Geoscience Reference
In-Depth Information
2
1.8
1.6
1.4
1.2
1
0.8
0.6
n
= 1
0.4
n
= 0
0.2
0
0
0.2
0.4
0.6
0.8
1
H
0
/
R
b
Figure 5.11.
Frequency in the form of the Rossby number Ro = 2
ω/f
0
of inertial waves synchronized with the vortex array vs. the
depth of the layer normalized by the radius of deformation. The solid lines show two modes,
n
=0and
n
= 1. The frequency was
found by solving (5.20) and (5.21), where the wave number
k
is determined by the spacing between the magnets
d
= 4.5 cm.
The turbulent vortical motion created by the electro-
magnetic forcing in the rotating layer of water should
coexist and interact with inertial waves. It is interesting
to see if inertial waves can be identified in this turbu-
lent flow and also to discuss the condition when they
can be emitted most effectively by vortices. The Rhines
scale was determined by synchronizing/matching the field
of vortices to the Rossby waves. Let us perform similar
analysis for turbulence on the
f
-plane by matching vor-
tices and inertial waves instead of Rossby waves. Assume
again that the vortices are arranged in a checkerboard
pattern with wave number
k
=
π/d
,where
d
is a diame-
ter of a vortex. We require the synchronization between
the vortex arrangement and inertial wave equating the
rotation rate of a vortex,
v
, and the frequency of the
wave,
ω
, obtained from the dispersion relation (5.20). We
also match the wave number of the vortex arrangement
to that of the ine
rtial wave suc
h that (5.21) is written in
to occur when the Rossby number is below unity, the
layer should be relatively deep. This depth will decrease
if the size of the vortices/spacing between the magnets is
decreased. Typical values of the Rossby number for the
flow shown in Figure 5.10 are lower than 0.5 (except in
a few large vortices). With depth
H
0
= 10cm such that
H
0
/R
b
= 0.45, this flow falls below the line in Figure 5.11,
which indicates that the condition for the most efficient
emission of inertial waves is not achieved.
In an attempt to identify the waves in this turbulent
flow, we performed a double Fourier transform (in time
and space) of the velocity components of the flow. The
transform was performed in one direction across the tank
and for the time interval of about 90 inertial periods with
seven samples per period. The resulting energy spectrum
is shown in Figure 5.12 in frequency and wave number
space. A line showing the dispersion relation (5.20) for
inertial waves of the lowest mode
n
= 0 is also shown
on the diagram. One can see that a peak of energy is
located at the (dimensionless) wave number 10 (the forcing
wave number) at zero frequency (since the forcing is con-
stant). Note that (steady) geostrophic motions lie along
the horizontal axis (
ω
= 0). Further examination of the
diagram in Figure 5.12 reveals that the peak at wave num-
ber 10 extends toward higher frequencies up to the limiting
curve given by the dispersion relation. There is also notice-
able energy at somewhat lower frequencies (about 0.1-0.2)
the form
γ
n
=
k
ω
2
/(f
0
−
ω
2
)
. Solving then the disper-
sion relation numerically, we can obtain the frequency
of the wave/vortex array for different parameters of the
system. Figure 5.11 shows the values of frequency for dif-
ferent values of the depth of the layer. Here the spacing
between the magnets used in our experiments determines
the typical size of the vortex
d
and hence the value of the
wave number
k
for inertial waves. Note that the graph in
Figure 5.11 indicates that in order for the synchronization
