Geoscience Reference
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f/2
f/2
z
h(x, y, t)
h(x, y)
z
δ E
y
h B (x, y)
x
x
Figure 7.1. Left: Schematic view of a homogeneous fluid layer over spatially variable topography in a rotating system. Right:
Bottom Ekman layer with thickness δ E and the geostrophic interior with depth h(x , y) under the rigid-lid approximation. Typically
δ E << h(x , y) in laboratory experiments.
7.2.2. Topography Effects
quasi-two-dimensional flows over topography. Before pre-
senting a complete dynamical model, we discuss first the
main inviscid and viscous contributions separately.
1. Inviscid Topography Effects . Ignoring the Ekman
layer, the vertical integration of the continuity equation
(using kinematical conditions at the surface and at the
bottom) implies that the horizontal divergence is given
by material changes of fluid columns:
∂u
The principal effect of topography on the dynamics of
rotating flows is by stretching or squeezing of vertical fluid
columns. It is assumed that the horizontal flow field (u , v)
is independent of the vertical coordinate z , in line with the
discussion in the previous section. In dimensional terms it
means that u
v(x , y , t) . In addition, the
flow remains in hydrostatic balance in the vertical direc-
tion. These assumptions allow the formulation of the flow
dynamics in terms of the vertical component of the rela-
tive vorticity ω = ∂v/∂x
u(x , y , t) and v
∂x + ∂v
1
h
Dh
Dt .
∂y =
(7.10)
∂u/∂y . Taking the curl of the
equations of motion (7.1), the evolution equation for ω is
Inserting this expression in the inviscid form of (7.9)
yields the material conservation of potential vorticity
q = + f )/h :
∂y + ∂u
+ f ) = ν
∂ω
∂t + u ∂ω
∂x + v ∂ω
∂x + ∂v
2 ω . (7.9)
Dq
Dt = 0.
∂y
(7.11)
Now it is required to find the horizontal velocity com-
ponents in terms of the relative vorticity in order to close
the system. For this purpose the continuity equation (7.2)
is integrated in the vertical direction over the full layer
depth, i.e., over h B
This property indicates that the relative vorticity of
a fluid column will change when experiencing depth
variations due to stretching/squeezing effects on fluid
columns. As a column moves toward a deep region,
it is stretched and gains positive vorticity. In contrast,
fluid motions toward shallower regions imply squeezing
effects and the production of negative vorticity. Since
most of laboratory experiments are characterized by
weak viscous effects, this dynamical behavior is funda-
mental when studying the effects of variable topography.
2. Viscous Topography Effects . The main effect of
the viscous Ekman boundary layer at the bottom is to
slow down the flow motion. This behavior arises as a
consequence of the Ekman pumping-suction condition
between the thin boundary layer and the rest of the fluid
column. Essentially, the Ekman theory states that the
flow inside the Ekman layer induces a nonzero vertical
velocity [see, e.g., Pedlosky , 1987]. By means of this mech-
anism, fluid is exchanged between the Ekman layer and
h + h B ,where h(x , y , t) is the
layer depth and h B (x , y) describes the spatially variable
bottom topography (Figure 7.1, left). Note that h con-
tains the free surface elevation associated with the flow
itself, which may be time dependent. In the discussion
below it is necessary to consider the rigid-lid approxima-
tion, which consists of neglecting the temporal variations
of h when compared with changes associated with topo-
graphic variations. This approximation filters out gravity
waves as if the top surface was flat, and the surface ele-
vation is no longer a dependent variable. In addition, a
thin Ekman layer of thickness δ E
z
( 2 ν/f ) 1 / 2 is consid-
ered at the solid bottom (Figure 7.1, right). Depending on
the boundary conditions imposed, the z integration of the
continuity equation may lead to different formulations of
 
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