Geoscience Reference
In-Depth Information
7.2.4. Experimental Considerations
In the presence of variable topography, the flow
response will strongly depend on the specific configura-
tion used at the bottom of the container. Regardless of
the shape of the topography, it is fundamental to have
an estimation of the effects associated with depth varia-
tions before starting an experiment. Such an estimation is
obtained by means of the
topographic β-effect
,whichis
dynamically equivalent to the planetary
β
-effect. In order
to understand this equivalence, it is useful to compare the
potential vorticity of the following two cases: (i) the lab-
oratory with variable topography and (ii) the planetary
β
-plane assuming a constant depth. In the inviscid limit,
conservation of potential vorticity in these two cases is
expressed by
Experiments on geophysical flows are usually
performed within a container rotating about a verti-
cal axis aligned with gravity. The depth scale
H
is usually
smaller than the horizontal scale
L
, but not necessarily.
This is an important point to keep in mind when dis-
cussing the use of the shallow-water equations versus the
quasi-geostrophic approximation. The experimental tank
is set to rotate steadily for a certain time until the fluid
inside reaches a state of solid-body rotation. How long
is this time? The
spin-up
process is essentially viscous:
When the container is set in motion, the fluid takes some
time to adjust to the rotation by viscous stresses exerted
by the lateral boundaries (through Stewartson layers) and
the solid bottom (Ekman layer). An additional Ekman
layer may be present at the upper boundary, due either
to the presence of a solid lid or to stresses generated
at the free surface (e.g., by wind). However, the effects
associated with this top Ekman layer will be ignored in
the rest of the chapter. The main damping contribution
during the spin up is due to bottom friction effects, which
become effective after a few Ekman periods
T
E
(7.14).
This is the appropriate time scale for adjustment to
solid-body rotation, and it has to be taken into account
before starting an experiment.
Once the fluid has reached a state of rest in the rotat-
ing system (i.e., a state of solid-body rotation), the actual
experiment is started: The fluid is set in motion by gen-
erating vortices, currents, or turbulent flows or by any
other desired initial flow. Several methods are carefully
described by
van Heijst and Clercx
[2009] and references
therein. In the following sections we shall give further
details of the initial forcing in different experiments.
We should stress here the requirement of generating a
low with a low to moderate Rossby number in order to
ensure that the main horizontal balance in the fluid is
geostrophic.
During the experiment, viscous topographic effects are
expected to become important at times of the order of
T
E
, again. Thus, experiments devoted to study flows in
the absence of bottom friction should have a duration
shorter than the Ekman period. In contrast, when bottom
friction effects are the main subject of study, it is neces-
sary to perform experiments during one or more Ekman
periods. The conventional way to include Ekman damp-
ing is by considering only the dominant linear term in
the vorticity equation (7.17) and neglecting the nonlin-
ear Ekman terms. This is an acceptable approximation for
most laboratory experiments. However, weak nonlinear
Ekman friction effects might be important in some cases,
for instance, when considering weak asymmetries between
cyclonic and anticyclonic regions (
Zavala Sansón and van
Heijst
[2000a]).
ω
+
f
h
q
lab
≡
= const.,
q
pla
≡
ω
+
βy
= const.
(7.20)
These relationships show that increasing (decreasing)
depth in the laboratory is equivalent to decreasing
(increasing)
y
in the ocean. Now assume the fluid depth in
the container to vary linearly along an arbitrary direction,
say
y
, such that
h
=
H
δ
B
y/W
,where
H
is the maxi-
mum fluid depth,
δ
B
is the height of the topography over
a horizontal distance
W
, and the topographic slope is
such that
δ
B
/W
−
1. As the Rossby number of the flow is
assumed small enough, conservation of potential vortic-
ity for fluid parcels in the rotating laboratory tank implies
that
q
lab
≈
ω
+
β
t
y
= const., with
β
t
=
f δ
B
WH
.
(7.21)
Thus, the first-order effect of the bottom topography in
the laboratory is equivalent to the planetary
β
-effect along
the meridional
y
direction. Therefore, it is usually referred
to as the topographic
β
-effect. Note that the “meridional”
direction in the laboratory depends on the orientation of
the topography.
In order to write the topography parameter
β
t
in nondi-
mensional terms, we must multiply by the horizontal scale
of the flow
L
and divide by the Coriolis parameter:
β
t
=
Lδ
B
/WH
. Different values can be estimated as the topog-
raphy varies in different locations. It is important to point
out that the appropriate parameter to compare the influ-
ence of topographic variations in experiments and in geo-
physical situations is the nondimensional parameter
β
t
,
and not the topographic slope
δ
B
/W
.
These arguments can be applied to the parabolic free
surface deformation of a rotating system, which also
produces an equivalent
β
-effect. In this case, the free
surface elevation
η
f
2
W
2
/
8
g
plays the role of
δ
B
,
and
W
is about half the length of the container. Thus,
β
η
≈
≈
f
3
W/
8
gH
. In many experiments the stretching and
squeezing effects due to the parabolic deformation of the
