Geoscience Reference
In-Depth Information
was confirmed experimentally by
McEwan et al.
[1972].
Finally, the energy transfer to a mode will be most effec-
tive if the frequency of the forcing is equal or close to that
of the wave itself (“resonance”). All these together then
lead to the concept of resonant triads [
Bretherton
, 1964;
Plumb
, 1977], in which triads can interact if their frequen-
cies align to maximize energy transfer form one scale to
others, which is expressed in the selection criteria for the
zonal and radial wave numbers as
zonal wave number for the first three cross-channel modes.
Here, the selection criteria are satisfied by choosing two
wave modes and calculating the difference between the
wave modes to identify candidates for triads. To deter-
mine whether there is the possibility for resonance, the
difference in the frequency of the chosen pair of waves
is calculated. The wave number difference and the fre-
quency difference are then used to place the circle in the
dispersion diagram. If a mode is found within that circle,
it satisfies the condition and can participate in the res-
onant triad interaction. As the selection rule applies to
both the zonal and cross-channel wave number, the graph
is in fact a projection of a three-dimensional graph with
axes
m
,
n
,
ω
, where the lines for the different cross-channel
modes are displaced in the third direction onto the plane
of the zonal modes only. So, in this picture one has to
ensure that the mode within the circle also satisfies the
second of the selection rules in equation (3.12). The exam-
ple shown is the triad of zonal wave numbers 4, 3, and
1, where the two selection rules are satisfied. The reso-
nance condition is not fully satisfied but still within a range
allowing for some energy transfer. For this case, linear
Rossby waves were used for illustration purposes. How-
ever, finite-amplitude baroclinic waves have a substantially
modified frequency. For one, the strongly nonlinear shape
of a finite-amplitude wave leads to the fact that a single
wave mode is not represented by a single mode but by a
superposition of the fundamental mode and its harmon-
ics, all moving with the same group velocity. In addition,
the frequency or angular velocity of a wave depends on the
wave amplitude. As a result, the resonance condition may
be satisfied for a certain range of wave amplitudes but not
for another.
m
±
m
=0,
n
±
n
=0,
m
±
n
±
(3.12)
respectively, and resonance condition for their frequencies
ω
−
ω
ω
−
ω
,
(3.13)
where
is the average drift frequency of the three
modes. True resonance is achieved if the left-hand side is
equal to zero but energy transfer can also take place
at nonzero but small values. In a steady wave case,
the nonlinear coupling would lead to entrainment of
the frequencies such that they do add up to zero, but
in cases with varying mode amplitudes the strength of
entrainment may also fluctuate. If the left-hand side is
nonzero when the coupling is weak, the waves may drift
apart, but if that drift is slow, i.e.,
ω
, their relative
phases will still be close enough to reestablish entrainment
when the amplitude of the driving mode becomes strong
enough again.
The basic form of nonlinear interactions through a
resonant triad is illustrated in Figure 3.7 in a diagram
following
Ablowitz and Segur
[1981, Section 4.2.b] using
the dispersion relation for linear Rossby waves against the
ω
0.12
3.4.2. Harmonic Forcing and Zonal Mean-Flow
Correction
0.1
One special case of the selection rules is where
m
=
m
,
in which case the “triads” are to feed energy to the first
harmonic of the wave,
m
=2
m
, and to the zonal flow,
i.e.,
m
= 0. To satisfy Fjørtoft's constraint of transfer
to larger and smaller scales, the energy transfer to the har-
monic requires the flow of energy from
(m
,2
)
to
(m
,1
)
and
(
2
m
,1
)
. Since finite-amplitude waves are never sinusoidal,
there is always strong energy transfer between a mode and
its harmonic.
Similarly, for the mean-flow correction, the transfer
requires
(m
,1
)
and
(m
,2
)
, though with the complication
that the self-interaction is of the form of sin
nπy
while the
modes satisfying the boundary conditions are of the form
cos
πy
; cf. equation (3.8). Since the resonance condition
is irrelevant for the zonal mean flow, this route for energy
transfer is always possible and only depends on the wave
amplitude, whereas the energy transfer through resonant
0.08
ω
−
ω
ʹ
0.06
m
−
m
ʹ
m
ʺ
0.04
ωʺ
0.02
n
=1
n
=2
n
=3
0
0
1
2
3
4
5
6
7
8
m
Figure 3.7.
Basic routes of energy transfer through an almost
resonant triads involving the first radial/cross-channel modes
of wave numbers
(m
,
n)
=
(
3, 1
)
and
(m
,
n
)
=
(
4, 1
)
and the
second radial mode of wave number
(m
,
n
)
=
(
1, 2
)
.
