This study investigates the subsurface sound channel or acoustic duct that appears seasonally along the U.S. Pacific Northwest coast below the surface mixed layer. The duct has a significant impact on sound propagation at mid-frequencies by trapping sound energy and reducing transmission loss within the channel. A survey of the sound-speed profiles obtained from archived mooring and glider observations reveals that the duct is more prevalent in summer to fall than in winter to spring and offshore of the shelf break than over the shelf. The occurrence of the subsurface duct is typically associated with the presence of a strong halocline and a reduced thermocline or temperature inversion. Furthermore, the duct observed over the shelf slope corresponds to a vertically sheared along-slope velocity profile, characterized by equatorward near-surface flow overlaying poleward subsurface flow. Two potential duct formation mechanisms are examined in this study, which are seasonal surface heat exchange and baroclinic advection of distinct water masses. The former mechanism regulates the formation of a downward-refracting sound-speed gradient that caps the duct near the sea surface, while the latter contributes to the formation of an upward-refracting sound-speed gradient that defines the duct's lower boundary.

Ocean processes along the U.S. Pacific Northwest coastline are dominated by the large-scale circulation within the California Current System (CCS). This dynamically complex eastern boundary current system includes the California Current, the California Undercurrent, seasonal currents such as the upwelling jet and the wintertime Davidson Current, and eddies.1,2 The general thermohaline structure of the CCS is regulated by the advection of different water masses originating from the subarctic and tropical eastern Pacific by the major currents within the CCS. The two primary source water masses are the Pacific Subarctic Upper Water (PSUW), which is relatively cold and fresh, and the Pacific Equatorial Water (PEW), which is relatively warm and salty.3 The PSUW enters the CCS from the north, carried by the equatorward-flowing California Current near the surface. The PEW enters the CCS from the south, carried by the poleward-flowing California Undercurrent in the subsurface. In coastal waters off the west coast of Canada, the overflowing of cold, fresh water of subarctic (e.g., PSUW) and riverine origins over warm, saline water of equatorial origin (e.g., PEW) contributes to the formation of a pronounced halocline, which is often accompanied by a weak thermocline or a temperature inversion (i.e., an increase in temperature with depth) in the absence of static instability.4,5 According to Roden,4 different processes can lead to the occurrence of temperature inversions along a halocline in different regions of the ocean. The occurrence of temperature inversions in deep subarctic waters is attributed to wintertime cooling followed by summertime heating of the sea surface driven by air-sea interactions. On the other hand, Roden4 attributes the temperature inversions in subtropical regions such as those off the coast of Southern California to baroclinic advection of water of different origins.

The presence of a strong halocline and a reduced thermocline or temperature inversion often results in a local minimum in sound speed.5 A secondary sound channel or acoustic duct whose axis lies at the depth of the local sound-speed minimum then forms within the halocline.5 The duct behaves as a waveguide that traps acoustic energy and is therefore important for long-distance sound propagation.5 Such subsurface ducts have been observed off the west coast of Canada,5 in the Beaufort Sea,6 and off the coast of New Jersey.7,8 These secondary ducts differ from the sound fixing and ranging (SOFAR) channel or the deep sound channel, which is typically found below the permanent thermocline at depths between 600 and 1200 m at low and middle latitudes.9 Seawater thermohaline properties, particularly salinity, have a dominant influence on the formation of the secondary duct.5 In comparison, the SOFAR channel occurs at a depth where pressure begins to have a dominant effect on sound speed.9 

In this paper, we report the observation of secondary subsurface acoustic ducts offshore of the U.S. Pacific Northwest coastline, occurring at shallower depths than the primary subsurface duct of the SOFAR channel. We examine the influence of these secondary subsurface ducts on acoustic propagation within the frequency range of 1–10 kHz, referred to as mid-frequency in this paper. Acoustic simulations based on the measured sound speed field show that the duct has a strong impact on sound propagation at 3.5 kHz. Analysis of in situ measurements of ocean properties identifies the formation of the shallow duct with the mixing of water masses driven by large-scale circulation within the northern CCS and surface heat exchange.

The remainder of this paper is structured as follows. Section II presents the observational data and describes the method for detection and characterization of the subsurface duct. Section III presents the results of the acoustic simulations conducted to investigate the duct's impact on sound propagation. Section IV discusses the temporal and spatial variations of the prevalence of the duct. Section V discusses the potential mechanisms that contribute to duct formation. Section VI provides concluding remarks and an outlook on future research.

The conductivity-temperature-depth (CTD) profile data analyzed in this study were recorded by the Washington Offshore profiler mooring of the Ocean Observatories Initiative's Coastal Endurance Array (OOI-CEA) observatory between May 2014 and March 2022 [Fig. 1(a)] and during recurrent deployments of two types of gliders: Seagliders operated by the Applied Physics Laboratory of the University of Washington (APL-UW) from July 2010 to March 2022 [Fig. 1(a)] and OOI-CEA Slocum coastal gliders from April 2014 to April 2021 [Fig. 1(b)].10 The OOI-CEA Washington Offshore mooring is located over the shelf slope at 46.85°N, 124.98°W, where the seafloor depth is 540 m [Fig. 1(a)].10 The wire-following profiler of the mooring, equipped with a CTD, a three-dimensional (3D) single-point velocity meter, and other scientific instruments, moves up and down through the water column between a depth of 30 and 500 m 6–8 times a day. The CTD data recorded during each up and down cast is used to construct a vertical profile of sound speed, which is computed as a function of temperature, salinity, and pressure using the 2010 International Thermodynamic Equation of Seawater (TEOS-10).11 To extend each profile to the sea surface, we first obtain concurrent temperature and salinity measurements from the CTD data recorded by the OOI-CEA Washington Offshore surface mooring at a nominal depth of 7 m. The surface and profiler moorings are located within 1 km of each other. We then interpolate temperature and salinity over the vertical range from the top of each profile to the surface mooring, from which we compute near-surface (i.e., depth < 30 m) sound speed. Subsequently, the extended sound-speed profiles between depths of 7 and 500 m are interpolated onto a uniform vertical grid of 1 m resolution. For gliders, the CTD data recorded during the descent and ascent of each dive are processed to yield two vertical sound-speed profiles, respectively, which are then interpolated onto a uniform vertical grid of 1 m resolution. For the acoustic simulations described in Sec. III, the glider sound-speed profiles are linearly extrapolated to the sea surface and the seafloor, using the sound-speed gradients calculated from the near-surface and near-bottom sections of the original sound-speed profiles, respectively.

We identify the subsurface duct, if present, in each vertical sound-speed profile using an automated algorithm developed using matlab. The algorithm first finds the local sound-speed minima in a sound-speed profile using the matlab function “findpeaks,” which locates the local extrema within an input signal vector (e.g., a sound-speed profile),12 and represents nearby ( Δ z < 5 m) groups of minima by their single most prominent member. From these candidates, the algorithm selects the sound-speed minimum whose depth, prominence, and width satisfy the criteria predetermined by visual inspection of a sample set of sound-speed profiles [Figs. 2(c) and 3(c)]. These criteria are as follows: the product of prominence and width is below 20 m2/s, and the depth is less than 150 m. Here, the prominence of a sound-speed minimum is a measure of its difference from an adjacent maximum (see MathWorks12 for the definitions of prominence and width of a local extremum).

In the sound-speed field determined from the CTD data recorded by a Seaglider during August 3 to 11, 2014, the subsurface duct appears as a horizontal layer of relatively low sound speed across the over 160 km long transect [Fig. 2(a)]. The duct's axis, as defined by the depth of the sound speed minimum, shoals from depths of 110 m offshore to 50 m nearshore [Fig. 2(a)]. We identity the eddy-like feature near the middle of the transect between 26 and 26.5 kg/m3 isopycnals as a California Undercurrent eddy (cuddy), which is a submesoscale coherent vortex that originates along the continental slope and travels offshore in the west-southwest direction.13 The warm and saline seawater trapped in a cuddy originates from the PEW transported by the California Undercurrent.14 As a result, a cuddy has relatively high sound speed, as observed in the Seaglider data [Fig. 2(a)]. At a depth of 150 m, the sound speed inside the cuddy is approximately 1484 m/s, while outside the core of the cuddy, the sound speed is approximately 1481 m/s [Fig. 2(a)].

The seaward propagation of a cuddy is potentially an important mechanism for transporting the warm, saline PEW carried by the California Undercurrent from the continental slope to offshore regions.15 Likewise, this process can contribute to the offshore formation of the subsurface duct by supplying the warm, saline water, which therefore has high sound speed, below the axis of the duct [Fig. 2(a)].

The sound-speed field determined from the CTD data recorded by an OOI-CEA coastal glider during October 6 to 16, 2018, shows a subsurface duct that extends from offshore of the continental slope to over the shelf [Fig. 3(a)]. The central axis of the duct shoals from 140 m depth in offshore regions to 30 m over the continental shelf. There are “pockets” of water of relatively high sound speed, approximately 1483 m/s, compared with 1480 m/s found within the duct; two of these pockets are located at ranges of approximately 50 and 150 km.

The subsurface ducts observed in Seaglider [Fig. 2(a)] and OOI-CEA coastal glider [Fig. 3(a)] sound-speed fields share the following characteristics. First, both ducts have axes located beneath the surface mixed layer and exhibit shoaling in the shoreward direction. Second, the axes of both ducts lie between the 25.5 and 26 kg/m3 isopycnals. Third, the lower boundaries of both ducts, which comprise water of relatively high sound speed, are bounded approximately between the 26 and 26.5 kg/m3 isopycnals. The latter isopycnal is known to be at the core, that is, the axis of peak flow velocity, of the California Undercurrent.2,16–18 These similarities suggest that the subsurface ducts in the two instances may originate from the same formation mechanisms, a possibility considered further in Sec. V.

The subsurface duct in the OOI-CEA glider sound-speed field is also observed in the sound-speed profiles determined from the CTD data recorded by the adjacent OOI-CEA Washington Offshore profiler mooring [Fig. 4(a)]. The local sound-speed minima that constitute the duct's axis are detected in most sound-speed profiles in summer to fall (June to November). In winter (e.g., December to February), the surface mixed layer deepens because of enhanced wind forcing and convection (Fig. 4). The deepening of the surface mixed layer causes the axis of the subsurface duct to also move down, and the duct disappears when vertical mixing is strong enough to eliminate the corresponding subsurface sound-speed minimum [Fig. 4(a)]. Subsequently, continued surface cooling from late winter to early spring (February to April) leads to the formation of a surface duct, of which the sound speed minimum is located at the sea surface [Figs. 4(a) and 4(b)].

The effect of subsurface ducts on mid-frequency sound propagation is demonstrated by simulating a case where an acoustic point source emitting 3.5 kHz monochromatic sound is inside a subsurface duct using the parabolic equation (PE) method.19 It numerically solves the one-way wave equation along a vertical slice of the ocean, including cylindrical spreading and assuming azimuthal isotropy, but ignoring out-of-plane effects, which are not expected to be significant in the problems under investigation.

The sound-speed field employed in this analysis is adapted from the Seaglider observations shown in Fig. 2(a) treated as a true realization of the ocean for a particular time. In reality, it took the glider approximately 10 days to collect data along this transect; therefore, the sound speed field displayed in Fig. 2(a) is not a true “snapshot” of the actual sound speed field and may contain spatial aliasing. The acoustic simulation reported in this paper is based on the following assumptions. (i) The sound speed field from the glider data is treated as a snapshot, and any range-dependency of the field due to the finite speed of the glider is treated as actual range-dependence. (ii) The glider data do not cover depths very close to the sea surface; the missing near-surface data are put in through linear interpolation using the sound-speed gradient computed from the near-surface portion of the profile. (iii) The glider data do not cover depths close to the seafloor; the missing near-bottom data are filled in through linear extrapolation, assuming a sound speed increase of 0.016 m/s for every meter increase in depth. The geoacoustic properties along the transect are not known. For this simulation effort, the sediment is assumed to be a homogeneous half-space with sound speed of 1510 m/s, sediment-to-water density ratio of 1.2, and attenuation coefficient of 0.05 dB/wavelength.

In the simulation, we place the acoustic source near the axis of the subsurface duct at the 100 m depth and 20 km from the western end of the glider transect (Fig. 5). It is important to note that the duct shoals and narrows above the cuddy. To the east of the cuddy, the axis of the duct shifts to a shallower depth of approximately 85 m [Fig. 5(a)]. The simulated sound level shows strong arrivals near the ranges of 60, 90, and 120 km [Fig. 5(b) and 5(c)]. They are the convergence zone arrivals, which are refracted upward by the high sound speed in deep water. These arrivals are not related to the subsurface duct. Here, we define sound level as 10 log 10 ( p ( r , d ) 2 ) in dB, where p ( r , d ) is the simulated acoustic pressure at range r and depth d resulting from a source transmitting sound with unit amplitude at a 1 m range in a homogeneous medium. The trapping of sound within the subsurface duct is most noticeable as the high sound field intensity near the 100 m depth between 20 and 40 km [Fig. 5(b) and (c)]. Guided by the duct, the trapped sound moves to a shallower depth between 90 and 120 km [Fig. 5(b)]. This simulation result suggests that when a sound source is placed inside a subsurface duct, there is significant reduction of transmission loss inside the duct and trapped sound energy follows the range-dependent duct. In order to accurately model sound propagation in and near subsurface ducts, it is essential to have reliable models or actual measurement of the presence and structure of such ducts.

A recent study has demonstrated the frequency-dependent nature of the effects of a subsurface duct on sound propagation.6 One important parameter used to characterize this frequency-dependency is the cutoff frequency ( f c)—the lowest frequency at which sound energy can be trapped inside a given duct. We estimated f c for our case by employing a standard eigenvalue finder20 to compute the first normal mode function over a range of frequencies. We then identify f c as the frequency above which the ducting effect becomes apparent, with the peak of the first-mode function predominantly confined within the vertical bounds of the subsurface duct. We applied this method to a sound-speed profile extracted from the glider sound-speed field used in the acoustic simulation [Fig. 6(a)]. For the computation of normal modes, we set the water depth at 987 m and used the same half-space bottom model. Our computation encompassed a frequency range spanning from 70 to 120 Hz in 10 Hz increments. The results exhibit significant ducting effects for frequencies above 90 Hz, which we consequently identified as the cut-off frequency for the duct [Fig. 6(b)].

Analysis of the long-term time series of sound-speed profiles constructed from the CTD data recorded by the OOI-CEA Washington Offshore profiler and surface moorings reveals strong seasonal variability in the formation of the subsurface duct over the shelf slope (Fig. 4). We compute the record-averaged monthly duct prevalence, which is the percentage of the profiles within a given month that include sound speed minima associated with the subsurface duct, from the mooring data recording from May 2014 to March 2022. The result shows that the duct is more prevalent in summer to fall (June to November) than it is in winter to spring (December to May) (Fig. 7). The duct prevalence reaches the maximum of 70% in July and the minimum of just above 10% in February. The other properties, such as the depth, temperature, salinity, and spiciness21 at the duct's axis, also show strong seasonal variability (Fig. 8). Here, we calculated spiciness at a reference pressure of 0 dbar (i.e., sea surface) using the TEOS-10.11 

The depth of the duct's axis increases from mid fall through winter (October to February) [Fig. 8(a)], which is likely driven by the seasonal deepening of the surface mixed layer. The temperature, salinity, and spiciness at the duct's axis increase from summer to winter. In spring, the water at the duct's axis is warmer, fresher, and of lower spiciness than the rest of the year [Fig. 8b–8(d)].

We analyze the sound-speed fields determined from the CTD data recorded by Seagliders and OOI-CEA coastal gliders to investigate the spatial variability of duct formation along the U.S. Pacific Northwest coastline. It should be noted that these glider observations may involve bias due to sampling along a relatively small number of targeted transects. Nevertheless, they offer valuable insights into the spatial distribution of subsurface ducts within our study site, particularly in the cross-slope direction. Analysis of the combined Seaglider and OOI-CEA coastal glider sound-speed observations from July 2010 to March 2022 shows a decrease in duct prevalence with shoaling seafloor depth (Fig. 9). Over the continental shelf, the duct is observed in <5% of the sound-speed profiles in regions shallower than 100 m. Offshore of the continental slope, the duct prevalence reaches a maximum of over 40%. The axis of the duct shoals onshore from between 80 and 100 m offshore of the continental slope to less than 60 m over the shelf [Fig. 10(a)]. The decrease in duct depth over the shelf coincides with an upward tilt of the isopycnals near the duct's axis as observed in the OOI-CEA glider data [Fig. 3(a)]. Over the shelf, the water at the duct's axis is warmer, fresher, and of lower (i.e., a larger negative value) spiciness than off the shelf [Fig. 10(b)–10(d)]. These differences in thermohaline properties indicate mixing of the water inside the duct with near surface water as the duct's axis shoals onshore. Offshore of the shelf break, the temperature, salinity, and spiciness at the duct's axis decrease with increasing seafloor depth in waters deeper than 375 m.

In a hydrostatically stable environment, a temperature inversion leads to the formation of a local sound-speed minimum and thus a subsurface duct. On the other hand, a temperature inversion is not a necessary condition for the formation of a subsurface duct, which requires the presence of an upward-refracting sound-speed gradient (increasing with depth) below the surface mixed layer. As explained in the following analysis, an upward-refracting sound-speed gradient can occur along a steep halocline, which is common for the Pacific Northwest, in the absence of a temperature inversion.

With a vertical axis ( z) that is positive upwards, an upward-refracting sound-speed gradient in a hydrostatically stable environment can be expressed as the following conditions:
d c / d z < 0 ,
(1)
d ρ / dz < 0.
(2)
Assuming the water column is horizontally homogeneous and stably stratified, the vertical gradient of in situ sound speed ( c) can be broken down into contributions from vertical gradients in conservative temperature ( θ), absolute salinity ( S A), and pressure ( p) as
d c / d z = c / θ d θ / d z + c / S A d S A / d z + c / p d p / d z .
(3)
Similarly, the vertical gradient of density can be written as
d ρ / d z = ρ / θ d θ / d z + ρ / S d S A / d z + ρ / p d p / d z .
(4)
Here, θ and S A, measured in degrees Celsius (°C) and parts per thousand (ppt), respectively, follow the definitions in TEOS-10.11 The unit for p is the decibar (dbar). We calculate the change of sound speed and density due to unit change in conservative temperature, absolute salinity, and pressure at θ = 8.12 ° C, S A = 33.49 ppt, and p = 106.78 dbar. These values represent the averages taken between the axis of the duct and that of the high-sound-speed layer beneath the duct identified in the sound-speed profiles derived from the CTD data collected by the OOI-CEA Washington Offshore profiler mooring (Fig. 4). The results are c / θ = 3.7421 m / s / ° C, c / S A = 1.2316 m / s / ppt, c / p = 0.0163 m / s / dbar, ρ / θ = 0.1548 k g / m 3 / ° C, ρ / S A = 0.7803 k g / m 3 / ppt, and ρ / p = 0.0046 k g / m 3 / dbar. These derivatives are obtained by computing the differences in sound speed and density due to unit change in θ, S A, and p around the aforementioned reference values using TEOS-10.11 Substituting these and d p / d z 1 dbar / m into Eqs. (3) and (4), the conditions in Eqs. (1) and (2) become
d θ / d z < 0.0044 C / m ( 0.3291 C / ppt ) d S A / d z ,
(5)
d θ / d z > 0.0297 C / m + ( 5.0407 C / ppt ) d S A / d z .
(6)
Because d S A / d z < 0 (salinity increases with depth) in the Northeast Pacific, Eqs. (5) and (6) can be rewritten as
d θ / d z < 0.0044 C / m + ( 0.3291 C / ppt ) d S A / d z ,
(7)
d θ / d z > 0.0297 C / m + 5.0407 C / ppt d S A / d z .
(8)

The right-hand side (RHS) of Eq. (7) is positive, and the RHS of Eq. (8) is negative. Therefore, an upward-refracting sound-speed gradient can form with d θ / d z being either positive or negative, which suggests a temperature inversion below the mixed layer (i.e., d θ / d z < 0 ) is not a necessary condition. This aligns with the observation that not all occurrences of subsurface ducts identified in mooring and glider data are accompanied by temperature inversions. For example, for the sound-speed profiles obtained from the CTD data collected by the OOI-CEA Offshore and Surface moorings from May 2014 to March 2022, approximately 63% of the profiles with subsurface ducts also include temperature inversions. This ratio becomes 61% for data collected by the Seagliders from July 2010 to March 2022 and is 66% for data collected by the OOI-CEA gliders from April 2014 to April 2021. From Eqs. (7) and (8), it is also evident that the range of d θ / d z that allows the occurrence of an upward-refracting sound-speed gradient expands with increasing d S A / d z. This suggests a steep halocline, such as the one observed off the Washington Coast and further north in the subarctic region of the Northeast Pacific, is a favorable condition for the occurrence of an upward-refracting sound-speed gradient below the mixed layer and thus the formation of a subsurface duct.

Our region of interest (Fig. 1) lies in the transition between the subarctic and the subtropical systems described by Roden,4 who attributes the occurrence of the shallow temperature inversions in the two systems to wintertime surface cooling and baroclinic advection of distinct water masses, respectively. When investigating the formation of the subsurface duct in our study area, it is sensible to consider the combined effects of these two processes.

Surface heat losses are particularly pronounced during November to March in the northern part of our study site. The deepening of the surface mixed layer driven by surface heat losses and convection likely leads to the reduced duct prevalence in late winter and early spring over the shelf slope of the Washington Coast by merging the mixed layer with the shallow temperature minimum (Fig. 7). On the other hand, the wintertime deepening and cooling of the mixed layer followed by the springtime surface warming likely facilitates the formation of the sound speed minima along the duct's axis in the coming summer. This idea of duct formation driven solely by local surface cooling and heating, however, is inconsistent with the observations from the OOI-CEA Washington Offshore profiler mooring. Specifically, the water at the duct's axis identified in the summertime sound-speed profiles is colder and fresher than the water present locally along the same isopycnals earlier in late winter to spring (Fig. 11). This relatively cold, fresh water within the duct is unlikely to have been formed locally by surface cooling but would need to have originated further north, possibly in relict entrainment and transition zones at the bottom of colder, fresher deep winter mixed layer, and be transported southward by horizontal advection within the CCS, with modification by shear and mixing en route.

The baroclinic advection of distinct water masses within our study site is largely driven by the southward-flowing, relatively cold and fresh California Current over the northward-flowing, warmer and saltier California Undercurrent. This process contributes to the formation of a strong halocline, along which a subsurface duct can form when the thermocline is sufficiently weak or inverted. The ocean currents over the upper shelf slope off the Washington Coast have strong seasonal variability, which is evident in the flow velocity measured by the current meter on the OOI-CEA Washington Offshore profiler mooring [Figs. 12(a) and 13(a)]. The currents flow primarily in the along-slope directions and are largely in geostrophic balance.17 Near the sea surface, the along-slope current is primarily poleward from late fall through winter and reverses its direction to become primarily equatorward during spring. The timing of this reversal varies inter-annually. For example, the reversal occurred in late April in 2019 [Fig. 12(a)] and late February in 2021 [Fig. 13(a)]. Previous studies on the CCS have referred to the poleward near-surface current as the Davidson Current.1 In comparison, the along-slope current below 200 m depth is primarily poleward, except for a brief period during spring to early summer (March to June) when the vertical extent of the equatorward surface current reaches its peak [Figs. 12(a) and 13(a)]. During this period of strong equatorward near-surface flow, the sound speed at the axis of the duct decreases and subsequently reaches a minimum in summer (June to August) [Figs. 12(c) and 13(c)]. This observation suggests that the relatively cold, fresh water within the duct likely originates from regions in the north and is transported to the mooring site during spring to early summer when the prevailing near-surface current is equatorward.

In addition to the equatorward California Current, the California Undercurrent, flowing poleward over the continental slope, plays a critical role in the formation of a subsurface duct by supplying the underlying water that is relatively warm, saline, and of high sound speed [Figs. 12(c) and 13(c)]. This observation is consistent with the previous finding by Dosso and Chapman5 that the formation of the ducts in coastal waters off the west coast of Canada is strongly influenced by the California Undercurrent. Moreover, the offshore increase in duct prevalence (Fig. 9) likely reflects the influence of the seaward extension of California Undercurrent driven by cross-slope eddy transport.12 During upwelling conditions, the onshore intrusion of warm, high-salinity water within the California Undercurrent beneath cold, low-salinity shelf water contributes to duct formation over the shelf. The low (<10%) duct prevalence in waters shallower than 150 m suggests the intrusion weakens towards shore (Fig. 7).

At the mooring site, the California Undercurrent is observed as the poleward along-slope flow around the 26.5 kg/m3 isopycnal [Fig. 12(a) and 13(a)]. During summer to fall, flow of the equatorward California Current above the poleward California Undercurrent results in a vertically sheared structure in the along-slope current within the upper water column [Fig. 12(a) and 13(a)]. This drives baroclinic advection of water of the relatively cold, fresh PSUW from the north and the relatively warm, saline PEW from the south, causing the halocline to steepen and the thermocline to weaken. These conditions, as discussed in Sec. V A, favor the formation of an upward-refracting sound-speed gradient and hence a subsurface duct, either directly through baroclinic advection or through preconditioning the thermocline by increasing sound speed at the bottom of the subsequent deep winter mixed layer further north. It is also worth noting that the timeframe and depth range within which the sound speed inside the duct reaches a minimum appear to coincide with an intensified negative shear in along-slope velocity [Fig. 12(b) and 12(c) and Fig. 13(b) and 13(c)].

The correspondence of duct occurrence with thermohaline and flow structures discussed above becomes clearer when examining time-averaged profiles of sound speed, along-slope velocity, temperature, and salinity (Fig. 14). These profiles are derived from the CTD and current-meter data collected by the OOI-CEA profiler mooring from May 2014 to March 2022. Specifically, in the presence of subsurface ducts, the along-slope current exhibits an intensified, negative vertical shear, with equatorward flow above the duct's axis and poleward flow below it [Fig. 14(b)]. The temperature profile exhibits a weakened thermocline [Fig. 14(c)], while the salinity profile remains largely unchanged [Fig. 14(d)].

The intricate nature of ocean dynamics in the northern CCS implies that the formation of the subsurface ducts in this region is unlikely to be solely attributed to either of the two aforementioned mechanisms: wintertime surface cooling and baroclinic advection of distinct water masses. Instead, a complete understanding of the origin of these ducts can only be achieved through a comprehensive analysis that integrates these mechanisms with other prevailing processes, such as coastal upwelling and downwelling, from a regional perspective rather than a localized one. The limitations of the observational data to this study, especially the reliance on measurements from a single location for ocean currents, have restricted our ability to obtain a synoptic view of the duct-formation processes within our study site. Nevertheless, we can still obtain valuable insights into the physical processes contributing to duct formation through an analytical approach.

We initiate our analysis by assuming an idealized environment with negligible diffusion and forcing (e.g., radiation, precipitation, evaporation, etc.). Consequently, the ensuring analysis does not directly address the first duct-formation mechanism, wintertime deepening and cooling of the surface mixed layer, as it involves air-sea heat exchange and turbulent vertical mixing. Further investigation of this mechanism is deferred to future studies. Under these simplifications, the conservation equations governing the conservative temperature ( θ) and absolute salinity ( S A) are as follows:
θ t + V H H θ + w θ z = 0 ,
(9)
S A t + V H H S A + w S A z = 0.
(10)
Here, V H is the horizontal velocity vector, H θ and H S A are the horizontal gradients of temperature and salinity, w is the vertical velocity, and z is the vertical axis that is positive upward. Subsequently, we approximate sound speed ( c) as a linear function of θ, S A, and p ,
c = c 0 + b θ θ θ 0 + b S S A S A 0 + b p p p 0 ,
(11)
where b θ, b S, and b p are all positive constant coefficients in the approximation of c near its value c 0 at locally representative state variables θ 0, S A 0, and p 0. The vertical sound-speed gradient can then be written as
c z = b θ θ z + b S S A z + b p p z .
(12)
We assume that the water column is in hydrostatic balance:
p z = g ρ 0 ,
(13)
where g is the gravitational acceleration and ρ0 is a constant reference density at θ0, S A 0, and p0. Here, density changes (Δρ) are neglected as Δρ ≪ ρ0 across the depth range of interest. Substituting (13) into (12) gives
c z = b θ θ z + b S S A z b p g ρ 0 .
(14)
Taking the time derivative of (14) yields the time rate of change of vertical sound-speed gradient in relation to θ and SA:
t ( c z ) = b θ 2 θ z t   + b S 2 S A z t
(15)
Applying b θ z to (9), b S z to (10), summing the results, and substituting (15) gives:
t ( c z ) + V H z H ( b θ θ + b S S A ) + V H H z ( b θ θ + b S S A ) + w z z ( b θ θ + b S S A ) + w 2 z 2 ( b θ θ + b S S A ) = 0.
(16)
From (11), applying the hydrostatic approximation (13), assuming a flat sea surface and constant atmospheric pressure, we derive the following relations:
H C = H ( b θ θ + b S S A ) ,
(17)
H c z = H z ( b θ θ + b S S A ) ,
(18)
2 c z 2 = 2 z 2 ( b θ θ + b S S A ) .
(19)
Substituting (14) and (17) to (19) into (16) and rearranging the terms gives
t ( c z ) = ( V H z H C + V H H c z ) [ w z ( c z + b p g ρ 0 ) + w z ( c z ) ] .
(20)

The left-hand side of Eq. (20) represents the time rate of change of the sound-speed gradient. The formation of a subsurface duct requires the sound-speed gradient to be downward refracting (positive) near the sea-surface and upward refracting (negative) below the surface mixed layer. As such, processes that contribute to the formation of a duct are the ones that increase the sound-speed gradient [i.e., / t ( c / z )>0] near the sea-surface, such as summertime surface warming, and the ones that reduce the sound-speed gradient below the surface mixed layer. The terms on the RHS of Eq. (20) are organized into two groups: the first group relates to the impact of horizontal flow on duct formation and the second group to vertical flow. In the following discussion, we examine the role of both groups in duct formation with a specific focus on subsurface processes, considering that surface heating is likely the predominant contributing mechanism near the sea surface.

The two terms in the first RHS group represent changes in sound-speed gradient over time, driven by vertical shear in horizontal flow and horizontal advection. We estimate the sum of these terms for the location of the OOI-CEA Profiler mooring using a combination of the mooring current-meter measurements and the climatology data from the World Ocean Atlas (WOA) 2018 for the 2005–2017 decadal period.22 Specifically, we estimate V H and V H / z by averaging the horizontal flow velocity measured from May 2014 to Mar 2022 and calculating the vertical gradient of the time-averaged velocity, respectively. Additionally, we estimate H c and H ( c / z ) from a three-dimensional, background sound-speed field obtained by interpolating the WOA climatology over a 40  × by 40 km2 area centered at the profiler mooring, spanning depths from 40 to 120 m. The results suggest that accounting for the minus sign, the estimated value of the first RHS group is generally negative within the depth range of 50–85 m (Fig. 15). This finding indicates that, within this depth range, the combined effect of vertically sheared horizontal flow and horizontal advection results in a reduction of the sound-speed gradient. This, in turn, contributes to the formation of an upward-refracting sound-speed gradient. Notably, the aforementioned depth range encompasses the typical depth of and the region immediately below the axis of the subsurface duct identified in the mooring observations [Fig. 14(a)]. This observation provides further evidence of the significant role played by horizontal flow and its vertical shear in the process of duct formation.

In the absence of reliable measurements of vertical flow velocity and its vertical structure, we are unable to directly estimate the combined value of the second RHS group of Eq. (20). However, we can still gain insights into the impact of vertical flow velocity on duct formation by examining the first and second vertical derivatives of sound speed derived from the WOA climatology.22 The results indicate that c / z stays positive within the depth range of 40–120 m, while 2 c / z 2 remains negative (Fig. 16). This, combined with the fact that bpgρ0 is a positive constant, suggests that the two terms within the second RHS group will contribute to the formation of an upward refracting sound-speed gradient [i.e., / t ( c / z )>0] if w / z is positive and w is negative within the same depth range. The first condition will be met in the presence of horizontal convergence, as w / z = ( u / x + v / y ), where u and v are the eastward and northward velocity components, for incompressible fluid. The second condition corresponds to downward advection of the vertical sound-speed gradient.

Our analysis of the hydrographic data recorded along the U.S. Pacific Northwest coastline leads to the identification of a secondary subsurface acoustic duct. A numerical simulation based on the sound-speed field determined from glider CTD data suggests that the presence of the duct has major impact on sound propagation at a mid-range frequency of 3.5 kHz in the upper ocean. Specifically, the ducting effect is evident in the trapping of sound energy and the consequent reduction in transmission loss within the duct. Glider observations show that the duct is a large-scale phenomenon that extends hundreds of kilometers from the outer continental shelf to regions offshore of the continental slope. The axis of the duct shoals onshore from between 80 and 100 m depth offshore of the continental slope to less than 60 m over the shelf. Analysis of the sound-speed profiles determined from glider CTD data suggests that the prevalence of the duct decreases onshore, from over 40% in regions offshore of the continental slope to less than 5% over the shelf. In addition, analysis of the long-term time series of sound-speed profiles determined from the CTD data recorded over the shelf slope off the Washington Coast suggests that the duct is more prevalent in summer to fall than in winter to spring. Furthermore, examination of concurrent OOI-CEA mooring observations of sound speed and flow velocity indicates that the duct observed over the shelf slope is associated with a vertically sheared along-slope velocity profile, characterized by equatorward near-surface flow overlaying poleward subsurface flow.

The subsurface duct typically occurs where there is a strong halocline in conjunction with a reduced thermocline or temperature inversion. Our investigation into duct formation focuses on two key mechanisms: seasonal surface heat exchange and baroclinic advection of distinct water masses. In the first mechanism, wintertime surface cooling provides a vertical pathway for cold, fresh surface water to sink and mix with the water in the subsurface through convection. The process is followed by summertime surface heating that leads to the formation of a downward-refracting sound-speed gradient that caps the duct near the sea surface. The second mechanism involves the equatorward near-surface flow of cold, fresh water with a low sound speed, overlaying the poleward subsurface flow of relatively warm, saltier water with a higher sound speed. This process contributes to the development of an upward-refracting sound-speed gradient below the surface mixed layer, defining the lower boundary of the duct. A better understanding of the spatiotemporal variability of duct formation requires detailed analysis of long-term observation of the sound speed field across the Pacific Northwest. This is left for future research, partly because of the lack of spatial and temporal coverage of existing observational data. Namely, the existing glider observations are unevenly distributed in space and discontinuous in time; the long-term mooring observation is limited to a single location. Given that the subsurface duct is a large-scale phenomenon having pronounced spatiotemporal variability, ocean models, supplemented by small-scale stochastic models, are a potentially important tool for providing true synoptic views of the sound speed field across a large spatial domain such as the region of interest for this study. An effort to evaluate the skill of available global and regional ocean models with respect to sound speed is under way and will remain a focus for our future research.

This work was supported by the Task Force Ocean Program, U.S. Office of Naval Research, Grant Nos. N00014-21-1-2419 and N00014-21-1-2524.

The data collected by the OOI Offshore profiler mooring and costal gliders are openly available in the Ocean Observatories Initiative (OOI)'s data portal.23 The data collected by the Seagliders are openly available in the Integrated Ocean Observing System (IOOS) Glider DAC.24 

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