when there is a conflict. By setting
0098 :varlink:`hFacMinDr` equal to or larger than the thinnest nominal layers,
0099 :math:`\min{(\Delta z_f)}`, but setting :varlink:`hFacMin` to some small
0100 fraction then the model will only lop thick layers but retain stability
0101 based on the thinnest unlopped thickness;
0102 :math:`\min{(\Delta z_f,hFacMinDr)}`.
0103
0104 .. admonition:: S/R :filelink::filelink:`INI_MASKS_ETC <model/src/ini_masks_etc.F>`
0105 :class: note
0106
0107 | :math:`h_c , h_w , h_s` : :varlink:`hFacC`, :varlink:`hFacW`, :varlink:`hFacS` ( :filelink:`GRID.h <model/inc/GRID.h>` )
0108 | :math:`h_c^{-1} , h_w^{-1} , h_s^{-1}` : :varlink:`recip_hFacC`, :varlink:`recip_hFacW`, :varlink:`recip_hFacS` ( :filelink:`GRID.h <model/inc/GRID.h>` )
0109
0110
