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U 2 /L compared with the Coriolis acceleration fU . Alter-
natively, the Rossby number can be interpreted as the
ratio of the inertial period and the advective time scale
= f 1 /(L/U) ; in this sense, a small Rossby number refers
to “slow”motions with respect to the rotation period. The
Ekman number E estimates the importance of viscous
effects with respect to Coriolis accelerations. The relative
vorticity of the velocity field is defined as ω =
shallow-layer flows. Sections 7.3-7.5 contain several
examples of laboratory experiments performed in facil-
ities with very different horizontal and vertical sizes.
Finally, Section 7.6 presents some concluding remarks.
7.2. THEORY AND EXPERIMENTAL
BACKGROUND
×
u and
scales as U/L . The nondimensional vorticity equation is
ω
7.2.1. Essential Balance in a Rotating System
ω = ( k + ω )
∂t + u ·
u + E
2 ω .
·
(7.5)
The motion of a homogeneous fluid in a steadily rotat-
ing tank presents a remarkable characteristic: When the
rotation axis is aligned with gravity, fluid motion is pre-
dominantly horizontal, i.e., in a plane perpendicular to the
rotation axis. This implies that fluid motion takes place in
the form of vertical columns that remain always parallel to
the angular velocity of the system. This phenomenon was
predicted by S. S. Hough since 1897 [ Gill , 1982, p. 506]
and by Proudman [1916], and it has been observed and
reported in numerous laboratory studies since the early
experiments of Taylor [1917, 1921].
In order to illustrate this behaviour, consider a homoge-
neous, incompressible fluid with density ρ and kinematic
viscosity ν moving in a steadily rotating system with con-
stant angular velocity . Fluid motion is governed by the
Navier-Stokes equations and by the continuity equation
representing conservation of mass:
Since rotation effects are fundamental in mesoscale and
large-scale geophysical flows, rotating tank experiments
are usually designed in such a way that the Rossby and
the Ekman numbers are typically very small, i.e.,
1
and E
1. Considering quasi-steady motions and ignor-
ing nonlinear and viscous effects, it is verified from (7.3)
that the flow is nearly in geostrophic balance,
u ∼−∇
P .
k
×
(7.6)
For this particular balance the vorticity equation (7.5) is
reduced to
u
k
·∇
0,
(7.7)
which is the celebrated Taylor-Proudman theorem. To
facilitate the physical understanding of (7.6) and (7.7),
consider a laboratory fluid tank steadily rotating about
the z axis of a Cartesian coordinate frame (x , y , z ) . Both
the angular velocity vector and the constant-gravity vector
are aligned in the vertical direction. It is straightforward
to show that the horizontal momentum equations indicate
that the horizontal accelerations are balanced by the pres-
sure gradients, and the z equation expresses that the flow
is so slow that it can be considered to remain in hydrostatic
balance. The vorticity equation (7.7) implies
u
∂z
u
∂t + u
1
ρ
2 u ,
·
u +2
×
u =
P + ν
(7.1)
·
u = 0,
(7.2)
where u and P are the velocity field and pressure, respec-
tively. (The pressure is actually the reduced pressure p
ρ , with p the mechanical pressure and an effective
potential representing conservative forces per unit mass.
This potential usually contains both the gravitational
potential g and the centrifugal potential c whose gra-
dient is the centrifugal acceleration
0,
(7.8)
c ≡−
×
×
(
r ) ,
that is, each component of the velocity (u , v , w ) is inde-
pendent of the coordinate parallel to the axis of rotation.
The most striking consequence of the Taylor-Proudman
theorem is that the horizontal divergence is zero, because
∂w /∂z = 0 in the continuity equation. This condition
states that there is no divergence or convergence of fluid
in any plane (x , y ) perpendicular to the axis of rotation.
If the vertical velocity component is zero at some level,
for instance the solid bottom tank, then it is zero for all
z . In this case the flow is purely two-dimensional, and the
motion takes place in the form of columns, always par-
allel to the rotation axis. Such a visually attractive effect
is easily observed in rotating tank experiments by adding
dye to the flow. What happens when these fluid columns
experience a change of depth?
with r being the position vector.)
Consider the system's angular velocity as = k , with
the rotation axis along direction k . The governing equa-
tions can be written in nondimensional form by intro-
ducing a length L , a time L/U , with U a characteristic
velocity, and considering the pressure scale as 2 UL :
u
∂t
u + k
+ u ·
u =
P + E
2 u ,
×
−∇
(7.3)
u = 0.
·
(7.4)
The nondimensional numbers in (7.3) are the Rossby num-
ber = U/fL ,where f =2 is the Coriolis parameter,
and the Ekman number E = ν/fL 2 . The Rossby num-
ber measures the importance of the advective acceleration
 
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