High-frequency bottom-pressure and acoustic variations in a sea strait: internal wave turbulence

Abstract

During a period of 3 days, an accurate bottom-pressure sensor and a four-beam acoustic Doppler current profiler (ADCP) were mounted in a bottom frame at 23 m in a narrow sea strait with dominant near-rectilinear tidal currents exceeding 1 m s⁻¹ in magnitude. The pressure record distinguishes small and short surface waves, wind- and ferry-induced near-surface turbulence and waves, large turbulent overturns and high-frequency internal waves. Typical low-frequency turbulent motions have amplitudes of 50 N m⁻² and periods of about 50 s. Such amplitudes are also found in independent estimates of non-hydrostatic pressure using ADCP data, but phase relationships between these data sets are ambiguous probably due to the averaging over the spread of the slanted acoustic beams. ADCP's echo amplitudes that are observed in individual beams show much better phase correspondence with near-bottom pressure, whether they are generated near the surface (mainly air bubbles) or near the bottom (mainly suspended sediment). These 50-s motions are a mix of turbulence and internal waves, but they are not due to surface wave interactions, and they are not directly related to the main tidal flow. Internal waves are supported by stratification varying between extremely strong thin layer and very weak near-homogeneous stratification. They are driven by the main flow over 2-m amplitude sand-wave topography, with typical wavelengths of 150 m.

Appendices

Appendix 1

1.1 ADCP beam spread variations and potential errors: echo intensity

Echo intensity is measured in each of the four beams individually. As a result, arrivals of distinct phenomena can be used for estimating advective phase speeds of structures passing. From the heading information of Marsdiep's ADCP (Fig. 2d), it is known that beam 3 was only 11° to the East of the cross-channel axis, so that the main current is more or less in the direction of beams 1 (pointing to the North Sea) and 2 (pointing to the Wadden Sea). This is indeed clearly visible in echoes between the beams, for example during flood (Fig. 13). Naturally, the non-simultaneous arrival of structures smaller than the beam spread has its impact on current estimates which are averaged over the (four) beam spread (Appendix 2).

Fig. 13

Comparison between relative echo intensity measured in beams 1 (a) and 2 (b) during a 600-s period of near-maximum flood flow (same as Fig. 11). Clearly, beam 1 is hit first, near the surface about 15 s before beam 2, and providing a more slanted pattern as expected for flood advection at a speed of 1.2 m s⁻¹ for beams pointing into (beam 1) and with (beam 2) the current

Appendix 2

2.1 ADCP beam spread variations and potential errors: current estimates

In an area like the Marsdiep, where small-scale topography and turbulent eddies O(10 m) can dominate flow conditions, one can expect current variations or inhomogeneities being differently measured in ADCP's vertically slanted acoustic beams that are separated horizontally over the same distance. An adequate means to verify potential errors due to averaging ADCP's current estimates over the beam spread is verification using the redundant fourth beam the contribution of [horizontal] current inhomogeneities Δu _ij  = u _i  − u _j , similar for Δv _ij , i ≠ j different beam numbers. After all, the definition of the vertical current average of the four beams reads:

$$ w = \sum\limits_{{i{ } = { }1}}^4 { - {b_i}/4\cos \theta } = \sum\limits_{{i{ } = { }1}}^4 {{w_i}/4} + ({u_1} - {u_2})\cos \theta /4 + ({v_3} - {v_4})\cos \theta /4, $$

where b _i denotes the truly measured velocities in the direction of each of the beams. The indexed current components are given in instrumental coordinates, as if beam 3 is pointing to the north. The fourth beam is invoked when considering the properly scaled (van Haren et al. 1994) subtraction of two beam pairs, the ‘error velocity’:

$$ e = \sum\limits_{{i = 1}}^2 {{b_i}/4\cos \theta } - \sum\limits_{{i = 3}}^4 {{b_i}/4\cos \theta } = \sum\limits_{{i{ } = { }1}}^2 { - {w_i}/4 + } \sum\limits_{{i{ } = { }3}}^4 {{w_i}/4} - ({u_1} - {u_2})\cos \theta /4 + ({v_3} - {v_4})\cos \theta /4. $$

As indicated by van Haren et al. (1994), addition w _p ≡ w + e and subtraction w _m ≡ w − e isolate potential horizontal current inhomogeneity effects on vertical current estimates to its u and v components, respectively.

As the main current is more or less in the direction of beams 1 and 2 (Appendix 1), w _p predominantly captures along-channel current inhomogeneities, which we expect to be the more vigorous if any directionality in overturning and short-scales waves occurs. Van Haren et al. (1994) found that near a sloping bottom, variations in the along-slope current component caused a measurable defect in observed w. Here, we see some differences in high-frequency w and w _p amounting up to 20 % in variance (Fig. 14).

Fig. 14

Enlargement of Fig. 4a, compared with original four-beam w at the same depths

Of some other concern in w observations is potential bias in tilt-sensor data, which may incorrectly transfer horizontal current data in w despite the internal correction for each individual acoustic measurement ping. According to the manufacturer, tilt should be measured better than ±0.5°. Here, its bias error is verified at the tidal frequency. Over time, tilt varies slowly by about 0.5° (Fig. 2e), but no measurable effects on w are found, not at semi-diurnal tidal frequencies, which are thus genuinely measured and are due to up/down motions over the sand waves, and certainly not at the much larger aspect ratio high-frequency IWT, which are definitely genuine.