Chapter 5. SPATIALLY-COHERENT STRUCTURES
The sketch in Figure 5-32 shows the flow geometry in the Romanova
(1984) and Voronovich et al. (1998a) theory. The problem is formulated
under a rigid-lid approximation. The shear is localized in the thin subsurface
layer of thickness h and has no inflection points in the velocity profile U(z).
The latter condition is to ensure that the flow is dynamically stable with
respect to perturbations in the inviscid limit.
For resonance to occur in the flow configuration shown in Figure 5-32,
the typical frequency of the vorticity waves should be of the same order as
that of the internal wave N 0
0
0
/
m a x
N
u H c N
N ,
(5.27)
where equation (5.27) is formulated in dimensional variables and N
c is a
dimensionless coefficient.
Figure 5-32. Flow geometry and notation in the theory of shear waves. (After Romanova,
1984.)
In an attempt to describe the nonlinear resonant interaction by means of
an asymptotic analysis, Voronovich et al. (1998a) derived the set of two
coupled equations for normalized wave amplitudes a and b of the internal
and vorticity modes, respectively.
0,
2
0 .
t
x
x x x
x
t
x
x
a
a a
b
b
bb a
'
­
®
¯
(5.28)
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