4.2 Reduction techniques for graphical
Ray dynamics 63
Figure 4.7: A propagation scenario for the ocean model with the bilinear profile. The envelope of the perturbed profile is indicated by dashed lines
increasing at subsequent intersections, and it is desirable to achieve some kind of com-pression. For this purpose the range modulo the wavelength of the range variation can be used in case of periodic perturbation, and the trajectories of a modified map M are plotted. This is shown for our previous example (left panel of Figure 4.2) in Figure 4.8. The map Mpreserves all important features of the Poincar´e return map.
Invariant curves indicate regular motion; the invariant curves and the area filling sec-tions of chaotic ray trajectories belong to nonintersecting regions in the space of M;
resonant trajectories are associated with fixed points ofM. Furthermore, the mapping is area preserving too. It can be easily justified in terms of the equivalent two-degrees-of-freedom system obtained by the transformation (4.26). When we are interested in the area preserving property associated with the surface of intersection spanned by the tangent of the ray angle and range, it is enough to retain equations in their degrees of freedom only, and formulate the Jacobian by these two variables. To analyse the evolution of the Jacobian by (4.23), the divergence too is taken in the retained degrees of freedom. It is found that the divergence is zero, since ¯p01 does not depend on ¯p1, and
¯
q20 = 1, and therefore the Jacobian is constantly unity similarly as with the Poincar´e return map.
The existence of the two maps discussed above, P and M, relies on the fact that for a periodic perturbation it is possible to make the 3D phase space cylindrical and so bounded. In any other case, i.e. non-periodic driving or no driving at all, these maps are not uniquely defined. OnlyM∗ is well defined, but there is no obvious way to compress the range variation and so it does not support graphical studies.
For an ocean model with no variation of sound speed with range, the ray trajectories give a unique partition of the ray tangent-depth phase space. Hence the dynamics can be conveniently studied by the 2D phase portrait. Figure 4.9 displays ray trajectories which sample the phase portrait for a Munk profile. [Note that instead of the ray
0 2 4 6 8 10 0
5 10 15
mod(r, R) [km]
ϕ[◦]
Figure 4.8: An alternative to the Poincar´e section of the ray trajectories. Initial conditions are: z0 =za and ϕ0 =−k/2, k = 2, . . . ,24◦
tangent, the more intuitive ray angle is plotted as introduced by (4.5).] Now, it is possible to reduce the dynamics using intersections with a surface in two ways. The surface of intersection is again at the channel axis, but both positive and negative going ray intersections are allowed. Then, the ray tangentpnmay be either positive or negative; the range rn is monotonically increasing.
The first technique is plotting the ray tangent,pn, the value at the current intersec-tion against the difference between this value and the one at the previous intersecintersec-tion, pn−pn−1. The second technique is plottingpn against the difference between the cor-responding range and the one at the previous intersection,rn−rn−1. Figure 4.10 shows representations of the unperturbed ray system with each reduction technique. The first reduction technique (Figure 4.10a), which ignores range, extracts information provided by the phase portrait (Figure 4.9). The points that represent sectioned trajectories lie on a straight line of gradient 1/2, due to the symmetry of the phase portrait. This follows frompn=−pn−1, thuspn−pn−1 =−2pn−1 = 2pn, and sopn/(pn−pn−1) = 1/2.
The second reduction technique (Figure 4.10b), involving the range values in turn, provides information about the lengths of the upper and lower loops of the ray paths:
they are monotonic smooth functions of the ray take-off angles.
We now apply these techniques to the harmonically driven ray system. Results are shown in Figure 4.11. Note that values are taken only at every other intersection, i.e.
Ray dynamics 65
−15 −10 −5 0 5 10 15
0 0.5 1 1.5 2 2.5 3 3.5
ϕ[◦]
depth[km]
Figure 4.9: Ray trajectories without perturbation. Initial conditions are: z0 =za and ϕ0 =−k/2, k= 2,3, . . . ,24◦
the plots are of p2n against either p2n −p2n−1 or r2n−r2n−1. These representations preserve two features of P and M. Closed loops and area filling points respectively indicate regular and chaotic motion. Resonant trajectories are associated with a finite
−20 0 20
−15
−10
−5 0 5 10 15
ϕn−ϕn−1 [km]
ϕn[◦ ]
(a)
10 15 20 25 30 35 40 45
−15
−10
−5 0 5 10 15
rn−rn−1 [km]
ϕn[◦ ]
(b)
Figure 4.10: Two representations of the unperturbed ray system: (a) the ray angle plotted against the difference in ray angle between subsequent intersections, and (b) the ray angle plotted against the difference in range between subsequent intersections.
Initial conditions are: z0 =za and ϕ0 =−k/2, k= 2,3, . . . ,24◦
0 10 20 30 0
5 10 15
ϕ2n−ϕ2n−1 [◦] ϕ2n[◦ ]
16 18 20 22
8 9 10 11 12
ϕ2n−ϕ2n−1 [◦] ϕ2n[◦]
5 10 15 20 25
0 5 10 15
r2n−r2n−1 [km]
ϕ2n[◦ ]
13 13.5 14 14.5 15
8 9 10 11 12 13
r2n−r2n−1 [km]
ϕ2n[◦]
(a)
(b)
Figure 4.11: Two representations of the perturbed ray system: (a) the ray angle plotted against the difference in ray angle between subsequent intersections, and (b) the ray angle plotted against the difference in range between subsequent intersections. Initial conditions are: z0 =za and ϕ0 = −k/2, k = 2,3, . . . ,24◦. Blow-ups of figures on the left are displayed on their right
number of points in the centre of islands. Both primary and secondary resonances are indicated. The third feature of nonintersecting regular and chaotic regions does not apply. Closed loops may intersect other loops or the area filled by points due to chaotic motion. That is, points on the plot need not belong to one trajectory uniquely, and so there are no associated unique mappings. In fact, the mappings between subsequent points are two-valued mappings with both reduction techniques. This was revealed by continued research since the submission of this thesis, which is presented in Appendix D.
Ray dynamics 67
5 10 15 20 25
0 5 10 15 20 25 30
∆rn [km]
∆ϕn[◦]
13 13.5 14 14.5 15
16.5 18 19.5 21
∆rn [km]
∆ϕn[◦]
Figure 4.12: A further representation of the perturbed ray system. Initial conditions are: z0 =za and ϕ0 =−k/2, k= 2,3, . . . ,24◦
It is worth noting that the loops and area filling points in Figure 4.11 are confined by envelopes. The envelopes lie along curves already seen in Figure 4.10, and their width clearly depends on the perturbation strength.
At last, a third representation, shown in Figure 4.12, is to plot the differences of successive ray tangents against the differences of the corresponding ranges, at every other intersection, i.e. p2n−p2n−1 = ∆pagainst r2n−r2n−1 = ∆r. This plot preserves the main features of the one seen in Figure 4.11b, however, the range on the vertical axis has been doubled due to taking differences of successive tangents. Another difference is that loops are replaced with simple curve segments, and no curve segment overlaps any other nor does it merge into the chaotic sea. The space associated with this representation is partitioned into nonintersecting regular and chaotic regions. In fact, the mapping that converts one point to the next is a two-valued mapping, but this time the ambiguous points belong to the same ray trajectory.
Quasiperiodic driving. If there are two or more internal waves whose periods are in-commensurable, the driving is said to be quasiperiodic. Otherwise, it is periodic with the period which is the smaller of the smallest common multiple or the largest common divisor of all the periods involved. For higher mode numbers this period can be very large, so that the application of mapsP and Mis not feasible. It is often the case for N = 2 already. Therefore, such systems need different treatment. For simplicity, we consider again just two waves, and describe the sound speed perturbation by (4.93).
Figure 4.13 shows the representation of the ray dynamics using the last reduction tech-nique introduced above. Regular and chaotic motions are respectively indicated by
5 10 15 20 0
10 20 30 40
∆rn [km]
∆ϕn[◦ ]
16.4 16.6 16.8 17
8.8 9.3 9.8 10.3 10.8
∆rn [km]
∆ϕn[◦]
Figure 4.13: The representation of the ray system with a two mode perturbation model applying the third graphical technique. The perturbation parameters are: A1 =A2 = 0.01, R1 = 7 [km], R2 = 13 [km], rϕ = 5 [km]. Initial conditions are: z0 = za and ϕ0 =−1,−2, . . . ,−12◦. The curve with ϕ0 =−5◦ is magnified
either curves similar to the Lisajous curves or area filling points. The curves of regular motion intersect either other such curves or merge into the chaotic sea, therefore, the property of nonintersecting regular and chaotic partition of the space associated with the representation does not hold with quasiperiodic driving. The blown-up curve in Figure 4.13, which is similar to a Lisajous curve, is not closed. It is generally found (results not presented here) that such curves are not closed for wavelengths that make relative primes, and closed when their largest common divisor is greater than one. In our example wavelengths of 7 and 13 km do make relative primes.
For the applied perturbation a fraction of the Poincar´e sections of periodically perturbed ray trajectories are regular, and the region that they belong to is immersed into a chaotic sea in phase space (Figure 4.2). This is consistent with predictions of the KAM-theorem and Chirikov’s resonance overlap criterion. We have seen that other representations of the dynamics, supposedly associated with mappings such asMand the third representation, provide graphical means for drawing the same conclusion.
More realistic models of the perturbation involve a sum of N range-periodic inter-nal wave modes. Most generally the corresponding Hamiltonian is assumed to have quasiperiodic range-dependency, which means that the internal wave lengths are al-lowed to be incommensurate. It was shown by Brown (1998) that for such systems the KAM-theorem has implications similar to those cited above for systems with a range-periodic Hamiltonian. The result was demonstrated for the simplest case,N = 2, when
Ray dynamics 69 multiple sectioning of the trajectories yielded figures similar to Poincar´e sections of pe-riodically perturbed trajectories (Figure 4.6). The procedure is detailed at the end of the previous section. It is pointed out here that given a certain range of simulation, the number of points to plot is increasing with increasing δ. The accuracy, however, whether these points in phase space are close to the sectioning surface is decreasing. In contrast, with the approach presented above, the points which constitute the magnified curve in Figure 4.13 lie on the surface of intersection at the axis depth to within the accuracy of a root-finding procedure, and the range of simulation solely determines the number of points to plot. Furthermore, Figure 4.13 supports the idea of nonintersect-ing regular and chaotic regions in the phase space of the ray equations, i.e. a dense set of stable solutions. Note the cascade of sections of regular orbits.
It is pointed out that increasing N implies higher dimensionality of the problem.
Brown (1998) associated this with the higher [2(N+ 1)] dimensional phase space of the equivalent autonomous system. Here it is manifested by the following. We consider regular trajectories in the representation of the third reduction technique in case of N
= 0, 1, 2. With no perturbation (N = 0), sectioning the trajectories results in single points; with harmonic perturbation (N = 1) the sections of trajectories are curve segments. In case of the simplest quasiperiodic perturbation (N = 2) the increasing dimensionality is indicated by the intersecting sections of regular trajectories (Figure 4.13).