Map Factors
The dynamic equations can be expressed using a generalized map factor,
,
which is defined as the ratio of the model grid distance to the true
distance on the Earth's surface.
Map factors are used
conjunction with the horizontal derivative terms within the dynamics equations.
In this model,
they can be used for conformal Lambert or Mercator projections (usual limited
area domain applications) or for a latitude-longitude (Plate Carrée)
projection (global scale applications). In
this latter case, the horizontal component is stretched so that the map
factor must be considered separately for the two horizontal
directions (
and
). See for example, Haltiner and Williams (1980)
and Richardson et al. (2007)
for detailed discussions of map factors.
The the actual distance along a grid cell wall is defined
as
(the analogous expression applies for
), where
is
a constant grid cell distance (usually
at the equator for
the Plate Carrée projection, or at the center of a conformal projection).
The mass fluxes across the grid faces
can then be expressed using map factors as
Again note that for Cartesian coordinates (no projection),
, so that Eq.s 286-287
are equivalent to Eq.s 124-125.
The continuity equation (Eq. 28)
neglecting sources
can be expanded using a general map factor as
Next, the prognostic flux-form equation for scalars
(Eq. 48), momentum
(Eq.s 30-32),
and geopotential (Eq. 35),
again
neglecting diabatic terms,
can be expanded, respectively, using
a general map factor as
where
represents a scalar (passive tracer, TKE,
temperature, cloud water, etc...).
Note that the horizontal diffusion terms,
,
include map factors. They are expressed as
 |
(292) |
where
corresponds to the quantity being diffused
and
represents the order of the diffusion operator, and
represents the corresponding diffusivity.
In addition to horizontal and vertical diffusion, the
and
momentum equations also include divergence damping terms
(Eq. 77, for more details see Section 2.2.1).
The default damping is fourth order, and as an example,
it is given for the
-momentum
equation by
![$\displaystyle D_{u\,d}^4 = - {\frac {\kappa_{d4\,u}}{\mu}}
{\frac{\partial^3 }{...
...eft[ {\frac{\partial }{\partial x}}
\left(\nabla \cdot{\bf V}\mu\right) \right]$](img686.svg) |
(293) |
thus it amounts to taking the second order
derivative of the gradient of the horizontal divergence.
The divergence (defined at mass points)
is expressed in finite difference form by Eq. 126,
where map factors
are introduced using the definitions in
Eq.s 286-287 so that
 |
(294) |
where
is the reference grid distance (m).
The gradient of the divergence
in the
-direction as
 |
(295) |
where
represents the divergence gradient in the
-direction and is
centered on the
points of the C-grid.
Next, the second order diffusion of the divergence gradient
in the
-direction is
computed as
![$\displaystyle {\frac{\partial^2 \varphi}{\partial x^2}}\Big\vert_{i,j} =
{\frac...
...ac{m_{x\,i-1,j}}{d}} \left( {\varphi}_{i,j} - {\varphi}_{i-1,j} \right)
\right]$](img689.svg) |
(296) |
Finally, the fourth order damping coefficient
(defined in Eq. 85)
has a spatial dependence owing to the
map factor and is expressed for the
momentum equation as
 |
(297) |
where
must be less than 1 for numerical stability.
The value
is selected to optimize the balance between effectively
filtering high frequency
noise and avoiding too much damping of the divergence
(see Section 2.2.1
for a detailed discussion).
The same form is used in the
-direction for the damping in
the
momentum equation.
Finally, for the lat-lon grid, the map factors are defined as
where
represents the radius of the Earth,
represents the latitude in radians,
and
and
are the constant
latitude and longitude grid thicknesses, respectively, in radians.
Thus,
varies strongly as a function of latitude.