6.4 Theoretical solution of A and B wave generation
6.4.2 Evolutionary mechanism
Beyond the source of generation the internal wave field can be presented as a superposition of diurnal and semidiurnal progressive waves:
ζ (x, z, t) = as1g1(z)exp(ks1x− σst+ ϕ1s) + ad1g1(z)exp(k1dx− σdt+ ϕ1d) (6.2) where as1and ad1, k1s and kd1, ϕ1sand ϕ1dare the wave amplitudes, wavenumbers, and the phases of the first mode internal waves. The superscripts s and d denote semidiurnal and diurnal har-monics, respectively. Note that with the assumption N2(z) >> σ2, which is valid for the whole
6.4. THEORETICAL SOLUTION OF A AND B WAVE GENERATION
Figure 6.8:a) Amplitude of the first baroclinic mode (a1) calculated for April, 2007. The solid lines correspond to the semidiurnal constituents, the dashed line to the diurnal.
Superposition of semidiurnal and diurnal internal waves at two control points: to the west (panel b) and to the east (panel c) of the eastern ridge with their amplitudes predicted by the analytical solution (panel a).
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Figure 6.9:The same as in Figure 6.8 but for July, 2010.
6.4. THEORETICAL SOLUTION OF A AND B WAVE GENERATION
Figure 6.10:Evolution of an initial two-wave profile (solid line) calculated using the KdV equation for the parameters typical to the SCS. Dashed and dotted lines represent the wave profiles after 25 and 50 h of wave propagation, respectively.
water column in the northern SCS, the vertical structure functions g1(z) for the semidiurnal and diurnal harmonics coincide, which is why only one symbol g1(z) was used for both.
Superposition of waves (as expressed by equation 6.2) with the amplitudes as1(t) and ad1(t) taken from Figures 6.8a and 6.9a produces the wave displacements that are presented in Figures 6.8b-c and 6.9b-6.8b-c. These time series show an alternation of large and small peaks whi6.8b-ch holds for the whole period in July, 2010 and for the most part of the neap-spring cycle in April, 2007.
Similar intermittency of the wave amplitude is expected in space as well. If so, the progressive wave with an irregular profile can steepen and disintegrate, as is shown in Figure 6.10. In this experiment the wave profiles were calculated using the KdV equation with the parameters of nonlinearity and dispersion typical for the northern SCS.
The number and intensity of ISWs emerging from an initial perturbation depend on several pa-rameters. Some observations (Ramp et al. 2010) suggest that ISWs are generated when the tidal velocity in the LS exceeds some critical level below which no ISW activity can be recorded.
Moreover, some observations (Yang et al. 2004) showed that the transition from no waves to large amplitude ISW regime is quite sharp, whereas some revealed quite a monotonous transi-tion from high to low ISW activities (Ebbesmeyer et al. 1991), which correlates with the gradual changes in the semidiurnal tidal forcing shown in Figure 6.2b.
As it follows from the inverse scattering problem, the number of the new-born ISWs in a two-layer system can be predicted by the following formula (Whitham 1974, Gerkema & Zimmer-man 1995):
where a is the nondimensionalised initial wave amplitude (by the water depth H), ε =Uσ L0εb
b is a parameter of nonlinearity (U0and σ are the strength and frequency of the barotropic forcing,
6.4. THEORETICAL SOLUTION OF A AND B WAVE GENERATION
respectively; Lbis the topographic length scale; εbis a nondimensionalised topographic height scale (by the water depth H) and is assumed εb<< 1), α =HH1 is the nondimensional depth of the interface (H1is the upper layer depth), δ = (σ Hc )2is the dispersion parameter (c is the long wave phase speed), and γ =p
α (1 − α ).
For the conditions of the northern SCS, the position of the interface should be taken between 500 and 700 m, in order to provide the best fit of all the basic wave parameters to observations.
For 700 m depth equation (6.3) predicts a 40 m threshold of the wave amplitude when two or more solitary waves can emerge from a propagating long wave. Hereafter 40 m amplitude is used as a boundary to distinguish A waves from B waves. Note that this value is consistent with Figure 6.10, in which the wave disintegration is shown as a direct evolutionary process.
Applying the amplitude threshold to the wave profiles shown in Figures 6.8b-c and 6.9b-c, every depression was marked as A wave if its amplitude is greater than 40 m, or B waves otherwise.
Similar to Figures 6.4 and 6.7 one capital and one lower case letter are used for marking the wave troughs within every 24 hours time span. Note that the displacements with amplitudes less than 10 m are marked as 0, implying that such waves are too weak to be disintegrated near the generation site (less than 3 wavelengths), and require longer distance for ISW formation (longer length of nonlinearity). The subscript of every particular wave trough in Figures 6.8 and 6.9 is similar to those shown in Figures 6.4 and 6.7.
The perfect correlation takes place between the type of the ISW events to the west of the LS shown in the left column in Figure 6.4 and the type of the waves predicted by the linear theory in Figure 6.8b. The name of every single wave event coincides in both figures. Note that the correlation holds for the whole neap-spring cycle. The perfect coincidence between wave events also takes place to the west of the LS (right column in Figure 6.4 and Figure 6.8c). Thus the ’evolutionary’ mechanism proposed here based on the steepening and disintegration of the propagating internal tidal waves is confirmed by the results of numerical modelling with a good accuracy.
Comparison of Figures 6.7 and 6.9 shows that the vast majority of the wave events coincide:
i.e., 82% at the western observational point, and 93% at the eastern one. The restricted number of cases shown by the shaded circular spots in Figure 6.9 has different type of waves than that in Figure 6.7. Such a discrepancy could be a consequence of a number of factors such as inaccurate calculation of the amplitude threshold, idealized topography, etc. However, the ’evolutionary’
method looks quite adequate and robust to give a reliable prediction of A and B waves.