Quantifying low-frequency acoustic propagation through realistic crossing nonlinear internal-wave trains in shallow water: nonlinear collision physics, front curvature, bottom loss, and multi-train environments

On continental shelves, low-frequency sound can become horizontally ducted between the crests of a nonlinear internal-wave train, and when two such trains cross, the ducted energy is scattered, cross-ducted, or reflected depending on the crossing angle. The existing quantitative theory of this process (Shmelev, Lin and Lynch, 2020) rests on four idealizations: straight, parallel, identical-amplitude wave fronts; a crude first-order treatment of the thermocline displacement inside the crossing region; a lossless sea bottom; and only two interacting trains over a flat seafloor. The open problem is to quantify how the acoustic outcome changes when these idealizations are removed — specifically the consequences of (i) fully nonlinear soliton–soliton collision dynamics that reshape the thermocline in the crossing region, (ii) natural curvature of the wave fronts, (iii) sediment attenuation interacting with crossing-induced modal energy transfer, and (iv) multiple trains and variable amplitudes. The common objective is a predictive account of the azimuthal variability, horizontal arrival-angle structure, and modal energy budgets of low-frequency sound in observed crossing-wave environments, establishing when the idealized straight-front regime classification applies and when it fails.

Background

Nonlinear internal waves (NIWs, also called internal solitons) are large-amplitude undulations of the thermocline — the layer of sharp vertical density and sound-speed gradient in the stratified ocean — commonly generated by tidal flow over shelf-break topography. They are routinely imaged by synthetic-aperture radar (SAR), which sees their sea-surface signature, and crossing NIW trains with a wide range of intersection angles are frequently observed on continental shelves, for example during the Shallow Water 2006 experiment (SW06) on the New Jersey shelf. In shallow water, low-frequency sound (tens to hundreds of hertz) propagates as vertical normal modes — solutions of the depth-separated waveguide problem — and, because an NIW crest depresses the thermocline and perturbs the modes' horizontal wavenumbers, a pair of parallel NIW crests can act as a horizontal (lateral) acoustic duct: a mode whose propagation direction makes a grazing angle $\chi$ with the crests that is smaller than the horizontal critical angle $\chi_{\mathrm{crit},n}=\cos^{-1}(k_{rn}^{\mathrm{crest}}/k_{rn}^{0})$ — the ratio of mode $n$'s horizontal wavenumber at the crest to its background value — remains trapped between the crests and can be carried along the front for tens of kilometers. Prior work characterized single NIW trains: mode coupling through a soliton packet (Preisig and Duda, 1997), radiation from the termination of a truncated internal-wave duct (Lin, Duda and Lynch, 2009), a fully three-dimensional coupled-mode solution (an expansion in vertical modes coupled in range and azimuth) for straight trains (Shmelev, Lynch, Lin and Schmidt, 2014), and ducting, refraction and dispersion by curved single trains (Lynch et al., 2010; Lin et al., 2013). Shmelev, Lin and Lynch (2020) extended this to two crossing trains using an idealized shelf waveguide (80 m depth, 100 Hz, thermocline depressions of the form $A\,\mathrm{sech}^2(r/W)$ with $A=25$ m, width parameter $W=70$ m, and $r$ the horizontal distance from the wave-crest center) and a wide-angle three-dimensional parabolic-equation (PE) model — a standard range-marching numerical method for range-dependent acoustic propagation — validated against the three-dimensional coupled-mode solution for a straight duct up to grazing angles of about $40^\circ$. They found that if the crossing angle $\gamma_{\mathrm{cross}}$ between the trains exceeds roughly $2\chi_{\mathrm{crit},n}$ (about $15^\circ$ for their parameters), strong three-dimensional mode coupling and scattering dominate and remove ducted energy, while for $\gamma_{\mathrm{cross}}<2\chi_{\mathrm{crit},n}$ geometric mechanisms — subcritical reflection from the leading crossing wave, horizontal tunneling through it followed by re-trapping (tunneled cross-ducting), and reflection cross-ducting — transfer energy into the crossing train, with ducted depth-integrated intensity reduced by about 6 dB at the smallest angles modeled ($2^\circ$–$4^\circ$). These conclusions rest on the four idealizations listed above: the first-order treatment of the thermocline displacement inside the crossing region is justified only to first order by laboratory observations of solitary-wave collisions (Hammack, Henderson, Guyenne and Yi, 2005), sediment attenuation was deliberately omitted to isolate the horizontal refraction and scattering mechanisms, front curvature was excluded from the crossing scenarios despite being known to matter for single trains, and only two identical trains over a flat bottom were modeled.

Problem Statement

Quantify the low-frequency acoustic consequences of realistic crossing nonlinear internal-wave trains in shallow water by removing the idealizations underlying the Shmelev–Lin–Lynch (2020) crossing-wave analysis, and determine the resulting limits of the idealized regime classification. Specifically: (i) replace the first-order approximation of the thermocline displacement in the crossing region with the fully nonlinear soliton–soliton collision dynamics of two interacting internal-wave trains, and determine how the collision-modified stratification changes the previously identified tunneling, subcritical-reflection, and mode-coupling outcomes; (ii) include the natural curvature of internal-wave fronts and determine how curvature modifies the crossing-angle regime boundary $\gamma_{\mathrm{cross}}\gtrless 2\chi_{\mathrm{crit},n}$ and the magnitude of cross-ducting and reflection; (iii) include sediment (bottom) attenuation and determine how realistic lossy environments reshape crossing-induced modal energy transfer, duct residence, and higher-order modal damping; and (iv) extend the analysis beyond two identical straight-front trains to multiple crossing trains and trains of variable amplitudes. The common objective uniting these strands is a quantitative account of how the crossing-angle regime classification (steep crossing: three-dimensional mode coupling and scattering; shallow crossing: subcritical reflection, tunneled cross-ducting, and reflection cross-ducting, with the boundary near $\gamma_{\mathrm{cross}}\approx 2\chi_{\mathrm{crit},n}$) and the associated ducted-intensity losses, modal energy budgets, azimuthal variability, and horizontal arrival-angle structure of low-frequency sound change under collision-modified stratification, curved fronts, sediment attenuation, and multiple or unequal-amplitude trains — establishing when the straight-front, two-train, lossless regime picture applies and when it fails. A qualifying solution must address all four strands (i)–(iv) quantitatively rather than a single idealization alone, and must return the resulting regime boundary together with the ducted-intensity losses and modal-energy budgets that constitute this common objective, either at the stated waveguide parameters or at explicitly justified alternatives.

References

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