Active intensity vortex and stagnation point singularities in a shallow underwater waveguide

The method of normal modes is central to understanding acoustic waveguide propagation in shallow water, for which sound propagation is influenced by both seabed properties and water depth.^1,2 Seabed, or geoacoustic, properties can be inferred from knowledge of acoustic normal modes through inversion techniques based on broadband^3,4 or narrowband signals.⁵ In studies involving narrowband signals, the inversion process can involve measurement of the complex pressure field as a function of range from a continuous-wave (CW) source, approximating a steady state condition. The Hankel transform relationship between this range-dependent pressure field and a depth-dependent Green's function⁶ is then used, for which peaks in the estimated Green's function are identified with modal wavenumbers k_n that form the basis for inversion studies.^5,7–9

One such study, the Nantucket Sound experiment by Frisk et al.,¹⁰ provides an exemplary demonstration of modal propagation and interference in shallow water and subsequent geoacoustic inversion. Measurements of the complex pressure field for frequencies 140 and 220 Hz were made in waters of depth 13.9 m as a function of range for ∼1.3 km (after which the depth increased). For the observations at 220 Hz, the pressure magnitude exhibited a spatial interference pattern owing to the coherent combination and consequent interference of modes 1 and 2 that dominate the observed field, manifesting an interference length scale or cycle distance $2 π / Δ k ∼$ 136 m, where $Δ k$ is the difference between the wavenumbers corresponding to modes 1 and 2.

At ranges corresponding to destructive interference, strong fades exist in the pressure level that can exceed 40 dB in measurement, while theoretically the pressure (potential energy) can vanish completely if the two modes are of equal amplitude (a small component of kinetic energy remains). While the two-mode interference pattern with one modal interference length scale is easy to interpret, more complex interference patterns, as would be encountered at higher source frequencies or in deeper water, are also exploited in geoacoustic inversion.^9,11,12

In these studies, complex acoustic pressure is the field observable with hydrophones, and it is fading in this quantity, or equivalently, potential energy, that is observed at ranges and depths corresponding to modal destructive interference. The fading occurs at or near ranges and depths corresponding to a point known as a dislocation, about which there exists an intensity vortex.¹³ 

This study examines the vector acoustic properties at dislocations and, more generally, within the vortex region encompassing an intensity vortex and stagnation point. We do this with CW measurements made on the Applied Physics Laboratory, University of Washington Intensity Vector Autonomous Recorder version 2 (IVAR-2),¹⁴ which was deployed during the 2022 campaign of the Seabed Characterization Experiments (SBCEX22). During one IVAR-2 deployment on the New England Mud Patch (NEMP), an area of seabed on the continental shelf with an extensive O(10 m) thick mud layer, a low-frequency narrowband source was towed, commencing ∼ 3 km from IVAR-2 and closing in distance. This generates a (complex harmonic) acoustic vector field as a function of range (pressure and three-axis velocity). At the source frequency, 43 Hz, only three trapped modes can propagate efficiently in the NEMP waveguide. Owing to the depth of the source (45 m), the source weakly excites one of these modes and we show subsequently that only two modes dominate the field. Thus, a relatively simple mode interference pattern of cycle distance $∼ 2 π / Δ k$ is also observed, corresponding to a vortex region repeating in range.

With the full vector acoustic field, one can identify the proximity and depth of the two, isolated singular points¹⁵ (the vortex and stagnation points) characteristic of a vortex region. The remainder of this paper examines these vortex regions in the context of the experimental measurements, and is organized as follows. In Sec. II, a mode-based simulation of vector acoustic properties is presented along with key first- and second-order quantities subsequently used to study details of the vortex region. For this, we use the most realistic description of the waveguide, based on measurements of the water column sound speed, water depth at time of measurement, and a plausible description of the seabed based on several published studies. The vector acoustic field observations from the towed source experiment are presented in Sec. III, including data analysis and comparison with modeled results. An inverse Hankel transform of modeled and measured complex pressure and vertical velocity yields an estimate of the horizontal wavenumber spectrum providing an additional model-data comparison. Section IV concludes with a summary and discussion.

Some elements of the experimental conditions (Sec. III) are introduced here as necessary to describe a realistic acoustic waveguide for modeling purposes. In particular, the waveguide description applies to bathymetry and seabed on the NEMP at position 40.4416°N, 70.5282° W (corresponding to the receiver location), along with a ∼ 2.5 km transect emanating from this location toward the northwest.

The towed source observations were made May 26, 2022 commencing near 00:00 UTC, and water column sound conductivity, temperature, and depth (CTD) measurements made close to this time put the minimum sound speed as 1484 m/s at approximately 25 m, with an approximately iso-speed surface layer of thickness ∼10 m and speed 1493 m/s, and bottom sound speed at depth 77 m was 1488 m/s [Fig. 1(a)]. Water density is set to 1027 kg/m³ based on the water temperature of $9 °$⁠.

A geoacoustic inversion is not within the scope of this study, and instead, we use a plausible description of the seabed based on several published studies. A salient feature of the NEMP seabed in this location is the low-speed mud layer commencing at the water–sediment interface with thickness of order 10 m. In the lower portions of this layer, there is evidence of increasing sand content and hence, sound speed constituting a transition layer of thickness O(1) m between the mud and higher-speed substrate below.^26–29 The seabed geoacoustic model will be informed by these studies. Furthermore, a recent study¹⁴ emerging from the same SBCEX22 field experiment involved a different set of vector acoustic observations on the NEMP using IVAR-2, made 2 weeks earlier [Leg B of research vessel (R/V) Neil Armstrong] at a site ∼ 2 km south and east from this site. Modeling of these observations was similarly informed, for which observations involved the ratio of kinetic to potential acoustic energy of underwater noise measured below a passing ship. We thus start with this nominal form of the seabed upon which further adjustments are made.

The seabed (Table I) consists of three layers below the water–sediment interface of thickness $L i , i =$ 1,3, with density ρ_i, sound speed c_i, and terminated by a half-space characterized by $ρ 4 , c 4$⁠. The mud layer is represented by the first two layers, and the transition layer by the third layer. Two sound speeds are listed for these layers, marking the upper and lower speeds between which the sound speed varies linearly in $1 / c 2 ( z )$⁠. Parameterized in this manner, the sound speed versus depth for the mud layer [Fig. 1(b)] is similar to the power law profile emerging from the study in Knobles et al.²⁶ We note that elastic effects are not addressed for simplicity and it is acknowledged that the attenuation values necessarily represent an effective attenuation for which coupling shear waves have contributed to the energy loss of compressional waves.³⁰ 

At the source frequency of 43 Hz, this waveguide has three trapped modes, with a wavenumber (k_n) having only a very small imaginary component [Fig. 1(c) and Table II]. Each trapped mode shows support below the water–sediment interface, well into the mud layer and deeper sediments. In the following analysis, the source depth is set to 45 m (corresponding to the towing depth in the experiment), which reduces the contribution from mode 2 such that the interference pattern has one dominant range scale (corresponding to the difference in wavenumbers between modes 1 and 3).

This effect is shown [Fig. 2(a)] by way of the active intensity magnitude $| I → |$ as a function of range and depth displayed in units of dB re W/m³, corresponding to the coefficient S_f equal to 22 Pa m, a value we subsequently justify. The thin horizontal line at 77 m identifies the water–sediment boundary, and a sequence of 12 interference features, or vortex regions, close to and occasionally below the water–sediment interface. Two singular points of intensity exist within each region and the white box identifies one region examined in closer detail. An interference length scale set by $2 π / Δ k$ based on the dominant modes 1 and 3 (Table II) is ∼ 252 m, whereas here, the range between these features varies between about 220 and 280 m owing to the influence of the weakly excited mode 2. Although such interference features would have a similar manifestation with pressure-based measures, for example, $| p | 2$ or potential energy E_p, details within each interference region differ in significant ways that are observable with a vector sensor.

Figure 2(b) shows the normalized gradient of acoustic pressure phase expressed in decibels, $10 log 10 | ∇ ϕ | / k$⁠, for the vortex regions near the seabed interface; the solid horizontal line again represents the water–sediment boundary at depth 77 m, and the dashed line represents the sampling depth and effective sampling range of the vector acoustic measurements (Sec. III). Displayed in this manner, large and positive (red) and negative (blue) decibel swings in this quantity map to the vortex center and stagnation point, respectively.^22,24

Expanded views (depth to range aspect ratio 6:25) of the white box in Fig. 2(a), show active intensity magnitude [Fig. 2(c)], and normalized phase gradient [Fig. 2(d)]. With these views, the two active intensity singular points are more visible, the upper representing an intensity vortex center (dislocation) and lower a stagnation point, separated in depth by 1.2 m.

Using the example identified by the box near range 757 m in Fig. 2(a), some salient features of the vortex region are further highlighted by comparing $| I → |$ with $2 c w E p$⁠, along with a third quantity (Fig. 3) with dimension equivalent to intensity but based entirely on velocity, which we express as kinetic energy [Eq. (5) $2 c w E k$]. Comparing levels of $| I → |$ with $2 c w E p$ reveals that the vortex region is composed of two singularities. One is the precise vortex center (single red-shaded funnel). This is the dislocation, where formally $| p | → 0$⁠,^22,24 which, in this case, is tracked by $2 c w E p → 0$⁠. The dislocation is co-located with one of two points; the first being where the active intensity $| I → | → 0$ (two blue-shaded funnels) and the second being the stagnation point. (Note that the tip of the red-shaded funnel appearing to extend farther than the two blue-shaded funnels is a consequence of the computational range and depth resolution equal to 0.01 m.) A well-defined length scale of the vortex region is the separation in depth between the vortex center and stagnation point, which, in this example, equals 1.2 m; though less, this scale is not inconsistent with a more generalized description that puts the linear dimension as $O ( 10 − 1 ) λ$⁠, where λ is acoustic wavelength.¹³ 

Referring again to the vortex region, a sample traverse in range over a depth zone located outside the vortex center in a direction away from the stagnation point, as indicated by $| I → | < 2 c w E p$⁠, will observe a large $| ∇ ϕ | / k$ (positive decibel measure). At the stagnation point $| ∇ ϕ | / k → 0$ taking $| I → |$ with it; however $| p |$⁠, and thus $2 c w E p$⁠, are maintained, although at low level, which puts $| I → | < 2 c w E p$ or equivalently a negative decibel measure for $| ∇ ϕ | / k$⁠. This negative measure is maintained for a sample traverse over a depth zone located outside the stagnation point in a direction away from the vortex center.

The collapse of active intensity at the stagnation point is via both $v r → 0$ and $I z → 0$⁠, but not necessarily v_z.^15,31 Thus, in the vicinity of the vortex region (i.e., the entirety of Fig. 3), the potential energy is exceeded by kinetic energy with the latter dominated by the vertical component $E k z$⁠, as seen by the yellow (kinetic related) and red (potential related) shaded contours.

A plan view of the same vortex region (Fig. 4) shows arrows (intensity streamlines) that identify the direction of active intensity $I →$ and indicate that sound energy is flowing around the vortex center in closed paths.²² The stagnation point serves as transition to open streamlines and continuation of energy flux down the waveguide. Within a small range increment corresponding to depths between the vortex and stagnation points, the energy flux is directed toward the source,¹³ although this is an extremely challenging measurement to confirm owing to the very low magnitude of $I →$⁠.

The background color in Fig. 4 corresponds to contours of circularity Θ, which is the normalized field indicator for the curl of intensity, $∇ × I →$⁠. Intensity curl is zero at the stagnation point and then rapidly increases in magnitude both above and below (with negative sign moving toward the vortex center and positive sign moving in the opposite direction; in this case, toward the seabed). We note that while the intensity streamlines of Fig. 4 intuitively confirm the sign of curl in the vortex region, they are less helpful in understanding the curl in the region below the saddle point, which is positive owing to the rapid increase in horizontal intensity moving away from the dislocation depth (i.e., the component of curl given by $∂ I r / ∂ z$⁠).

The illustrative modeling results discussed in the previous sections were based on the seabed description in Table I to be representative of the experiment data. As the modeling suggests, this range-sampling traverses eight vortex regions, and while each of these vortex regions appear similar in structure, each is centered at a different depth. Model estimates of the measurements [taken along the dashed line in Fig. 2(b)] will be presented alongside experimental data to guide the interpretation of first-order, second-order, and normalized field indicators of the vector field.

The towed source observations were made May 26, 2022 commencing near 00:00 UTC, covering approximately 2500–500 m closing toward the IVAR-2 receiver along a bearing 110°. The source (Geospectrum, Halifax Canada, model M-72) was towed at a mean depth of 45 m, transmitting simultaneous tones at 37, 40, and 43 Hz. A record of towing depth over this range shows a standard deviation of 0.6 m, with, on occasion, differences from the mean exceeding 1 m. The observations discussed here are limited to 43 Hz, owing to lower received level associated with other tonals; however, future work may incorporate these data. The estimated source level at 43 Hz was 146 dB re μPa. The value of S_f (Sec. II) used in modeling equals 22 Pa and is equivalent to a source level of 144 dB, which is acceptably close given reasonable uncertainties in source and receiving system calibration at such low frequency.

The vector field measurements were made with IVAR-2, a system that records four coherent channels of acoustic data continuously; one channel for acoustic pressure and three for acoustic acceleration from which acoustic velocity is obtained. The system is based on a M20-105 sensor (Geospectrum, Halifax Canada) from which data are recorded on a multi-channel AMAR-G4 recorder (JASCO, Halifax Canada). The system configuration, framework, and additional instrumentation installed are described in the recent study¹⁴ emerging from the same SBCEX22 field experiment for which observations were made 2 weeks earlier [Leg B of (R/V) Neil Armstrong] using IVAR-2 at a site 2 km south and east from the coordinates of this current measurement site (Sec. II). As in the previous study, the nominal height above the seabed of the center-point of the sensor was 1.45 m.

We document here only the necessary receiver sensitivities applicable to the 43 Hz signal of interest that originate from calibration data supplied for the M20-105 sensor interpolated to 43 Hz. For the dynamic (pressure) channel this is –177.0 dB re V/μPa. For the kinematic channels, these are expressed in an equivalent-pressure format, equal to –184.3.0 dB re V/μPa for the two horizontal (x, y) velocity channels and –185.3.0 dB re V/μPa for the vertical (z) velocity channel. These values are equivalent to an acoustic acceleration sensitivity of ∼ 30 V/g, where g is gravitational acceleration. Additionally, to arrive at the final estimates corresponding to 43 Hz narrowband complex velocities, a phase correction of 85.4° is applied to the vertical channel and 76.6° to the two horizontal channels.

Baseband demodulation is carried out on all four channels to achieve a time series of complex, narrowband pressure and velocity to be denoted as P(t) and $V x , y , z ( t )$⁠, respectively. Following from the global positioning system (GPS) location of R/V Armstrong (Fig. 5), the 43 Hz source approaches IVAR-2 for roughly 30 min at a steady rate of 1.6 (+/−0.1) m/s. This corresponds to a steady Doppler up-shift, around 43.045 Hz. Owing to the different vertical propagation angles of the trapped modes, the energy is spread in a band (from 43.045 to 43.035 Hz) below the maximum shift. To improve the signal-to-noise ratio (SNR) of the signal, we apply a very narrowband filter (0.01 Hz bandwidth), which provides a signal approximately 25 dB above the in-band noise. Note that this requires a demodulation about the Doppler-shifted frequency, and that when constructing the heterodyne signal, one must also account for the difference between the true transmit frequency (43 Hz) and demodulation frequency (43.04 Hz).

For range registration representing the mapping of, for example, P(t) to P(R), where R is an estimated range between IVAR-2 and source towed from R/V Armstrong, the known constant towing speed combined with GPS location information of R/V Armstrong are used (Fig. 5). This is reasonably straightforward but not without some degree of uncertainty. By shifting the range registration, and comparing with modeled values as a function of range, we estimate this error to be approximately ± 10 m.

Equations (14) and (15) are applied to the pressure results in Fig. 6(a) and vertical velocity results in Fig. 6(c), respectively, following closely the procedural road map in Frisk et al.⁹ The transformed variables for pressure and vertical velocity, respectively, are $p ( r ; z , z s ) , v z ( r ; z , z s )$ (model), and $P ( R ) , V z ( R )$ (IVAR-2 observations) for which the estimated range R is precisely mirrored by modeling range r, and a Hanning window is applied over the range between ∼ 500 and 2500 m. The effective aperture length, R ∼ 2000 m, limits the modal wavenumber resolution but is sufficient for this application.

For the modeled pressure, the operation yields the expected horizontal wavenumber spectrum [Fig. 7(a), magenta line] for which the three peaks coincide with wavenumbers for modes 1, 2, and 3 (Table II). The same operation applied to the measured pressure data [Fig. 7(a), black line] shows reasonable agreement insofar as three major peaks are resolved. Wavenumber real values corresponding to the three peaks are noted in the figure and agree with corresponding real values for horizontal wavenumbers for modes 1–3 (Table II), to within processing resolution.

Applying the same operation to the acoustic vertical velocity model and data provides another view of the normalized depth-dependent Green's function in the form of a vertical derivative [Fig. 7(b)], with horizontal velocity yielding effectively the same result as pressure. Recall that mode 2 is weakly excited owing to the source depth being very close to a zero-crossing, and for vertical velocity, mode 2 is further reduced owing to the receiver also being very close to the zero-crossing [see Fig. 1(c)]. Furthermore, for vertical velocity, the contribution of mode 1 is less than mode 3, opposite pressure and horizontal velocity where mode 1 is dominant. This difference is an effect of the steeper (i.e., more vertical) propagation angle of mode 3 relative to mode 1.

The two components of active acoustic intensity (Fig. 8) represent a coherent combination of pressure and velocity and both generally reinforce the general model-data concurrence first shown in Fig. 6 in terms of second-order quantities. For the eight dislocations passing near the vector sensor, we can see the approach and departure of each through the oscillation in vertical intensity. As acoustic energy flows around these dislocations, the vertical intensity flux is toward the sea surface as the dislocation approaches, and then shifts down toward the seabed as it departs. Note that the vertical intensity is zero at both the maximum and minimum of the horizontal intensity.

An interesting effect in horizontal active intensity [Fig. 8(b)] is observable for the model result at a range just beyond 1500 m. Here, the intensity magnitude falls to less than a picowatt in amplitude but points in direction toward the source (i.e., the magenta line dips below the dashed horizontal line and is not clearly visible on this scale).

The two non-dimensional intensity-based ratios, Θ [Eq. (9)] and $| ∇ ϕ | / k$ [Eq. (12)], though presenting a somewhat greater challenge to model-data comparison (Fig. 9), nonetheless provide observational confirmation of some properties of the vortex region. For additional context, refer to the image of $| ∇ ϕ | / k$⁠, which identifies the eight dislocations that are towed past the sensor [Fig. 2(b)]. Observe the 1 dB magnitude contours of the log of the $| ∇ ϕ | / k$ ratio (black outlines) that form a boundary of the vortex region, and which resemble the image of a butterfly with red wings pointed toward the surface (⁠ $| ∇ ϕ | > k$⁠) and a blue tail extending downward (⁠ $| ∇ ϕ | < k$⁠).

For the eight vortex regions near the seabed, the orientation of singular points have the vortex above the stagnation point, and the opposite orientation occurs for the corresponding set of regions near the sea surface. For each of these regions traversed by the IVAR-2 sensor, the sampling happens at a different depth relative to the two singular points (i.e., the vortex center and stagnation point).

As represented in Fig. 9, the two non-dimensional ratios, $| ∇ ϕ | / k$ and Θ, share a common response of manifesting a local minima or maxima during the sample traverse through each vortex region, but otherwise respond differently. A sample traverse can pass (1) above the pair of singular points, (2) below the pair of singular points, or (3) between the singular points (where horizontal intensity is pointed toward the source), and each condition produces a characteristic signature in $| ∇ ϕ | / k$ [Fig. 9(a)].

The signature of a sample traverse passing above one of these dislocations (which is farther away from the stagnation point) is the “volcano” appearance in $| ∇ ϕ | / k$ (expressed in decibels) where this quantity approaches a positive peak yet has a middle segment that is slightly depressed. A clear signature of such sampling occurs at ranges 1300, 1550, 2050, and 2300 m. The signature of a sample traverse passing below the stagnation point (and farther away from the dislocation) puts $| ∇ ϕ | < k$⁠, which is observed for the sample traverse through the 750 m vortex region.

For circularity Θ [Fig. 9(b)], the response mirrors the properties of $∇ × I →$ as suggested in Fig. 4, where, in this case, a traverse above or below the stagnation point produces a local maxima or minima, respectively. Note that over the 5 m horizontal scale of Fig. 4, Θ appears constant for a given depth manifesting the local, but smoothly varying maxima or minima observed in Fig. 9(b).

While model and data are nominally consistent in the sampling of eight vortex regions, there is a more noticeable difference for the vortex region near 1500 m. Here, the model points to a sampling traverse within the small depth zone (1.2 m) between the vortex center and stagnation point, and both model and data are consistent insofar as Θ passes through a local maxima. However, the model result for $| ∇ ϕ | / k$ falls rapidly (negative in dB measure) and horizontal intensity is directed back toward the source (Fig. 4). This brief occurrence of $| ∇ ϕ | < k$ at the center of the volcano-like feature in the model is not observed in the measurements and we suspect that an insufficient SNR may be limiting the ability to resolve a near-zero horizontal intensity measure concurrent with rapid changes in $| ∇ ϕ | / k$⁠.

Vector acoustic measurements of the underwater sound field originating from a narrowband towed source have yielded measures (Figs. 6–9) beyond those typically observed with pressure-based observations. The experiment took place on the NEMP, and interpretative modeling based on the normal mode code ORCA utilized a geoacoustic model (Table I) that is both well-centered within the range of models emerging from studies conducted at this location and consistent with measurements. The model-data comparisons were based on first-order variables of acoustic pressure and velocity along with inverse Hankel transforms yielding normalized horizontal wavenumber spectra, second-order variables in the form of horizontal and vertical intensity, and non-dimensional intensity-based ratios. We note that the inverse Hankel transform operation applied to acoustic vertical velocity vs range, as was done here, yields a separate normalized horizontal wavenumber spectrum associated with the vertical derivative of the depth-dependent Green's function; this information complements the results based on acoustic pressure.

The source was towed at a depth of 45 m, speed 1.54 m/s closing in range from ∼ 3000 to ∼ 500 m from the IVAR-2 receiver, which sampled 1.45 m above the seabed (referred to as the sampling traverse). At the source frequency (43 Hz) and water depth (77 m), the NEMP waveguide supported three trapped modes, with mode 2 weakly excited owing to the towed source depth; the 3-mode feature in this frequency range mirrors previous observations made near this site using underwater ship noise³² and broadband explosive sources.^3,33 Although being highly consistent with the geoacoustic profile used in Table I, the low-frequency observations discussed here do not resolve all properties of the seabed and its defining low-speed mud layer.

We define more broadly a vortex region that encompasses two singular points for active acoustic intensity: the vortex point that is co-located with a dislocation and the stagnation point. This study has emphasized details of the vortex region near the seabed, with measures of Θ [Eq. (9)] and $| ∇ ϕ | / k$ [Eq. (12)] providing a nuanced view. These measures give a degree of observational confirmation of some properties of the vortex region, for example, information on the particular sampling traverse through the vortex region.

Range registration, or the mapping of observations as a function of time to a function of range between source and receiver, is not without some degree of uncertainty, which we estimate as ± 10 m. Nonetheless, an interesting spatial series, for example, Θ as a function of range [Fig. 9(b)] for a single frequency, emerges that can be contrasted with Θ as a function of frequency for a fixed range.²⁰ 

While model and data are nominally consistent in the sampling of eight vortex regions, there is a more noticeable difference for the region near 1500 m. Here, the model points to a sampling traverse within the small depth zone (1.2 m) located between the vortex center and stagnation point, and both model and data are consistent insofar as Θ passes through a local maxima. The model result for $| ∇ ϕ | / k$ falls rapidly (negative in dB measure) and horizontal intensity is directed back toward the source (Fig. 4). This brief occurrence of $| ∇ ϕ | < k$ is not observed in the measurements and we suspect that an insufficient SNR may be limiting the ability to resolve a near-zero horizontal intensity measure along with rapid changes in $| ∇ ϕ | / k$⁠.

The precise model-based estimate of the separation for the vortex region at range 757 m (Figs. 2 and 3) is 1.2 m. It is interesting to see a small lengthening of this separation for regions that extend into the seabed, e.g., the region at range just beyond 2500 m (Fig. 2), lengthening to ∼ 1.6 m. This increase does not scale with the change in sediment acoustic wavelength, and a more applicable rule plausibly includes the density ratio. Although perhaps more comprehensive descriptions are possible, it is difficult to construct a general description concerning the topology of singular points within the natural ocean waveguides as studied here. However, limiting to a 2-mode case and an idealized waveguide described by pressure release surface and rigid seabed, such a description exists.³¹ 

The normal mode-based modeling indicates that the position (range and depth) of the vortex regions and their formation depends on the precise cancellation of the field and thus are sensitive to wavenumber. For example, extending the mud layer depth from the current 10 by 1 m, or increasing the half-space speed by 20 m/s, produces a significant shift in wavenumbers, disrupting the model-data agreement shown in the horizontal wavenumber spectra (Fig. 7) based on the inverse Hankel tranform of pressure and vertical velocity fields.

The role of shear in the seabed was not exhaustively studied given limited sampling in frequency and range, and it is likely that the attenuation used for the sediment basement (0.25 dB/λ) represents an effective attenuation for which coupling to shear waves has contributed to the energy loss of compressional waves. For example, introducing a shear speed of ∼300 m/s in the sediments below the mud layer³⁴ reduces the real part of the wavenumbers in Table II (starting with the fourth significant digit), but increases the imaginary part. Nonetheless, sediment attenuation appears to influence the depth of vortex regions over increasing range. For example, in Fig. 2(b) each set of three vortex regions (at approximate ranges 250, 500, 750, and repeating again at 1000, 1250, 1500 m, etc.) occurs ∼2 m deeper than the previous set, representing an early, subtle manifestation of mode stripping. This drop is also clear in the data, for example, at range ∼ 750 m the vortex region was above the sample traverse with $| ∇ ϕ | / k < 1$⁠, while at ∼ 1500 and ∼2250 m the opposite occurs; the vortex is below the traverse with $| ∇ ϕ | / k > 1$⁠. Without sediment attenuation, the model shows that the sets repeat at the same approximate depth with no change in separation depth between vortex center and stagnation point. Thus, the primary effect of attenuation for this case appears to be the vertical position of the vortex region.

A longer range aperture would have provided more information relating to sediment attenuation. This, however, was not possible given the multiple tasks planned for the experiment. Although a geoacoustic inversion is beyond the scope of this paper, we note that higher frequency CW tones spanning 50–1500 Hz were also transmitted by the towed source, and future analysis of these observations may well find use in determining the frequency dependence in the sediment geoacoustic properties.

This work was supported by the Office of Naval Research (USA).

The authors have no conflicts of interest to disclose.

The observed and subsequently processed data that support the findings in this study, and in the form displayed here, are available from the corresponding author upon reasonable request.