In shallow continental-shelf seas, solitary internal waves displace the thermocline by meters and thereby perturb the arrival times of acoustic normal modes — the observable on which ocean acoustic tomography relies. Li, Zhang, Badiey, and Song (AIP Conf. Proc. 1495, 2012; journal version with Nan, Sci. Sin.-Phys. Mech. Astron. 43, 2013) derived a compact criterion in a two-layer approximation: a mode whose equivalent grazing angle lies between the thermocline critical angle and the bottom critical angle carries energy through the whole water column, including the thermocline, and its arrival time fluctuates strongly under internal-wave activity, while a mode with a smaller grazing angle propagates mainly below the thermocline with stable timing; the same mode index can be sensitive at one frequency and insensitive at another. The criterion was validated only against SWARM'95 airgun data (modes 1 and 2 at 50 Hz, since the airgun spectrum was concentrated below about 200 Hz; a 15.0 km track running within about 2° of the soliton front, treated as range-independent; and a synchronization-based arrival-time calibration). The open problem is to establish quantitatively the criterion's domain of validity — for higher-order modes excited at higher frequencies, for range-dependent tracks, continuous sound-speed profiles, and non-solitary internal-wave fields. No prior work is known that validates, extends, or refutes the criterion in these regimes.

Background

Continental-shelf seas in summer are vertically stratified: a warm surface mixed layer sits above cold bottom water, and the transition is marked by a thermocline, a thin layer of rapid temperature decrease. Because sound speed in seawater depends on temperature, such a water column acts as an acoustic waveguide. At low frequencies (tens to hundreds of hertz) and kilometer ranges, propagation in this waveguide is well described by normal modes: the acoustic field decomposes into a finite set of modes, the mm-th having a depth-dependent eigenfunction Ψm(z)\Psi_m(z), a horizontal wavenumber kmk_m, a phase velocity Vp(m)=ω/kmV_p^{(m)}=\omega/k_m, and a group velocity Vg(m)V_g^{(m)} that sets the travel speed of a pulse in that mode; the modal arrival time over range RR is Tm=R/Vg(m)T_m = R/V_g^{(m)}.

Solitary internal waves (solitons) are large-amplitude, nonlinear internal-wave packets, ubiquitous on continental shelves, that vertically displace the thermocline by several meters as they travel at speeds of order 1 m/s. Since the 1990s they have been known to perturb sound propagation strongly: Zhou, Zhang, and Rogers (1991) reported resonant scattering by solitons causing more than 20 dB of extra attenuation in the Yellow Sea; Traykovski (1996) established the equivalence of modal and ray treatments of internal-wave travel-time perturbations; and in the SWARM'95 experiment (Shallow-Water Acoustics in a Random Medium, New Jersey shelf, 1995; Apel et al., 1997) Headrick et al. (2000) modeled mode arrivals and their fluctuation statistics from airgun transmissions.

Ocean acoustic tomography infers the ocean's sound-speed (temperature) structure, or source parameters, from measured acoustic travel times. When modal arrival times are the observable, internal-wave (IW) induced timing fluctuations are both an error source and a potential probe of the IW field; either way one must know which modes, at which frequencies, have trustworthy timing.

Li, Zhang, Badiey, and Song (2012; journal version with Nan, 2013) gave this question a physically transparent answer in a two-layer idealization of the stratification (sound speed c1c_1 above the thermocline and c2c_2 below, with c1>c2c_1>c_2). Defining the thermocline critical angle θc=arccos⁡(c2/c1)\theta_c=\arccos(c_2/c_1) and each mode's equivalent grazing angle by cos⁡θ2(m)=km/μ2=c2/Vp(m)\cos\theta_2^{(m)} = k_m/\mu_2 = c_2/V_p^{(m)} with μ2=ω/c2\mu_2=\omega/c_2, their criterion states: a mode with θc<θ2(m)<θb\theta_c < \theta_2^{(m)} < \theta_b (where θb\theta_b is the critical angle of the seabed) has its energy distributed through the entire water column, including the thermocline, and its arrival time fluctuates strongly as a soliton displaces the thermocline — equivalently, modes with a larger fraction of their energy inside the thermocline are more sensitive; a mode with θ2(m)<θc\theta_2^{(m)} < \theta_c propagates mainly below the thermocline and keeps a stable arrival time. Because kmk_m, and hence θ2(m)\theta_2^{(m)}, depends on frequency, the same mode number can be sensitive at one frequency and insensitive at another — in the SWARM'95 environment, mode 1 at 50 Hz is sensitive while mode 1 at 100 Hz is not. The criterion tells a tomography system designer which transmitted frequencies and source depths excite modes whose timing is, or is not, perturbed by IWs.

The criterion's experimental basis, however, is narrow. Using SWARM'95 data — an airgun source at 30 m depth fired every 60 s for about 82 min at 15.0 km range from a 16-element vertical line array, with four strong solitons recorded by a moored thermistor chain (a string of temperature sensors) during the analysis window — the authors could extract and compare with model predictions only modes 1 and 2 at 50 Hz. The model predicted that mode 7 at 350 Hz fluctuates by up to about 100 ms (sometimes arriving earlier than mode 1), but the airgun spectrum was concentrated below about 200 Hz, so no experimental mode-7 (350 Hz) arrival could be obtained. The acoustic track ran nearly parallel to the soliton front (about a 2° angle), so the environment was treated as horizontally range-independent, and the measured, continuous sound-speed profiles were replaced by the two-layer step. Finally, because the airgun firing and the receiver recording were not strictly synchronized, observed arrival times had to be calibrated via

Tm=Tmexp−T1exp(100 Hz)+T1cal(100 Hz)T_m = T_m^{\mathrm{exp}} - T_1^{\mathrm{exp}}(100\ \mathrm{Hz}) + T_1^{\mathrm{cal}}(100\ \mathrm{Hz})

which uses the insensitivity of mode 1 at 100 Hz. What has and has not been established beyond this narrow envelope is the subject of the open problem below.

Problem Statement

Establish, quantitatively, the domain of validity of the grazing-angle (thermocline-energy) criterion for internal-wave-induced normal-mode arrival-time fluctuations, beyond the envelope in which the criterion was demonstrated. The scientific objective is to convert the criterion of Li, Zhang, Badiey, and Song — derived by theoretical analysis in a two-layer, range-independent idealization for solitary internal waves and validated against the low-frequency SWARM'95 data set — into a quantitatively bounded statement about when it correctly predicts modal timing variability and when, and why, it fails. The two components below address the sources of the criterion's limited validation. Specifically:

(1) Higher frequency and higher mode order. Does the criterion correctly predict arrival-time sensitivity for higher-order modes excited at higher source frequencies than the validated low-frequency SWARM'95 set — the regime in which the source paper's own model predicted strong fluctuations but which could not be checked experimentally because of the limited bandwidth of the airgun source?

(2) More general environments. Does the criterion remain quantitatively accurate for range-dependent tracks (an acoustic path not nearly parallel to the internal-wave front, so the horizontally invariant treatment fails), for realistic continuous sound-speed profiles in place of the two-layer step, and for non-solitary internal-wave fields such as internal tides (internal waves at tidal frequencies) and random internal-wave spectra — regimes in which mode coupling (energy transfer between modes, which invalidates the uncoupled, adiabatic picture in which each mode keeps its identity along the track) and three-dimensional propagation can compete with the two-layer picture on which the criterion rests?

An answer must delimit where the criterion holds and where it fails across these axes, using field data, calibrated simulation, or theory combined with either, in service of the criterion's claimed tomographic application to source depth location and frequency selection in shallow-water tomography.

References

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