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Chapter 4. Subgrid-scale orographic
drag
IFS documentation Front Page
Chapter 1. Overview
Chapter 2. Radiation
Chapter 3. Turbulent diffusion and interactions with the surface
Chapter 4. Subgrid-scale orographic drag
Chapter 5. Convection
Chapter 6. Clouds and large-scale precipitation
Chapter 7. Land suface parametrization
Chapter 8. Methane oxidation
Chapter 9. Climatological data
REFERENCES
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4.2 Description of the scheme
Following Baines and Palmer (1990), the subgrid-scale orography over one grid-point region is represented by four parameters
, 
, 
and 
which stand for the standard deviation, the anisotropy, the slope
and the geographical orientation of the orography, respectively. These four
parameters have been calculated from the US Navy (USN) (
) data-set.The scheme uses values of low-level wind velocity and static stability which are partitioned into two parts. The first part corresponds to the incident flow which passes over the mountain top, and is evaluated by averaging the wind, the Brunt-Väisälä frequency and the fluid density between
and 
above the model mean orography. Following
Wallace et al. (1983), 
is interpreted as the envelope of the subgrid-scale mountain peaks
above the model orography. The wind, the Brunt-Väisälä frequency
and the density of this part of the low-level flow will be labelled 
, 
and 
, respectively. The second part is the `blocked' flow, and its evaluation
is based on a very simple interpretation of the non-dimensional mountain
height 
. To first order in the mountain amplitude, the obstacle excites
a wave, and the sign of the vertical displacement of a fluid parcel is controlled
by the wave phase. If a fluid parcel ascends the upstream mountain flank
over a height large enough to significantly modify the wave phase, its vertical
displacement can become zero, and it will not cross the mountain summit.
In this case the blocking height, 
, is the highest level located below the mountain top for which the
phase change between 
and the mountain top exceeds a critical value 
, i.e.
|
(4.9) |
In the inequality (4.9), the wind speed,
, is calculated by resolving the wind,
, in the direction of the flow 
. Then, if the flow veers or backs with height, (4.9)
will be satisfied when the flow becomes normal to 
. Levels below this `critical' altitude define the low-level blocked
flow. The inequality (4.9) will also be satisfied below inversion
layers, where the parameter 
is very large. These two properties allow the new parametrization
scheme to mimic the vortex shedding observed when pronounced inversions
occur (Etling 1989). The upper limit in the
equality (4.9) was chosen
to be 
, which is above the subgrid-scale mountain
tops. This ensures that the integration in equality (4.9)
does not lead to an underestimation of 
, which can occur because of the limited vertical resolution when
using 
as an upper limit (a better representation
of the peak height), but this upper limit could be relaxed given better
vertical resolution.In the following subsection the drag amplitudes will be estimated combining formulae valid for elliptical mountains with real orographic data. Considerable simplifications are implied and the calculations are, virtually, scale analyses relating the various amplitudes to the sub-grid parameters.
4.2.1 Blocked-flow drag
Within a given layer located below the blocking level
, the drag is given by Eq. (4.5). At a given altitude 
, the intersection between the mountain and the layer
approximates to an ellipse of eccentricity
 ,
|
(4.10) |
where, by comparison with Eq. (4.6), it is also supposed that the level
(i.e. the model mean orography) is at
an altitude 
above the mountain valleys. If the flow direction is taken into account,
the length 
can be written approximately as
|
(4.11) |
where
is the angle between the incident flow direction and the
normal ridge direction, 
, For one grid-point region and for uniformly distributed subgrid-scale
orography, the incident flow encounters 
obstacles is normal to the ridge 
, whereas if it is parallel to the ridge 
it encounters 
obstacles, where 
is the length scale of the grid-point region. If we sum up these
contributions, the dependence of Eq. (4.11) on 
and 
can be neglected, and the length 
becomes
 .
|
(4.12) |
Furthermore, the number of consecutive ridges (i.e. located one after the other in the direction of the flow) depends on the obstacle shape: there are approximately
successive obstacles when the flow is along the ridge, and 
when it is normal to the ridge. If we take this into account,
together with the flow direction, then
 .
|
(4.13) |
Relating the parameters
and 
to the subgrid-scale orography parameters 
and 
and, allowing the drag coefficient to vary
with the aspect ratio of the obstacle as seen by the incident flow, we have
 ,
|
(4.14) |
and the drag per unit area and per unit height can be written
 .
|
(4.15) |
The drag coefficient is modulated by the aspect ratio of the obstacle to account for the fact that
is twice as large for flow normal to an elongated obstacle as it
is for flow round an isotropic obstacle. The drag tends to zero when the
flow is nearly along a long ridge because flow separation is not expected
to occur for a configuration of that kind. It can be shown that the term
is similar to a later form used for the directional dependence of
the gravity-wave stress. For simplicity, this later form has been adopted,
i.e.
|
(4.16) |
where the constants
and 
are defined below. The difference between Eq. (4.15) and Eq. (4.16) has been shown to have only
a negligible Eq. (4.11)impact on all aspects of the model's behaviour,In practice, Eq. (4.16) is suitably resolved and applied to the component from of the horizontal momentum equations. This equation is applied level by level below
and, to ensure numerical stability,
a quasi-implicit treatment is adopted whereby the wind velocity 
in Eq. (4.16) is evaluated at the updated
time 
, while the wind amplitude, 
, is evaluated at the previous time step.4.2.2 Gravity-wave drag
This gravity-wave part of the scheme is based on the work of Miller et al. (1989) and Baines and Palmer (1990), and takes into account some three-dimensional effects in the wave stress amplitude and orientation. For clarity and convenience, a brief description is given here. On the assumption that the subgrid-scale orography has the shape of one single elliptical mountain, the mountain wave stress can be written as (Phillips 1984)
|
(4.17) |
where
, 
and 
is a constant of order unity. Furthermore, when 
or 
are significantly smaller than the length 
, characteristic of the gridpoint region size, there are,
typically, 
ridges inside the grid-point region.
Summing all the associated forces we find the stress per unit area, viz.
|
(4.18) |
where
has been replaced by 
, and 
by 
.It is worth noting that, since the basic parameters
, 
, 
are evaluated for the layer between 
and 
above the mean orography that defines the model's
lower boundary, there will be much less diurnal cycle in the stress than
in previous formulations that used the lowest model levels for this evaluation.
The vertical distribution of the gravity-wave stress will determine the
levels at which the waves break and slow down the synoptic flow. Since this
part of the scheme is active only above the blocked flow, this stress is
now constant from the bottom model level to the top of the blocked flow,
. Above 
, up to the top of the model, the stress
is constant until the waves break. This occurs when the total Richardson
number, 
, falls below a critical value 
, which is of order unity. When the non-dimensional mountain height
is close to unity, this algorithm will usually predict wave breaking at
relatively low levels; this is not surprising since the linear theory of
mountain gravity waves predicts low-level breaking waves at large non-dimensional
mountain heights (Miles and Huppert 1969). In reality,
the depth over which gravity-wave breaking occurs is more likely to be related
to the vertical wavelength of the waves. For this reason, when low-level
wave breaking occurs in the scheme, the corresponding drag is distributed
(above the blocked flow), over a layer of thickness 
, equal to a quarter of the vertical wavelengths of the waves, i.e.
|
(4.19) |
Above the height
are waves with an amplitude such that 
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05.04.2002 |
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