57
with V = e/llax + e a/ay. The generic equation (29) is not more complicated than (7).
Following Mellor and nlumberg (1985), the horizontal diffusion term has not been transformed
to its exact sigma-coordinate counterpart. In fact, a much simpler expression is chosen
(Deleersnijder and Wolanski, 1990; Deleersnijder, 1992).
The equations are discretized, in the sigma space, according to the finite volume technique
(Peyret and Taylor, 1983). The grid size is 10 km in both horizontal directions, and each water
column is divided into 10 sigma-levels. The vertical eddy diffusivities are computed from a
turbulence closure which is similar to that described in Section 4, except that the turbulence
macro-scale is obtained from an algebraic, empirical formula, and that the stability functions are
those of Nihoul and Djenidi (1987) (see also Deleersnijder, 1992). According to an appropriate
model calibration procedure, the horizontal viscosity Au is taken to be 500 m 2 s-l, while the
horizontal diffusivity is A = 75 m 2 S-l.
c
Upsloping and upwelling. In the sigma coordinate, the equations of the model do not
explicitly involve the "physical" vertical velocity w. Computing the latter is then part of the
postprocessing of the model results.
As suggested by Waleffe (1985) and Deleersnjder (1989), it is useful to split w into two
contributions, w = wus + wuw'
The upsloping velocity, w us ' reads
a1]
wus = a - - u.[ (1- a)Vh - aV1]]
at
(30)
It may be seen that a particle moving with a velocity equal to u + wuse z does not cross any
iso-a surface, implying that this particle remains at the same relative height in the water column.
Since the bottom and the surface are iso-a surfaces, wus may be regarded as the vertical
velocity adapted to the slope of the surface and the bottom. Since the present analysis pertains
to steady state model results, the first term in the right-hand member of (30) is zero.
The upwelling velocity,
(31)
is the velocity with which the water crosses the iso-a surfaces. Therefore, wuw may be
interpreted as the vertical velocity associated with proper up- or down-welling motions.
This decompositon of the vertical velocity provides an interesting analysis tool, for it renders
it possible to distinguish between the part of the vertical velocity that is necessary for the flow
to accommodate to the geometry of the basin and the extra vertical velocity related to actual upor down-wellings.A significant drawback must nevertheless be highlighted. Definitions (30)
and (31) are purely arbitrary: it is indeed possible to put forward several alternative expressions
of wus and wuw that could be equally valid as regards the distinction between the vertical
motions that are associated with the geometry of the basin and those that are not. What justifies
(30) and (31) is only that they take advantage in a very natural way of the use of the sigmacoordinate system. In a certain sense, (30) and (31) are inherent in the sigma-transformation.
In accordance with the present reasoning, we will examine separately the upsloping and
upwelling velocities.
To understand the space distribution of the upsloping velocity, it is desirable to identify the
dominant terms of definition (30). The sea surface elevation does not exceed ± 0.4 m
with V = e/llax + e a/ay. The generic equation (29) is not more complicated than (7).
Following Mellor and nlumberg (1985), the horizontal diffusion term has not been transformed
to its exact sigma-coordinate counterpart. In fact, a much simpler expression is chosen
(Deleersnijder and Wolanski, 1990; Deleersnijder, 1992).
The equations are discretized, in the sigma space, according to the finite volume technique
(Peyret and Taylor, 1983). The grid size is 10 km in both horizontal directions, and each water
column is divided into 10 sigma-levels. The vertical eddy diffusivities are computed from a
turbulence closure which is similar to that described in Section 4, except that the turbulence
macro-scale is obtained from an algebraic, empirical formula, and that the stability functions are
those of Nihoul and Djenidi (1987) (see also Deleersnijder, 1992). According to an appropriate
model calibration procedure, the horizontal viscosity Au is taken to be 500 m 2 s-l, while the
horizontal diffusivity is A = 75 m 2 S-l.
c
Upsloping and upwelling. In the sigma coordinate, the equations of the model do not
explicitly involve the "physical" vertical velocity w. Computing the latter is then part of the
postprocessing of the model results.
As suggested by Waleffe (1985) and Deleersnjder (1989), it is useful to split w into two
contributions, w = wus + wuw'
The upsloping velocity, w us ' reads
a1]
wus = a - - u.[ (1- a)Vh - aV1]]
at
(30)
It may be seen that a particle moving with a velocity equal to u + wuse z does not cross any
iso-a surface, implying that this particle remains at the same relative height in the water column.
Since the bottom and the surface are iso-a surfaces, wus may be regarded as the vertical
velocity adapted to the slope of the surface and the bottom. Since the present analysis pertains
to steady state model results, the first term in the right-hand member of (30) is zero.
The upwelling velocity,
(31)
is the velocity with which the water crosses the iso-a surfaces. Therefore, wuw may be
interpreted as the vertical velocity associated with proper up- or down-welling motions.
This decompositon of the vertical velocity provides an interesting analysis tool, for it renders
it possible to distinguish between the part of the vertical velocity that is necessary for the flow
to accommodate to the geometry of the basin and the extra vertical velocity related to actual upor down-wellings.A significant drawback must nevertheless be highlighted. Definitions (30)
and (31) are purely arbitrary: it is indeed possible to put forward several alternative expressions
of wus and wuw that could be equally valid as regards the distinction between the vertical
motions that are associated with the geometry of the basin and those that are not. What justifies
(30) and (31) is only that they take advantage in a very natural way of the use of the sigmacoordinate system. In a certain sense, (30) and (31) are inherent in the sigma-transformation.
In accordance with the present reasoning, we will examine separately the upsloping and
upwelling velocities.
To understand the space distribution of the upsloping velocity, it is desirable to identify the
dominant terms of definition (30). The sea surface elevation does not exceed ± 0.4 m