Geoscience Reference
In-Depth Information
(a)
(b)
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
1
1.5
2
2.5
3
1
1.5
2
2.5
3
f
(rad / s)
f
(rad / s)
(c)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
1
1.5
2
2.5
3
f
(rad / s)
Figure 12.9.
Contour plots of (a) the time of wave breaking in seconds, (b) the breaking wavelength in centimeters, and (c) the
breaking wave amplitude in centimeters computed in numerical solutions of the QG model equations (12.34a) and (12.34b). The
contours have been interpolated linearly between data points, which are indicated by crosses.
2
quickly. This reflects the fact that the continuous slope
used in our numerical solutions results in a finite width PV
front, whereas the theory describes a discontinuous front.
The resulting wave envelops a smaller net relative vortic-
ity, particularly during its initial formation, and therefore
steepens less rapidly. The long-wave approximation used
to derive (12.28) also fails when the gradient of the inter-
face
R(θ
,
t)
becomes
typically by a factor of
3
. This indicates that PV
is imperfectly conserved in our experimental channel,
particularly given the similarities in the wavelength at
breaking (see Section 12.6.3). This may be due to stronger
bottom friction acting over the experimental shelf and
slope, as discussed in Section 12.3.1.
∼
(
1
)
, and this may exaggerate the
rate at which the wave steepens and breaks.
Figure 12.10a shows that
T
B
is again qualita-
tively similar in our experiments and numerical
solutions. In Figure 12.11a we plot the relative
error in the numerically computed breaking times,
T
B
=
T
B
numerical
/T
B
experiment
−
O
12.6.3. Breaking Wavelength
We now turn our attention to the length of the lee wave
at breaking. Apart from being a useful way to quantify
the closeness of our QG solutions and laboratory exper-
iments, the breaking wavelength determines the scale of
the eddies that form behind the protrusion. We define
L
B
as the distance along the shelf line between the azmiuthal
1. The QG model tends
to underpredict
T
B
relative to the laboratory experiments,
