## 1. Introduction

Air bubbles are injected into the surface layer of the ocean by breaking waves. Vertical currents associated with wave motion or Langmuir cells can carry the bubbles to depths of more than twice the wave height, or 10–15 m (Thorpe and Stubbs 1979; Thorpe 1982; Farmer and Li 1995). A similar, but more extreme, phenomenon has been observed in tidal fronts, where gas bubbles can reach 160-m depth (Farmer et al. 2002; Baschek et al. 2006). Gas bubbles are active tracers with typical rise speeds of 0.001–0.3 m s^{−1} and can therefore be used as indicators of the strength of the vertical currents in these environments.

In this note, we present a new way of looking at bubbles by combining their physical and chemical properties with acoustical measurements to compute bubble behavior in this environment. Observations of gas bubbles in a tidal front in the Fraser Estuary, British Columbia, Canada (Fig. 1), are interpreted with a bubble model (Thorpe 1982) that includes dissolution and compression due to changes in hydrostatic pressure. The strength of the vertical current that carries the bubbles to the measurement depth is estimated. The results are compared with simultaneous acoustic Doppler current profiler (ADCP) measurements along cross-frontal transects.

## 2. Observations

Measurements of gas bubbles and currents were taken in a tidal front in the Fraser Estuary at Boundary Pass (48.78°N, 123.0°W) in September 2000 on research vessel CCGS *Vector*. A 100-kHz Biosonics echo sounder measured the acoustic backscatter intensity of the water column. The transducer was mounted at a depth of 2 m. The receiver bandwidth was set to 5 kHz, the transmission interval to 0.5 s, the pulse width to 0.3 ms, and the gain to 18 dB. A 150-kHz ADCP Workhorse from RDI was mounted on a strut at the side of the ship at a depth of 1 m. The instrument has a bandwidth of 39 kHz and a beam angle of 20°. Measurements were taken every 1 s, and the data were then averaged to create 10-s ensembles. The instrument operated in bottom-tracking mode, and the bin size was 4 m. The first useful measurement depth is at 5 m. The small surface wave motion in these protected waters and the high vertical current component allowed stable vertical current measurement using the standard ADCP four-beam solution. A comparison of both ADCP beam pairs showed that the results were accurate in spite of strong horizontal current gradients (Baschek 2003). The ship’s position and heading were provided by a differential GPS receiver and a flux gate compass.

An example of the flow field and acoustical backscatter intensity in the tidal front is shown in Fig. 2b. A sharp sill with a mean slope of 30° rises from 200- to 60-m water depth. An energetic tidal front is formed during flood tide over the sill crest by a hydraulically controlled sill flow (Baschek et al. 2006). Even at low wind speeds, the associated surface convergence zone causes wave breaking and the injection of gas bubbles due to wave–current interaction (Baschek 2005; Fig. 3). These bubbles are drawn down by a strong current with a vertical component in the range of *w* = −0.3 m s^{−1} to *w* = −0.7 m s^{−1} (Fig. 2a) and were detected by echo sounder measurements to depths of more than 160 m. The downwelling flow is subject to shear instability and overturning, and the measurements are sometimes blocked by dense bubble clouds.

The echo sounder image shows the acoustic backscatter intensity of the water column. Very strong backscatter indicates areas of bubble entrainment, since the acoustical cross section of a bubble is about 1000 times its geometrical one (Medwin 1977). We have verified the presence of bubbles in the downwelling regions of tidal fronts with independent acoustical resonator measurements (Farmer et al. 1998) within the upper 50 m of the water column (Baschek 2003). It can therefore be assumed that most of the high backscatter intensity in the water column is due to gas bubbles and not zooplankton, fish, or turbulence.

*p*is the hydrostatic pressure and

*f*the transmitting frequency of the echo sounder. For our calculations we use a mean density of the subducting water mass of

^{−3}and a polytropic coefficient of

*ν*= 1.4 for air bubbles (Medwin 1977; Hwang and Teague 2000). The resonance radius

*r*

_{res}is plotted in Fig. 2c for

*f*= 100 kHz.

## 3. Results

*r*and number of moles

*n*of the gases O

_{2}, N

_{2}, Ar, and CO

_{2}along the “path” of a bubble, which is determined by its rise speed

*w*, the vertical velocity

_{b}*w*, and the turbulent vertical velocity

*w*

_{turb}: Respectively,

*D*,

_{j}*S*, and Nu

_{j}*are the diffusivity, solubility, and Nusselt number of a gas*

_{j}*j*(Woolf and Thorpe 1991);

*R*,

*T*,

*g*, and

*γ*are the universal gas constant, temperature, gravitational acceleration, and surface tension coefficient;

*P*is the gas pressure inside the bubble and

*P*the gas pressure in the water far away from the bubble, which is determined by the gas saturation. The three coupled ordinary differential equations [(2)–(4)] were solved in MATLAB with an explicit Runge–Kutta formula.

^{w}The following calculations were carried out for different bubbles with initial radii *r*_{0} using a representative vertical velocity of *w* = −0.5 m s^{−1} below the injection depth of 0.1 m, a temperature of 10°C, and a gas saturation of 100% for all gases. The resulting bubble “paths” are shown in Fig. 2c (dashed lines). They intersect the resonant radius curve (solid line), which indicates the size of the bubbles that are detected by the echo sounder at a given depth. Subject to the limitations of our measurements, this allows the estimation of the radius *r*_{0} of these bubbles at the time of injection. The calculation is carried out backward in time, calculating the gas exchange across the bubble surface while considering the bubble’s rise speed (Fig. 4a) relative to the surrounding fluid at each step. The results of this calculation, shown in the figure as the intersection of the dashed curves with the surface, represents a conservative value, as the nominal vertical velocity of *w* = −0.5 m s^{−1} used in the calculations is closer to the upper bound of our measurements. The bubbles measured at 160-m depth would have had a radius of *r*_{0} = 1.7 mm at the sea surface. For a velocity of *w* = −0.4 m s^{−1}, the initial radius would have been *r*_{0} = 2.3 mm, and for *w* = −0.3 m s^{−1} it would have been *r*_{0} = 4.4 mm.

To test the sensitivity of the model with respect to the parameters used we calculated the bubble radius *r*_{0} at the surface by changing one parameter at a time and comparing the results to the reference run with *w* = −0.5 m s^{−1}, *T* = 10°C, and a gas saturation of 100%. A reduction of the diffusivities of all gases by 90%, as it may be caused by surfactants, would result in a decrease of *r*_{0} by 9%; a reduction of the diffusivities by 50% would reduce *r*_{0} by 32%. The effect of temperature variation is minimal. The temperature difference within the subducting water mass that carries the bubbles downward is <0.5°C, as indicated by the CTD measurements, which corresponds to a decrease of *r*_{0} by only 0.5%. A change of the polytropic coefficient (1) to the smallest possible value of *ν* = 1.0 causes a decrease of *r*_{0} by 6%. The effect of turbulence was estimated by multiplying a random walk function (normal distribution, mean = 0, standard deviation = 1) with the typical turbulent vertical velocity of 0.05 m s^{−1} and displacement of 3 m measured in tidal fronts (Gargett and Moum 1995) and adding it to the vertical velocity at each time step (2). The average of several model runs yields a reduction of *r*_{0} by 2%. Increasing the gas saturation by 2% to then 102% for all gases has an effect on *r*_{0} of <1%.

The model can also be used to estimate the minimal vertical velocity that is necessary to draw down a bubble from the sea surface to a certain depth where it is detected by the echo sounder. Bubbles that have a rise speed *w*_{b0} that is greater than the vertical current *w* return to the sea surface; bubbles with a rise speed of |*w*_{b0}| ≤ |*w*| are drawn down by the current. The largest bubble that is drawn down reaches the greatest depth. Its radius corresponds to a rise speed of *w*_{b0} = −*w* since the rise speed of dirty bubbles steadily increases with radius (Fig. 4a). This means that the minimal vertical current required to explain the presence of a bubble of radius *r*_{res} at the depth of measurement is equal to the initial rise speed *w*_{b0} of that bubble at its injection depth.

The path of this largest bubble is shown in Fig. 4b for different velocities and an injection depth of 0.1 m. The bubble rise speed is shown in Fig. 4a, and the relationship between minimal vertical current and depth is given in Fig. 4c, showing that bubbles at 160-m depth are drawn down by a vertical current of at least −0.27 m s^{−1}. This is consistent with the current measurements in the tidal front showing values between −0.3 and −0.7 m s^{−1}.

Although simplifying assumptions are required to overcome our measurement limitations in this complicated environment, our model analysis provides a first-order estimate of the bubble radius at the injection site that can be used to infer the vertical velocities of the front near the surface. Additional measurements of bubble size distribution and accurate vertical velocities from the sea surface downward would help to verify the model results.

## Acknowledgments

We are grateful for the support of the officers and crew of research vessel CCGS *Vector*. Many thanks to the Ocean Acoustics Group at the Institute of Ocean Sciences, Canada, for their assistance during the experiments. The work formed a part of research projects carried out with the support of the Office of Naval Research.

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