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Chapter 4. Subgrid-scale orographic
drag
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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
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4.1 General principles
The influence of subgridscale orography on the momentum of the atmosphere, and hence on other parts of the physics, is represented by a combination of lower-troposphere drag created by orography assumed to intersect model levels, and vertical profiles of drag due to the absorbtion and/or reflection of vertically propagating gravity waves generated by stably stratified flow over the subgridscale orography. The scheme is described in detail in Lott and Miller (1996).
The scheme is based on ideas presented by Baines and Palmer (1990), combined with ideas from bluff-body dynamics. The assumption is that the mesoscale-flow dynamics can be described by two conceptual models, whose relevance depends on the non-dimensional height of the mountain via.
|
(4.1) |
where
is the maximum height of the obstacle, 
is the wind speed and 
is the Brunt-Väisälä frequency of the incident flow.At small
all the flow goes over the mountain and gravity waves
are forced by the vertical motion of the fluid. Suppose that the mountain
has an elliptical shape and a height variation determined by a parameter
in the along-ridge direction and by a parameter 
in the cross-ridge direction, such that 
, then the geometry of the mountain can be written in the form
 .
|
(4.2) |
In the simple case when the incident flow is at right angles to the ridge the surface stress due to the gravity wave has the magnitude
|
(4.3) |
provided that the Boussinesq and hydrostatic approximations apply. In Eq. (4.3)
is a function of the mountain sharpness (Phillips 1984), and for the mountain
given by Eq. (4.2), 
. The term 
is a function of the mountain anisotropy, 
, and can vary from 
for a two-dimensional ridge to 
for a circular mountain.At large
, the vertical motion of the fluid is limited and part
of the low-level flow goes around the mountain. As is explained in
Section 4.2, the depth, 
, of this blocked layer, when 
and 
are independent of height, can be expressed
as
|
(4.4) |
where
is a critical non-dimensional mountain height of order
unity. The depth 
can be viewed as the upstream elevation of the isentropic surface
that is raised exactly to the mountain top. In each layer below 
the flow streamlines divide around the obstacle,
and it is supposed that flow separation occurs on the obstacle's flanks.
Then, the drag, 
, exerted by the obstacle on the flow at these levels can be written
as
|
(4.5) |
Here
represents the horizontal width of the obstacle as seen by
the flow at an upstream height 
and 
, according to the free streamline theory of
jets in ideal fluids, is a constant having a value close to unity (Kirchoff 1876; Gurevitch 1965). According to observations,
can be nearer 2 in value when suction
effects occur in the rear of the obstacle (Batchelor 1967). In the
proposed parametrization scheme this drag is applied to the flow, level
by level, and will be referred to as the drag of the `blocked' flow, 
. Unlike the gravity-wave-drag scheme, the total stress exerted by
the mountain on the `blocked' flow does not need to be known a priori.
For an elliptical mountain, the width of the obstacle, as seen by the flow
at a given altitude 
, is given by
|
(4.6) |
In Eq. (4.6), it is assumed that the level
is raised up to the mountain top, with
each layer below 
raised by a factor 
. This leads, effectively, to a reduction of the obstacle width,
as seen by the flow when compared with the case in which the flow does not
experience vertical motion as it approaches the mountain. Then applying
Eq. (4.5) to the fluid layers below 
, the stress due to the blocked-flow drag is obtained
by integrating from 
to 
, viz.
 .
|
(4.7) |
However, when the non-dimensional height is close to unity, the presence of a wake is generally associated with upstream blocking and with a downstream foehn. This means that the isentropic surfaces are raised on the windward side and become close to the ground on the leeward side. It we assume that the lowest isentropic surface passing over the mountain can be viewed as a lower rigid boundary for the flow passing over the mountain, then the distortion of this surface will be seen as a source of gravity waves and, since this distortion is of the same order of magnitude as the mountain height, it is reasonable to suppose that the wave stress will be given by Eq. (4.3), whatever the depth of the blocked flow,
, although it is clearly an upper limit
to use the total height, 
. Then, the total stress is the sum of a wave stress, 
, and a blocked-flow stress whenever the non-dimensional mountain
height 
, i.e.
 .
|
(4.8) |
The addition of low-level drag below the depth of the blocked flow,
, enhances the gravity-wave stress term in Eq. (4.8) substantially. In the present scheme the value of
is allowed to vary with the aspect ratio of the obstacle, as in the
case of separated flows around immersed bodies (Landweber
1961), while at the same time setting the critical number 
equal to 0.5 as a constant intermediate value. Note also that for
large 
, Eq. (4.8) overestimates the drag in the
three-dimensional case, because the flow dynamics become more an more horizontal,
and the incidence of gravity waves is diminished accordingly. In the scheme
a reduction of this kind in the mountain-wave stress could have been introduced
by replacing the mountain height given in Eq. (4.3) with a lower `cut-off' mountain
height, 
. Nevertheless, this has not been done
partly because a large non-dimensional mountain height often corresponds
to the slow flows for which the drag given by Eq.
(4.8) is then, in any case, very small.Next Section
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05.04.2002 |
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