Formulation of the Homogeneous Model
29
(2.2.6)
in the stretching term in (2.2.5).
The density of the layer is constant and so the spatially variable part of the
pressure field, which is in hydrostatic balance with the gravitational force, is
independent of z. The horizontal, geostrophic velocity is therefore independent
of z, and thus an integral of (2.2.5) over the layer thickness, D, yields:
(2.2.7)
where WB is the velocity pumped out of the lower boundary layer and WE is the
Ekman velocity pumped out of (or into) the upper mixed layer.
We use standard Ekman layer theory to represent WB as (see Pedlosky
1987):
{JE
WB = l(
(2.2.8)
where {JE is much less than D. WE can be written in terms of the wind stress as
in Chapter 1 although it is convenient to leave it in its present form. It is
important to remember, however, that it is given entirely in terms of the wind
stress, and that its curl is independent of the motion and is thus a specified
function of horizontal position and provides the forcing for the circulation.
If (2.2.2) is used to write the velocity and vorticity in terms of the stream
function, we obtain the governing equation for the homogeneous model:
(2.2.9)
In (2.2.9) the second term on the left side is the Jacobian of the stream function
with the relative vorticity ( = '\1 2 rjl and represents the advection of relative
vorticity by the motion field, i.e.:
/X + v a( = _ ar/J a'\1
2
r/J + ar/J a'\1
2
r/J = J( r/1, '\12r/J)
ax
ay
ay ax
ax ay
(2.2.10)
The second term on the right side of (2.2.9) represents the effect of the
vorticity change due to vortex stretching produced by the vertical velocity
pumped out of the bottom boundary layer, and we refer to this as the bottom
friction term. The coefficient:
{JE
r=fo2D
is an inverse time scale for vorticity decay due to bottom friction.
(2.2.11)
We would have obtained the same equation if we had chosen to think
about the model as a purely two-dimensional model with the stress as a body
force. Conceptually the development given above has the advantage of emphasizing the important physical fact that the vertical velocity, though weak, is
29
(2.2.6)
in the stretching term in (2.2.5).
The density of the layer is constant and so the spatially variable part of the
pressure field, which is in hydrostatic balance with the gravitational force, is
independent of z. The horizontal, geostrophic velocity is therefore independent
of z, and thus an integral of (2.2.5) over the layer thickness, D, yields:
(2.2.7)
where WB is the velocity pumped out of the lower boundary layer and WE is the
Ekman velocity pumped out of (or into) the upper mixed layer.
We use standard Ekman layer theory to represent WB as (see Pedlosky
1987):
{JE
WB = l(
(2.2.8)
where {JE is much less than D. WE can be written in terms of the wind stress as
in Chapter 1 although it is convenient to leave it in its present form. It is
important to remember, however, that it is given entirely in terms of the wind
stress, and that its curl is independent of the motion and is thus a specified
function of horizontal position and provides the forcing for the circulation.
If (2.2.2) is used to write the velocity and vorticity in terms of the stream
function, we obtain the governing equation for the homogeneous model:
(2.2.9)
In (2.2.9) the second term on the left side is the Jacobian of the stream function
with the relative vorticity ( = '\1 2 rjl and represents the advection of relative
vorticity by the motion field, i.e.:
/X + v a( = _ ar/J a'\1
2
r/J + ar/J a'\1
2
r/J = J( r/1, '\12r/J)
ax
ay
ay ax
ax ay
(2.2.10)
The second term on the right side of (2.2.9) represents the effect of the
vorticity change due to vortex stretching produced by the vertical velocity
pumped out of the bottom boundary layer, and we refer to this as the bottom
friction term. The coefficient:
{JE
r=fo2D
is an inverse time scale for vorticity decay due to bottom friction.
(2.2.11)
We would have obtained the same equation if we had chosen to think
about the model as a purely two-dimensional model with the stress as a body
force. Conceptually the development given above has the advantage of emphasizing the important physical fact that the vertical velocity, though weak, is