Gridstructure in the UKMO UM DC Gary Dietachmayer FAME Meeting, Oct5 2006

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Grid-structure in the UKMO UM DCGary
DietachmayerFAME Meeting, Oct-5 2006
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Sources
http//r2/access/docs/pum6.0doc/UM_docs/papers/pd
f/p015.pdf (6.0)
http//www.metoffice.com/research/nwp/publications
/papers/unified_model/index.html (6.1), and now
(6.3)
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Overview

• Governing equations, prognostic variables,
  coordinate system
• Coordinate transformations rotated pole and
  general eta coordinate
• Grid-structure, boundary locations
• Examples of vertical grids

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Governing Equations / Base Coordinates
Start with full Navier-Stokes equations, in
spherical coords
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Governing Equations / Base Coordinates
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Governing Equations / Base Coordinates
Key variables for grid structure u, v, w, ?
(pressure)
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Coordinate Transformations Rotated Pole
For limited-area models, rotating the coordinate
pole achieves two things

• Moves the pole singularity out of the
  computational domain
• Allows us to minimise the variation in
  longitudinal resolution across the domain

Is conceptually simple only preferred-direction
in the system is Coriolis force
Moving the pole alone does not define the
transformation completely
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Coordinate Transformations Rotated Pole
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Coordinate Transformations Rotated Pole
Ie., no change in structure, but more complex f.
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Coordinate Transformations Eta coord
Too hard to deal with non-cube computational
domains, seek vertical coordinate transformation
that flattens surface topography.
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Coordinate Transformations Eta coord
Transform from r to eta-space in standard way
As occurs in LAPS, d?/dt replaces w in advective
and flux calculations, and the single pressure
gradient terms split into two
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Coordinate Transformations Eta coord
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Grid-staggering the 1D SW equations
Non-staggered grid
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Grid-staggering the 1D SW equations
Staggered grid
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Grid-staggering pros and cons
Pros generally considered overwhelming, LAPS is
an exception to the rule.

• More compact differencing more accuracy
• More compact differencing no grid-decoupling
• Allows natural implementation of conservative
  schemes

• More operations per grid point slower,
  particularly for SL
• More difficult to implement higher-order scheme

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Grid-staggering Arakawa C-grid
Rule Take pressure (?) as central point,
displace all velocity components half a
grid-space in their component direction (eg., u
is displaced half a grid-space in longitude). In
pseudo 3D
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Grid-staggering Arakawa C-grid
Looking down on the grid (longitude/latitude
plane)
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Grid-staggering Arakawa C-grid
Maths to Fortran translation in UM round
half-levels up to integer values in the actual
code
Location of other variables gt
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Grid-staggering North and South Poles
Poles coincide with half-integral j indices
(j1/2, j M 1/2)
At poles, all values of ?, ?, ?, m, and w are all
set equal. Pole-u values are diagnosed from the
surrounding v values.
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Grid-staggering Limited-area domains
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Grid-staggering Charney-Phillips grid
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Grid-staggering Vertical BCs
Note that diagnosing w at the ground requires
knowledge of u and v at the ground.
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Examples of vertical grids Linear
Simplest possible coordinate, very similar to
hydrostatic sigma coords. Like sigma,
eta-surfaces maintain their dependence on surface
topography, even in the upper parts of the domain
dont flatten out as fast as wed like. (See
hybrid coords for hydrostatic models.)
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Examples of vertical grids Blended
Linear/Quadratic
Split domain up into lower and upper parts,
separated by the interface r R_i on eta Eta_I.
Lower domain (quadratic)
Upper domain (linear)
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Examples of vertical grids Blended
Linear/Quadratic
Determine unknown A by matching vertical
resolution (dr/d?) across the interface.
Upper/linear Lower/Quadratic
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Examples of vertical grids QUADn levels
The blended linear-quadratic grid can be
simplified considerably by the choice of a
particular eta interface value
In the 6.1 documentation, this is used in the
operational system. Referred to as QUADn levels
where n I is the level number of the interface
surface r R_i.